%------------------------------------------------------------------------------
% File     : ITP218_3 : TPTP v8.2.0. Released v8.0.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem VEBT_List_Assn 00019_000444
% Version  : [Des22] axioms.
% English  :

% Refs     : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
%          : [Des22] Desharnais (2022), Email to Geoff Sutcliffe
% Source   : [Des22]
% Names    : 0062_VEBT_List_Assn_00019_000444 [Des22]

% Status   : Theorem
% Rating   : 1.00 v8.1.0
% Syntax   : Number of formulae    : 5304 (1350 unt;1108 typ;   0 def)
%            Number of atoms       : 11350 (4691 equ)
%            Maximal formula atoms :   15 (   2 avg)
%            Number of connectives : 8543 (1568   ~; 172   |; 826   &)
%                                         (1035 <=>;4942  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   23 (   6 avg)
%            Maximal term depth    :   11 (   2 avg)
%            Number of FOOLs       :  822 ( 215 fml; 607 var)
%            Number of X terms     :  140 (   0  []; 140 ite;   0 let)
%            Number of types       :  260 ( 259 usr)
%            Number of type conns  : 1207 ( 672   >; 535   *;   0   +;   0  <<)
%            Number of predicates  :   70 (  67 usr;   2 prp; 0-3 aty)
%            Number of functors    :  782 ( 782 usr; 177 con; 0-5 aty)
%            Number of variables   : 12293 (11731   !; 562   ?;12293   :)
% SPC      : TX0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            from the van Emde Boas Trees session in the Archive of Formal
%            proofs - 
%            www.isa-afp.org/browser_info/current/AFP/Van_Emde_Boas_Trees
%            2022-02-17 16:49:12.073
%------------------------------------------------------------------------------
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    fun_a_fun_int_o: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__fun_Itf__b_M_Eo_J_J,type,
    fun_nat_fun_b_o: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__fun_Itf__a_M_Eo_J_J,type,
    fun_nat_fun_a_o: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__Assertions__Oassn_J,type,
    fun_nat_assn: $tType ).

tff(ty_n_t__fun_It__Int__Oint_Mt__fun_Itf__b_M_Eo_J_J,type,
    fun_int_fun_b_o: $tType ).

tff(ty_n_t__fun_It__Int__Oint_Mt__fun_Itf__a_M_Eo_J_J,type,
    fun_int_fun_a_o: $tType ).

tff(ty_n_t__fun_It__Int__Oint_Mt__Assertions__Oassn_J,type,
    fun_int_assn: $tType ).

tff(ty_n_t__fun_It__Assertions__Oassn_Mt__Nat__Onat_J,type,
    fun_assn_nat: $tType ).

tff(ty_n_t__fun_It__Assertions__Oassn_Mt__Int__Oint_J,type,
    fun_assn_int: $tType ).

tff(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    set_set_nat: $tType ).

tff(ty_n_t__fun_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
    fun_set_nat_o: $tType ).

tff(ty_n_t__fun_It__Set__Oset_It__Int__Oint_J_M_Eo_J,type,
    fun_set_int_o: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__Set__Oset_I_Eo_J_J,type,
    fun_nat_set_o: $tType ).

tff(ty_n_t__fun_It__Int__Oint_Mt__Set__Oset_I_Eo_J_J,type,
    fun_int_set_o: $tType ).

tff(ty_n_t__Set__Oset_It__Product____Type__Ounit_J,type,
    set_Product_unit: $tType ).

tff(ty_n_t__List__Olist_It__List__Olist_Itf__b_J_J,type,
    list_list_b: $tType ).

tff(ty_n_t__List__Olist_It__List__Olist_Itf__a_J_J,type,
    list_list_a: $tType ).

tff(ty_n_t__fun_Itf__b_Mt__List__Olist_Itf__b_J_J,type,
    fun_b_list_b: $tType ).

tff(ty_n_t__fun_Itf__b_Mt__List__Olist_Itf__a_J_J,type,
    fun_b_list_a: $tType ).

tff(ty_n_t__fun_Itf__a_Mt__List__Olist_Itf__b_J_J,type,
    fun_a_list_b: $tType ).

tff(ty_n_t__fun_Itf__a_Mt__List__Olist_Itf__a_J_J,type,
    fun_a_list_a: $tType ).

tff(ty_n_t__fun_It__Product____Type__Ounit_M_Eo_J,type,
    fun_Product_unit_o: $tType ).

tff(ty_n_t__Set__Oset_It__List__Olist_Itf__b_J_J,type,
    set_list_b: $tType ).

tff(ty_n_t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
    set_list_a: $tType ).

tff(ty_n_t__fun_It__List__Olist_Itf__b_J_M_Eo_J,type,
    fun_list_b_o: $tType ).

tff(ty_n_t__fun_It__List__Olist_Itf__a_J_M_Eo_J,type,
    fun_list_a_o: $tType ).

tff(ty_n_t__List__Olist_It__Assertions__Oassn_J,type,
    list_assn: $tType ).

tff(ty_n_t__fun_Itf__b_Mt__fun_Itf__b_M_Eo_J_J,type,
    fun_b_fun_b_o: $tType ).

tff(ty_n_t__fun_Itf__b_Mt__fun_Itf__a_M_Eo_J_J,type,
    fun_b_fun_a_o: $tType ).

tff(ty_n_t__fun_Itf__b_Mt__Assertions__Oassn_J,type,
    fun_b_assn: $tType ).

tff(ty_n_t__fun_Itf__a_Mt__fun_Itf__b_M_Eo_J_J,type,
    fun_a_fun_b_o: $tType ).

tff(ty_n_t__fun_Itf__a_Mt__fun_Itf__a_M_Eo_J_J,type,
    fun_a_fun_a_o: $tType ).

tff(ty_n_t__fun_Itf__a_Mt__Assertions__Oassn_J,type,
    fun_a_assn: $tType ).

tff(ty_n_t__fun_Itf__b_Mt__Set__Oset_I_Eo_J_J,type,
    fun_b_set_o: $tType ).

tff(ty_n_t__fun_Itf__a_Mt__Set__Oset_I_Eo_J_J,type,
    fun_a_set_o: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__Nat__Onat_J,type,
    fun_nat_nat: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__Int__Oint_J,type,
    fun_nat_int: $tType ).

tff(ty_n_t__fun_It__Int__Oint_Mt__Nat__Onat_J,type,
    fun_int_nat: $tType ).

tff(ty_n_t__fun_It__Int__Oint_Mt__Int__Oint_J,type,
    fun_int_int: $tType ).

tff(ty_n_t__Set__Oset_It__Assertions__Oassn_J,type,
    set_assn: $tType ).

tff(ty_n_t__fun_It__Assertions__Oassn_M_Eo_J,type,
    fun_assn_o: $tType ).

tff(ty_n_t__fun_I_Eo_Mt__Assertions__Oassn_J,type,
    fun_o_assn: $tType ).

tff(ty_n_t__fun_It__Set__Oset_I_Eo_J_M_Eo_J,type,
    fun_set_o_o: $tType ).

tff(ty_n_t__fun_I_Eo_Mt__fun_I_Eo_M_Eo_J_J,type,
    fun_o_fun_o_o: $tType ).

tff(ty_n_t__List__Olist_It__Nat__Onat_J,type,
    list_nat: $tType ).

tff(ty_n_t__List__Olist_It__Int__Oint_J,type,
    list_int: $tType ).

tff(ty_n_t__fun_Itf__b_Mt__Nat__Onat_J,type,
    fun_b_nat: $tType ).

tff(ty_n_t__fun_Itf__b_Mt__Int__Oint_J,type,
    fun_b_int: $tType ).

tff(ty_n_t__fun_Itf__a_Mt__Nat__Onat_J,type,
    fun_a_nat: $tType ).

tff(ty_n_t__fun_Itf__a_Mt__Int__Oint_J,type,
    fun_a_int: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mtf__b_J,type,
    fun_nat_b: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mtf__a_J,type,
    fun_nat_a: $tType ).

tff(ty_n_t__fun_It__Int__Oint_Mtf__b_J,type,
    fun_int_b: $tType ).

tff(ty_n_t__fun_It__Int__Oint_Mtf__a_J,type,
    fun_int_a: $tType ).

tff(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
    set_nat: $tType ).

tff(ty_n_t__Set__Oset_It__Int__Oint_J,type,
    set_int: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_M_Eo_J,type,
    fun_nat_o: $tType ).

tff(ty_n_t__fun_It__Int__Oint_M_Eo_J,type,
    fun_int_o: $tType ).

tff(ty_n_t__fun_I_Eo_Mt__Nat__Onat_J,type,
    fun_o_nat: $tType ).

tff(ty_n_t__fun_I_Eo_Mt__Int__Oint_J,type,
    fun_o_int: $tType ).

tff(ty_n_t__Product____Type__Ounit,type,
    product_unit: $tType ).

tff(ty_n_t__List__Olist_Itf__b_J,type,
    list_b: $tType ).

tff(ty_n_t__List__Olist_Itf__a_J,type,
    list_a: $tType ).

tff(ty_n_t__fun_Itf__b_Mtf__b_J,type,
    fun_b_b: $tType ).

tff(ty_n_t__fun_Itf__b_Mtf__a_J,type,
    fun_b_a: $tType ).

tff(ty_n_t__fun_Itf__a_Mtf__b_J,type,
    fun_a_b: $tType ).

tff(ty_n_t__fun_Itf__a_Mtf__a_J,type,
    fun_a_a: $tType ).

tff(ty_n_t__Set__Oset_Itf__b_J,type,
    set_b: $tType ).

tff(ty_n_t__Set__Oset_Itf__a_J,type,
    set_a: $tType ).

tff(ty_n_t__List__Olist_I_Eo_J,type,
    list_o: $tType ).

tff(ty_n_t__fun_Itf__b_M_Eo_J,type,
    fun_b_o: $tType ).

tff(ty_n_t__fun_Itf__a_M_Eo_J,type,
    fun_a_o: $tType ).

tff(ty_n_t__Typerep__Otyperep,type,
    typerep: $tType ).

tff(ty_n_t__Assertions__Oassn,type,
    assn: $tType ).

tff(ty_n_t__Set__Oset_I_Eo_J,type,
    set_o: $tType ).

tff(ty_n_t__fun_I_Eo_M_Eo_J,type,
    fun_o_o: $tType ).

tff(ty_n_t__Num__Onum,type,
    num: $tType ).

tff(ty_n_t__Nat__Onat,type,
    nat: $tType ).

tff(ty_n_t__Int__Oint,type,
    int: $tType ).

tff(ty_n_tf__b,type,
    b: $tType ).

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

% Explicit typings (849)
tff(sy_c_ATP_058Lamm__a____,type,
    aTP_Lamm_a: set_fu3555266933746557007_nat_o > fun_fu5135969500543783852at_o_o ).

tff(sy_c_ATP_058Lamm__aa____,type,
    aTP_Lamm_aa: set_nat > fun_nat_o ).

tff(sy_c_ATP_058Lamm__ab____,type,
    aTP_Lamm_ab: set_Pr1261947904930325089at_nat > fun_Pr4796943543331071514_nat_o ).

tff(sy_c_ATP_058Lamm__ac____,type,
    aTP_Lamm_ac: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ad____,type,
    aTP_Lamm_ad: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ae____,type,
    aTP_Lamm_ae: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__af____,type,
    aTP_Lamm_af: fun_int_int ).

tff(sy_c_ATP_058Lamm__ag____,type,
    aTP_Lamm_ag: assn > fun_Pr6460944699146269935_nat_o ).

tff(sy_c_ATP_058Lamm__ah____,type,
    aTP_Lamm_ah: fun_Pr6460944699146269935_nat_o ).

tff(sy_c_ATP_058Lamm__ai____,type,
    aTP_Lamm_ai: assn > fun_as3747249873083764768_nat_o ).

tff(sy_c_ATP_058Lamm__aj____,type,
    aTP_Lamm_aj: assn > fun_as3747249873083764768_nat_o ).

tff(sy_c_ATP_058Lamm__ak____,type,
    aTP_Lamm_ak: fun_fu1678786105377694981_nat_o ).

tff(sy_c_ATP_058Lamm__al____,type,
    aTP_Lamm_al: fun_fu1678786105377694981_nat_o ).

tff(sy_c_ATP_058Lamm__am____,type,
    aTP_Lamm_am: fun_Pr6460944699146269935_nat_o > fun_Pr6460944699146269935_nat_o ).

tff(sy_c_ATP_058Lamm__an____,type,
    aTP_Lamm_an: fun_Pr6626665806141890376t_assn ).

tff(sy_c_ATP_058Lamm__ao____,type,
    aTP_Lamm_ao: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__ap____,type,
    aTP_Lamm_ap: nat > fun_nat_set_nat ).

tff(sy_c_ATP_058Lamm__aq____,type,
    aTP_Lamm_aq: nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__ar____,type,
    aTP_Lamm_ar: nat > fun_nat_set_nat ).

tff(sy_c_Assertions_Oassn_OAbs__assn,type,
    abs_assn: fun_fu3790787410453625514o_assn ).

tff(sy_c_Assertions_Oassn_ORep__assn,type,
    rep_assn: fun_as3747249873083764768_nat_o ).

tff(sy_c_Assertions_Oentails,type,
    entails: fun_assn_fun_assn_o ).

tff(sy_c_Assertions_Oentailst,type,
    entailst: assn > fun_assn_o ).

tff(sy_c_Assertions_Oin__range,type,
    in_range: fun_Pr6460944699146269935_nat_o ).

tff(sy_c_Assertions_Oin__range__rel,type,
    in_range_rel: fun_Pr7920380315054183110_nat_o ).

tff(sy_c_Assertions_Ois__pure__assn,type,
    is_pure_assn: fun_assn_o ).

tff(sy_c_Assertions_Oone__assn__raw,type,
    one_assn_raw: fun_Pr6460944699146269935_nat_o ).

tff(sy_c_Assertions_Oone__assn__raw__rel,type,
    one_assn_raw_rel: fun_Pr7920380315054183110_nat_o ).

tff(sy_c_Assertions_Oproper,type,
    proper: fun_fu5135969500543783852at_o_o ).

tff(sy_c_Assertions_Opure__assn,type,
    pure_assn: fun_o_assn ).

tff(sy_c_Assertions_Opure__assn__raw_001t__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_001t__Nat__Onat,type,
    pure_a825153325127701367it_nat: $o > fun_Pr6460944699146269935_nat_o ).

tff(sy_c_Assertions_OrelH,type,
    relH: ( set_nat * heap_e7401611519738050253t_unit * heap_e7401611519738050253t_unit ) > $o ).

tff(sy_c_Assertions_Otimes__assn__raw,type,
    times_assn_raw: fun_fu1678786105377694981_nat_o ).

tff(sy_c_Assertions_Otimes__assn__raw__rel,type,
    times_assn_raw_rel: fun_Pr3358449090565925198_nat_o ).

tff(sy_c_Assertions_Owand__assn,type,
    wand_assn: fun_as3983628580203722482n_assn ).

tff(sy_c_Assertions_Owand__raw,type,
    wand_raw: fun_fu1678786105377694981_nat_o ).

tff(sy_c_Assertions_Owand__raw__rel,type,
    wand_raw_rel: fun_Pr3358449090565925198_nat_o ).

tff(sy_c_Automation_OFI,type,
    fi: ( list_P8527749157015355191n_assn * assn * assn * assn * assn ) > fun_assn_o ).

tff(sy_c_Automation_OFI__QUERY,type,
    fI_QUERY: ( assn * assn ) > fun_assn_o ).

tff(sy_c_Automation_OFI__RESULT,type,
    fI_RESULT: ( list_P8527749157015355191n_assn * assn * assn ) > fun_assn_o ).

tff(sy_c_Automation_OSLN,type,
    sln: assn ).

tff(sy_c_BNF__Cardinal__Order__Relation_OnatLeq,type,
    bNF_Ca8665028551170535155natLeq: set_Pr1261947904930325089at_nat ).

tff(sy_c_BNF__Greatest__Fixpoint_OShift_001t__Int__Oint,type,
    bNF_Gr1870224194279859149ft_int: ( set_list_int * int ) > set_list_int ).

tff(sy_c_BNF__Greatest__Fixpoint_OShift_001t__Nat__Onat,type,
    bNF_Gr1872714664788909425ft_nat: ( set_list_nat * nat ) > set_list_nat ).

tff(sy_c_BNF__Greatest__Fixpoint_OShift_001t__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J,type,
    bNF_Gr4113829767105464016n_assn: ( set_li5131720305576846103n_assn * produc6575502325842934193n_assn ) > set_li5131720305576846103n_assn ).

tff(sy_c_BNF__Greatest__Fixpoint_OShift_001t__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J,type,
    bNF_Gr2819271973565845774_nat_o: ( set_li6341169313164406869_nat_o * fun_Pr6460944699146269935_nat_o ) > set_li6341169313164406869_nat_o ).

tff(sy_c_BNF__Greatest__Fixpoint_OShift_001tf__a,type,
    bNF_Greatest_Shift_a: ( set_list_a * a ) > set_list_a ).

tff(sy_c_BNF__Greatest__Fixpoint_OShift_001tf__b,type,
    bNF_Greatest_Shift_b: ( set_list_b * b ) > set_list_b ).

tff(sy_c_BNF__Greatest__Fixpoint_OSucc_001t__Int__Oint,type,
    bNF_Gr6350390219475566417cc_int: ( set_list_int * list_int ) > set_int ).

tff(sy_c_BNF__Greatest__Fixpoint_OSucc_001t__Nat__Onat,type,
    bNF_Gr6352880689984616693cc_nat: ( set_list_nat * list_nat ) > set_nat ).

tff(sy_c_BNF__Greatest__Fixpoint_OSucc_001t__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J,type,
    bNF_Gr3216292445876487756n_assn: ( set_li5131720305576846103n_assn * list_P8527749157015355191n_assn ) > set_Pr5949110396991348497n_assn ).

tff(sy_c_BNF__Greatest__Fixpoint_OSucc_001t__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J,type,
    bNF_Gr2876145917579644042_nat_o: ( set_li6341169313164406869_nat_o * list_f5222602229612168821_nat_o ) > set_fu3555266933746557007_nat_o ).

tff(sy_c_BNF__Greatest__Fixpoint_OSucc_001tf__a,type,
    bNF_Greatest_Succ_a: ( set_list_a * list_a ) > set_a ).

tff(sy_c_BNF__Greatest__Fixpoint_OSucc_001tf__b,type,
    bNF_Greatest_Succ_b: ( set_list_b * list_b ) > set_b ).

tff(sy_c_Binomial_Ogbinomial_001t__Int__Oint,type,
    gbinomial_int: int > fun_nat_int ).

tff(sy_c_Binomial_Ogbinomial_001t__Nat__Onat,type,
    gbinomial_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Finite__Set_Ocard_001_Eo,type,
    finite_card_o: set_o > nat ).

tff(sy_c_Finite__Set_Ocard_001t__Int__Oint,type,
    finite_card_int: set_int > nat ).

tff(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
    finite_card_nat: set_nat > nat ).

tff(sy_c_Finite__Set_Ocard_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    finite711546835091564841at_nat: set_Pr1261947904930325089at_nat > nat ).

tff(sy_c_Finite__Set_Ocard_001t__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J,type,
    finite5795495896937551127_nat_o: set_fu3555266933746557007_nat_o > nat ).

tff(sy_c_Finite__Set_Ofinite_001_Eo,type,
    finite_finite_o: fun_set_o_o ).

tff(sy_c_Finite__Set_Ofinite_001t__Assertions__Oassn,type,
    finite_finite_assn: set_assn > $o ).

tff(sy_c_Finite__Set_Ofinite_001t__Int__Oint,type,
    finite_finite_int: fun_set_int_o ).

tff(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Int__Oint_J,type,
    finite3922522038869484883st_int: set_list_int > $o ).

tff(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Nat__Onat_J,type,
    finite8100373058378681591st_nat: set_list_nat > $o ).

tff(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
    finite_finite_nat: fun_set_nat_o ).

tff(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    finite6177210948735845034at_nat: set_Pr1261947904930325089at_nat > $o ).

tff(sy_c_Finite__Set_Ofinite_001t__Product____Type__Ounit,type,
    finite4290736615968046902t_unit: set_Product_unit > $o ).

tff(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Nat__Onat_J,type,
    finite1152437895449049373et_nat: set_set_nat > $o ).

tff(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    finite9047747110432174090at_nat: set_se7855581050983116737at_nat > $o ).

tff(sy_c_Finite__Set_Ofinite_001t__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J,type,
    finite4610005178611006360_nat_o: fun_se7459991829846211660at_o_o ).

tff(sy_c_Fun_Ocomp_001_Eo_001_Eo_001t__Int__Oint,type,
    comp_o_o_int: ( fun_o_o * fun_int_o ) > fun_int_o ).

tff(sy_c_Fun_Ocomp_001_Eo_001_Eo_001t__Nat__Onat,type,
    comp_o_o_nat: ( fun_o_o * fun_nat_o ) > fun_nat_o ).

tff(sy_c_Groups_Ominus__class_Ominus_001t__Assertions__Oassn,type,
    minus_minus_assn: fun_as3983628580203722482n_assn ).

tff(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
    minus_minus_int: fun_int_fun_int_int ).

tff(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
    minus_minus_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_I_Eo_J,type,
    minus_minus_set_o: ( set_o * set_o ) > set_o ).

tff(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Assertions__Oassn_J,type,
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tff(sy_c_List_Oproduct_001t__Assertions__Oassn_001t__Assertions__Oassn,type,
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tff(sy_c_List_Oproduct_001t__Int__Oint_001t__Int__Oint,type,
    product_int_int: ( list_int * list_int ) > list_P5707943133018811711nt_int ).

tff(sy_c_List_Oproduct_001t__Int__Oint_001t__Nat__Onat,type,
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tff(sy_c_List_Oproduct_001t__Nat__Onat_001t__Int__Oint,type,
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tff(sy_c_List_Oproduct_001t__Nat__Onat_001t__Nat__Onat,type,
    product_nat_nat: ( list_nat * list_nat ) > list_P6011104703257516679at_nat ).

tff(sy_c_List_Oproduct__lists_001t__Int__Oint,type,
    product_lists_int: list_list_int > list_list_int ).

tff(sy_c_List_Oproduct__lists_001t__Nat__Onat,type,
    product_lists_nat: list_list_nat > list_list_nat ).

tff(sy_c_List_Oproduct__lists_001t__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J,type,
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tff(sy_c_List_Oproduct__lists_001tf__a,type,
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tff(sy_c_List_Oproduct__lists_001tf__b,type,
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tff(sy_c_List_Oremdups_001t__Nat__Onat,type,
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tff(sy_c_List_Oremove1_001_Eo,type,
    remove1_o: ( $o * list_o ) > list_o ).

tff(sy_c_List_Oremove1_001t__Int__Oint,type,
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tff(sy_c_List_Oremove1_001t__Nat__Onat,type,
    remove1_nat: ( nat * list_nat ) > list_nat ).

tff(sy_c_List_Oremove1_001t__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J,type,
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tff(sy_c_List_Oremove1_001tf__a,type,
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tff(sy_c_List_Oremove1_001tf__b,type,
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tff(sy_c_List_Orev_001t__Assertions__Oassn,type,
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tff(sy_c_List_Orev_001t__Int__Oint,type,
    rev_int: list_int > list_int ).

tff(sy_c_List_Orev_001t__List__Olist_It__Int__Oint_J,type,
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tff(sy_c_List_Orev_001t__List__Olist_It__Nat__Onat_J,type,
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tff(sy_c_List_Orev_001t__Nat__Onat,type,
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tff(sy_c_List_Orev_001t__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J,type,
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tff(sy_c_List_Orev_001tf__a,type,
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tff(sy_c_List_Orev_001tf__b,type,
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tff(sy_c_List_Orotate1_001t__Assertions__Oassn,type,
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tff(sy_c_List_Orotate1_001t__Int__Oint,type,
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tff(sy_c_List_Orotate1_001t__Nat__Onat,type,
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tff(sy_c_List_Orotate1_001t__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J,type,
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tff(sy_c_List_Orotate1_001tf__a,type,
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tff(sy_c_List_Orotate1_001tf__b,type,
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tff(sy_c_List_Osorted__wrt_001t__Int__Oint,type,
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tff(sy_c_List_Osorted__wrt_001t__Nat__Onat,type,
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tff(sy_c_List_Osorted__wrt_001t__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J,type,
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tff(sy_c_List_Osorted__wrt_001tf__a,type,
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tff(sy_c_List_Osorted__wrt_001tf__b,type,
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tff(sy_c_List_Osubseqs_001t__Int__Oint,type,
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tff(sy_c_List_Osubseqs_001t__Nat__Onat,type,
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tff(sy_c_List_Osubseqs_001tf__a,type,
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tff(sy_c_List_Osubseqs_001tf__b,type,
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tff(sy_c_List_Otake_001t__Nat__Onat,type,
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tff(sy_c_List_Oupt,type,
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tff(sy_c_List_Oupto,type,
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tff(sy_c_List_Oupto__aux,type,
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tff(sy_c_List_Oupto__rel,type,
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tff(sy_c_Misc_Olist__collect__set_001t__Assertions__Oassn_001_Eo,type,
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tff(sy_c_Misc_Olist__collect__set_001t__Assertions__Oassn_001t__Nat__Onat,type,
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tff(sy_c_Misc_Olist__collect__set_001t__Int__Oint_001_Eo,type,
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tff(sy_c_Misc_Olist__collect__set_001t__Int__Oint_001t__Nat__Onat,type,
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tff(sy_c_Misc_Olist__collect__set_001t__Nat__Onat_001_Eo,type,
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tff(sy_c_Misc_Olist__collect__set_001t__Nat__Onat_001t__Nat__Onat,type,
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tff(sy_c_Misc_Olist__collect__set_001tf__a_001_Eo,type,
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tff(sy_c_Misc_Olist__collect__set_001tf__a_001t__Nat__Onat,type,
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tff(sy_c_Misc_Olist__collect__set_001tf__b_001_Eo,type,
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tff(sy_c_Misc_Olist__collect__set_001tf__b_001t__Nat__Onat,type,
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tff(sy_c_Misc_Omerge_001t__Int__Oint,type,
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tff(sy_c_Misc_Omerge_001t__Nat__Onat,type,
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tff(sy_c_Misc_Omerge__list_001t__Int__Oint,type,
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tff(sy_c_Misc_Omerge__list_001t__Nat__Onat,type,
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tff(sy_c_Misc_Omergesort_001t__Assertions__Oassn,type,
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tff(sy_c_Misc_Omergesort_001t__Int__Oint,type,
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tff(sy_c_Misc_Omergesort_001t__Nat__Onat,type,
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tff(sy_c_Misc_Omergesort_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_Misc_Omergesort__by__rel_001t__Assertions__Oassn,type,
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tff(sy_c_Misc_Omergesort__by__rel_001t__Nat__Onat,type,
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tff(sy_c_Misc_Omergesort__by__rel_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_Misc_Omergesort__by__rel_001tf__a,type,
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tff(sy_c_Misc_Omergesort__by__rel_001tf__b,type,
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tff(sy_c_Misc_Orevg_001t__Int__Oint,type,
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tff(sy_c_Misc_Orevg_001t__Nat__Onat,type,
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tff(sy_c_Misc_Orevg_001tf__a,type,
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tff(sy_c_Misc_Orevg_001tf__b,type,
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tff(sy_c_Misc_Oslice_001t__Int__Oint,type,
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tff(sy_c_Misc_Oslice_001t__Nat__Onat,type,
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tff(sy_c_Misc_Oslice_001tf__b,type,
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tff(sy_c_Nat_OSuc,type,
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tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Assertions__Oassn_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Int__Oint_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
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tff(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Int__Oint,type,
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tff(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Int__Oint,type,
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tff(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
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tff(sy_c_Order__Relation_OunderS_001t__Nat__Onat,type,
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tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Assertions__Oassn_J,type,
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tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__Assertions__Oassn,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
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    aa_a_fun_nat_o: ( fun_a_fun_nat_o * a ) > fun_nat_o ).

tff(sy_c_aa_001tf__a_001t__fun_It__Nat__Onat_Mt__Assertions__Oassn_J,type,
    aa_a_fun_nat_assn: ( fun_a_fun_nat_assn * a ) > fun_nat_assn ).

tff(sy_c_aa_001tf__a_001t__fun_Itf__a_M_Eo_J,type,
    aa_a_fun_a_o: ( fun_a_fun_a_o * a ) > fun_a_o ).

tff(sy_c_aa_001tf__a_001t__fun_Itf__a_Mt__Assertions__Oassn_J,type,
    aa_a_fun_a_assn: ( fun_a_fun_a_assn * a ) > fun_a_assn ).

tff(sy_c_aa_001tf__a_001t__fun_Itf__b_M_Eo_J,type,
    aa_a_fun_b_o: ( fun_a_fun_b_o * a ) > fun_b_o ).

tff(sy_c_aa_001tf__a_001t__fun_Itf__b_Mt__Assertions__Oassn_J,type,
    aa_a_fun_b_assn: ( fun_a_fun_b_assn * a ) > fun_b_assn ).

tff(sy_c_aa_001tf__b_001_Eo,type,
    aa_b_o: ( fun_b_o * b ) > $o ).

tff(sy_c_aa_001tf__b_001t__Assertions__Oassn,type,
    aa_b_assn: ( fun_b_assn * b ) > assn ).

tff(sy_c_aa_001tf__b_001t__Int__Oint,type,
    aa_b_int: ( fun_b_int * b ) > int ).

tff(sy_c_aa_001tf__b_001t__Nat__Onat,type,
    aa_b_nat: ( fun_b_nat * b ) > nat ).

tff(sy_c_aa_001tf__b_001t__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J,type,
    aa_b_P6909873165786669230n_assn: ( fun_b_4490291949330546613n_assn * b ) > produc6575502325842934193n_assn ).

tff(sy_c_aa_001tf__b_001t__fun_It__Int__Oint_M_Eo_J,type,
    aa_b_fun_int_o: ( fun_b_fun_int_o * b ) > fun_int_o ).

tff(sy_c_aa_001tf__b_001t__fun_It__Int__Oint_Mt__Assertions__Oassn_J,type,
    aa_b_fun_int_assn: ( fun_b_fun_int_assn * b ) > fun_int_assn ).

tff(sy_c_aa_001tf__b_001t__fun_It__List__Olist_Itf__b_J_Mt__List__Olist_Itf__b_J_J,type,
    aa_b_f8935218930876307398list_b: ( fun_b_5542551703336567573list_b * b ) > fun_list_b_list_b ).

tff(sy_c_aa_001tf__b_001t__fun_It__Nat__Onat_M_Eo_J,type,
    aa_b_fun_nat_o: ( fun_b_fun_nat_o * b ) > fun_nat_o ).

tff(sy_c_aa_001tf__b_001t__fun_It__Nat__Onat_Mt__Assertions__Oassn_J,type,
    aa_b_fun_nat_assn: ( fun_b_fun_nat_assn * b ) > fun_nat_assn ).

tff(sy_c_aa_001tf__b_001t__fun_Itf__a_M_Eo_J,type,
    aa_b_fun_a_o: ( fun_b_fun_a_o * b ) > fun_a_o ).

tff(sy_c_aa_001tf__b_001t__fun_Itf__a_Mt__Assertions__Oassn_J,type,
    aa_b_fun_a_assn: ( fun_b_fun_a_assn * b ) > fun_a_assn ).

tff(sy_c_aa_001tf__b_001t__fun_Itf__b_M_Eo_J,type,
    aa_b_fun_b_o: ( fun_b_fun_b_o * b ) > fun_b_o ).

tff(sy_c_aa_001tf__b_001t__fun_Itf__b_Mt__Assertions__Oassn_J,type,
    aa_b_fun_b_assn: ( fun_b_fun_b_assn * b ) > fun_b_assn ).

tff(sy_c_fNot,type,
    fNot: fun_o_o ).

tff(sy_c_member_001_Eo,type,
    member_o: $o > fun_set_o_o ).

tff(sy_c_member_001t__Assertions__Oassn,type,
    member_assn: ( assn * set_assn ) > $o ).

tff(sy_c_member_001t__Int__Oint,type,
    member_int2: int > fun_set_int_o ).

tff(sy_c_member_001t__List__Olist_It__Int__Oint_J,type,
    member_list_int: ( list_int * set_list_int ) > $o ).

tff(sy_c_member_001t__List__Olist_It__Nat__Onat_J,type,
    member_list_nat: ( list_nat * set_list_nat ) > $o ).

tff(sy_c_member_001t__List__Olist_It__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J_J,type,
    member852475432509897056n_assn: ( list_P8527749157015355191n_assn * set_li5131720305576846103n_assn ) > $o ).

tff(sy_c_member_001t__List__Olist_It__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_J,type,
    member877585773093452190_nat_o: ( list_f5222602229612168821_nat_o * set_li6341169313164406869_nat_o ) > $o ).

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

tff(sy_c_member_001t__List__Olist_Itf__b_J,type,
    member_list_b: ( list_b * set_list_b ) > $o ).

tff(sy_c_member_001t__Nat__Onat,type,
    member_nat2: nat > fun_set_nat_o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J,type,
    member7957490590177025114n_assn: ( produc6575502325842934193n_assn * set_Pr5949110396991348497n_assn ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    member8440522571783428010at_nat: ( product_prod_nat_nat * set_Pr1261947904930325089at_nat ) > $o ).

tff(sy_c_member_001t__Product____Type__Ounit,type,
    member_Product_unit: ( product_unit * set_Product_unit ) > $o ).

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

tff(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    member2643936169264416010at_nat: ( set_Pr1261947904930325089at_nat * set_se7855581050983116737at_nat ) > $o ).

tff(sy_c_member_001t__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J,type,
    member7544390268018397336_nat_o: fun_Pr6460944699146269935_nat_o > fun_se7459991829846211660at_o_o ).

tff(sy_c_member_001tf__a,type,
    member_a2: ( a * set_a ) > $o ).

tff(sy_c_member_001tf__b,type,
    member_b2: ( b * set_b ) > $o ).

tff(sy_v_P,type,
    p: fun_a_fun_b_assn ).

tff(sy_v_l_H,type,
    l: list_b ).

% Relevant facts (4193)
tff(fact_0_pure__assn__eq__conv,axiom,
    ! [Pa: $o,Q: $o] :
      ( ( aa_o_assn(pure_assn,(Pa)) = aa_o_assn(pure_assn,(Q)) )
    <=> ( (Pa)
      <=> (Q) ) ) ).

% pure_assn_eq_conv
tff(fact_1_list__assn_Osimps_I1_J,axiom,
    ! [Pa: fun_a_fun_a_assn] : vEBT_L4319891404334229443sn_a_a(Pa,nil_a,nil_a) = one_one_assn ).

% list_assn.simps(1)
tff(fact_2_list__assn_Osimps_I1_J,axiom,
    ! [Pa: fun_a_fun_nat_assn] : vEBT_L2368420405249185739_a_nat(Pa,nil_a,nil_nat) = one_one_assn ).

% list_assn.simps(1)
tff(fact_3_list__assn_Osimps_I1_J,axiom,
    ! [Pa: fun_a_fun_int_assn] : vEBT_L2365929934740135463_a_int(Pa,nil_a,nil_int) = one_one_assn ).

% list_assn.simps(1)
tff(fact_4_list__assn_Osimps_I1_J,axiom,
    ! [Pa: fun_b_fun_a_assn] : vEBT_L1532435822361553410sn_b_a(Pa,nil_b,nil_a) = one_one_assn ).

% list_assn.simps(1)
tff(fact_5_list__assn_Osimps_I1_J,axiom,
    ! [Pa: fun_b_fun_b_assn] : vEBT_L1532435822361553411sn_b_b(Pa,nil_b,nil_b) = one_one_assn ).

% list_assn.simps(1)
tff(fact_6_list__assn_Osimps_I1_J,axiom,
    ! [Pa: fun_b_fun_nat_assn] : vEBT_L3603864734205580236_b_nat(Pa,nil_b,nil_nat) = one_one_assn ).

% list_assn.simps(1)
tff(fact_7_list__assn_Osimps_I1_J,axiom,
    ! [Pa: fun_b_fun_int_assn] : vEBT_L3601374263696529960_b_int(Pa,nil_b,nil_int) = one_one_assn ).

% list_assn.simps(1)
tff(fact_8_list__assn_Osimps_I1_J,axiom,
    ! [Pa: fun_nat_fun_a_assn] : vEBT_L6400351906427472169_nat_a(Pa,nil_nat,nil_a) = one_one_assn ).

% list_assn.simps(1)
tff(fact_9_list__assn_Osimps_I1_J,axiom,
    ! [Pa: fun_nat_fun_b_assn] : vEBT_L6400351906427472170_nat_b(Pa,nil_nat,nil_b) = one_one_assn ).

% list_assn.simps(1)
tff(fact_10_list__assn_Osimps_I1_J,axiom,
    ! [Pa: fun_nat_fun_nat_assn] : vEBT_L8301102511889123557at_nat(Pa,nil_nat,nil_nat) = one_one_assn ).

% list_assn.simps(1)
tff(fact_11_is__pure__assn__pure,axiom,
    ! [Pa: $o] : aa_assn_o(is_pure_assn,aa_o_assn(pure_assn,(Pa))) ).

% is_pure_assn_pure
tff(fact_12_list__ex1__simps_I1_J,axiom,
    ! [Pa: fun_a_o] : ~ aa_list_a_o(list_ex1_a(Pa),nil_a) ).

% list_ex1_simps(1)
tff(fact_13_list__ex1__simps_I1_J,axiom,
    ! [Pa: fun_b_o] : ~ aa_list_b_o(list_ex1_b(Pa),nil_b) ).

% list_ex1_simps(1)
tff(fact_14_list__ex1__simps_I1_J,axiom,
    ! [Pa: fun_Pr3571187171380813162assn_o] : ~ aa_lis1404944545740551261assn_o(list_e7761433933450087034n_assn(Pa),nil_Pr5671120429643327159n_assn) ).

% list_ex1_simps(1)
tff(fact_15_list__ex1__simps_I1_J,axiom,
    ! [Pa: fun_nat_o] : ~ aa_list_nat_o(list_ex1_nat(Pa),nil_nat) ).

% list_ex1_simps(1)
tff(fact_16_list__ex1__simps_I1_J,axiom,
    ! [Pa: fun_int_o] : ~ aa_list_int_o(list_ex1_int(Pa),nil_int) ).

% list_ex1_simps(1)
tff(fact_17_bind__simps_I1_J,axiom,
    ! [F: fun_a_list_a] : bind_a_a(nil_a,F) = nil_a ).

% bind_simps(1)
tff(fact_18_bind__simps_I1_J,axiom,
    ! [F: fun_a_list_b] : bind_a_b(nil_a,F) = nil_b ).

% bind_simps(1)
tff(fact_19_bind__simps_I1_J,axiom,
    ! [F: fun_a_list_nat] : bind_a_nat(nil_a,F) = nil_nat ).

% bind_simps(1)
tff(fact_20_bind__simps_I1_J,axiom,
    ! [F: fun_a_list_int] : bind_a_int(nil_a,F) = nil_int ).

% bind_simps(1)
tff(fact_21_bind__simps_I1_J,axiom,
    ! [F: fun_b_list_a] : bind_b_a(nil_b,F) = nil_a ).

% bind_simps(1)
tff(fact_22_bind__simps_I1_J,axiom,
    ! [F: fun_b_list_b] : bind_b_b(nil_b,F) = nil_b ).

% bind_simps(1)
tff(fact_23_bind__simps_I1_J,axiom,
    ! [F: fun_b_list_nat] : bind_b_nat(nil_b,F) = nil_nat ).

% bind_simps(1)
tff(fact_24_bind__simps_I1_J,axiom,
    ! [F: fun_b_list_int] : bind_b_int(nil_b,F) = nil_int ).

% bind_simps(1)
tff(fact_25_bind__simps_I1_J,axiom,
    ! [F: fun_nat_list_a] : bind_nat_a(nil_nat,F) = nil_a ).

% bind_simps(1)
tff(fact_26_bind__simps_I1_J,axiom,
    ! [F: fun_nat_list_b] : bind_nat_b(nil_nat,F) = nil_b ).

% bind_simps(1)
tff(fact_27_member__rec_I2_J,axiom,
    ! [Y: a] : ~ aa_a_o(member_a(nil_a),Y) ).

% member_rec(2)
tff(fact_28_member__rec_I2_J,axiom,
    ! [Y: b] : ~ aa_b_o(member_b(nil_b),Y) ).

% member_rec(2)
tff(fact_29_member__rec_I2_J,axiom,
    ! [Y: produc6575502325842934193n_assn] : ~ aa_Pro2491721552408928739assn_o(member743271071679453132n_assn(nil_Pr5671120429643327159n_assn),Y) ).

% member_rec(2)
tff(fact_30_member__rec_I2_J,axiom,
    ! [Y: nat] : ~ aa_nat_o(member_nat(nil_nat),Y) ).

% member_rec(2)
tff(fact_31_member__rec_I2_J,axiom,
    ! [Y: int] : ~ aa_int_o(member_int(nil_int),Y) ).

% member_rec(2)
tff(fact_32_is__pure__assnE,axiom,
    ! [A: assn] :
      ( aa_assn_o(is_pure_assn,A)
     => ~ ! [P: $o] : A != aa_o_assn(pure_assn,(P)) ) ).

% is_pure_assnE
tff(fact_33_is__pure__assn__def,axiom,
    ! [A: assn] :
      ( aa_assn_o(is_pure_assn,A)
    <=> ? [P2: $o] : A = aa_o_assn(pure_assn,(P2)) ) ).

% is_pure_assn_def
tff(fact_34_gen__length__code_I1_J,axiom,
    ! [N: nat] : gen_length_a(N,nil_a) = N ).

% gen_length_code(1)
tff(fact_35_gen__length__code_I1_J,axiom,
    ! [N: nat] : gen_length_b(N,nil_b) = N ).

% gen_length_code(1)
tff(fact_36_gen__length__code_I1_J,axiom,
    ! [N: nat] : gen_le1182329135810803268n_assn(N,nil_Pr5671120429643327159n_assn) = N ).

% gen_length_code(1)
tff(fact_37_gen__length__code_I1_J,axiom,
    ! [N: nat] : gen_length_nat(N,nil_nat) = N ).

% gen_length_code(1)
tff(fact_38_gen__length__code_I1_J,axiom,
    ! [N: nat] : gen_length_int(N,nil_int) = N ).

% gen_length_code(1)
tff(fact_39_list__assn_Osimps_I4_J,axiom,
    ! [Uu: fun_b_fun_nat_assn,V: nat,Va: list_nat] : vEBT_L3603864734205580236_b_nat(Uu,nil_b,aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,V),Va)) = bot_bot_assn ).

% list_assn.simps(4)
tff(fact_40_list__assn_Osimps_I4_J,axiom,
    ! [Uu: fun_nat_fun_nat_assn,V: nat,Va: list_nat] : vEBT_L8301102511889123557at_nat(Uu,nil_nat,aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,V),Va)) = bot_bot_assn ).

% list_assn.simps(4)
tff(fact_41_list__assn_Osimps_I4_J,axiom,
    ! [Uu: fun_int_fun_nat_assn,V: nat,Va: list_nat] : vEBT_L77084186935402305nt_nat(Uu,nil_int,aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,V),Va)) = bot_bot_assn ).

% list_assn.simps(4)
tff(fact_42_list__assn_Osimps_I4_J,axiom,
    ! [Uu: fun_b_fun_int_assn,V: int,Va: list_int] : vEBT_L3601374263696529960_b_int(Uu,nil_b,aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,V),Va)) = bot_bot_assn ).

% list_assn.simps(4)
tff(fact_43_list__assn_Osimps_I4_J,axiom,
    ! [Uu: fun_nat_fun_int_assn,V: int,Va: list_int] : vEBT_L8298612041380073281at_int(Uu,nil_nat,aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,V),Va)) = bot_bot_assn ).

% list_assn.simps(4)
tff(fact_44_list__assn_Osimps_I4_J,axiom,
    ! [Uu: fun_int_fun_int_assn,V: int,Va: list_int] : vEBT_L74593716426352029nt_int(Uu,nil_int,aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,V),Va)) = bot_bot_assn ).

% list_assn.simps(4)
tff(fact_45_list__assn_Osimps_I4_J,axiom,
    ! [Uu: fun_a_fun_b_assn,V: b,Va: list_b] : vEBT_L4319891404334229444sn_a_b(Uu,nil_a,aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,V),Va)) = bot_bot_assn ).

% list_assn.simps(4)
tff(fact_46_list__assn_Osimps_I4_J,axiom,
    ! [Uu: fun_b_fun_a_assn,V: a,Va: list_a] : vEBT_L1532435822361553410sn_b_a(Uu,nil_b,aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,V),Va)) = bot_bot_assn ).

% list_assn.simps(4)
tff(fact_47_list__assn_Osimps_I4_J,axiom,
    ! [Uu: fun_a_fun_nat_assn,V: nat,Va: list_nat] : vEBT_L2368420405249185739_a_nat(Uu,nil_a,aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,V),Va)) = bot_bot_assn ).

% list_assn.simps(4)
tff(fact_48_list__assn_Osimps_I4_J,axiom,
    ! [Uu: fun_a_fun_int_assn,V: int,Va: list_int] : vEBT_L2365929934740135463_a_int(Uu,nil_a,aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,V),Va)) = bot_bot_assn ).

% list_assn.simps(4)
tff(fact_49_list__assn_Osimps_I3_J,axiom,
    ! [Uu: fun_nat_fun_a_assn,V: nat,Va: list_nat] : vEBT_L6400351906427472169_nat_a(Uu,aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,V),Va),nil_a) = bot_bot_assn ).

% list_assn.simps(3)
tff(fact_50_list__assn_Osimps_I3_J,axiom,
    ! [Uu: fun_nat_fun_b_assn,V: nat,Va: list_nat] : vEBT_L6400351906427472170_nat_b(Uu,aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,V),Va),nil_b) = bot_bot_assn ).

% list_assn.simps(3)
tff(fact_51_list__assn_Osimps_I3_J,axiom,
    ! [Uu: fun_nat_fun_nat_assn,V: nat,Va: list_nat] : vEBT_L8301102511889123557at_nat(Uu,aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,V),Va),nil_nat) = bot_bot_assn ).

% list_assn.simps(3)
tff(fact_52_list__assn_Osimps_I3_J,axiom,
    ! [Uu: fun_nat_fun_int_assn,V: nat,Va: list_nat] : vEBT_L8298612041380073281at_int(Uu,aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,V),Va),nil_int) = bot_bot_assn ).

% list_assn.simps(3)
tff(fact_53_list__assn_Osimps_I3_J,axiom,
    ! [Uu: fun_int_fun_a_assn,V: int,Va: list_int] : vEBT_L4155206938757026253_int_a(Uu,aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,V),Va),nil_a) = bot_bot_assn ).

% list_assn.simps(3)
tff(fact_54_list__assn_Osimps_I3_J,axiom,
    ! [Uu: fun_int_fun_b_assn,V: int,Va: list_int] : vEBT_L4155206938757026254_int_b(Uu,aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,V),Va),nil_b) = bot_bot_assn ).

% list_assn.simps(3)
tff(fact_55_list__assn_Osimps_I3_J,axiom,
    ! [Uu: fun_int_fun_nat_assn,V: int,Va: list_int] : vEBT_L77084186935402305nt_nat(Uu,aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,V),Va),nil_nat) = bot_bot_assn ).

% list_assn.simps(3)
tff(fact_56_list__assn_Osimps_I3_J,axiom,
    ! [Uu: fun_int_fun_int_assn,V: int,Va: list_int] : vEBT_L74593716426352029nt_int(Uu,aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,V),Va),nil_int) = bot_bot_assn ).

% list_assn.simps(3)
tff(fact_57_list__assn_Osimps_I3_J,axiom,
    ! [Uu: fun_a_fun_b_assn,V: a,Va: list_a] : vEBT_L4319891404334229444sn_a_b(Uu,aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,V),Va),nil_b) = bot_bot_assn ).

% list_assn.simps(3)
tff(fact_58_list__assn_Osimps_I3_J,axiom,
    ! [Uu: fun_b_fun_a_assn,V: b,Va: list_b] : vEBT_L1532435822361553410sn_b_a(Uu,aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,V),Va),nil_a) = bot_bot_assn ).

% list_assn.simps(3)
tff(fact_59_list_Oinject,axiom,
    ! [X21: nat,X22: list_nat,Y21: nat,Y22: list_nat] :
      ( ( aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X21),X22) = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Y21),Y22) )
    <=> ( ( X21 = Y21 )
        & ( X22 = Y22 ) ) ) ).

% list.inject
tff(fact_60_list_Oinject,axiom,
    ! [X21: int,X22: list_int,Y21: int,Y22: list_int] :
      ( ( aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X21),X22) = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Y21),Y22) )
    <=> ( ( X21 = Y21 )
        & ( X22 = Y22 ) ) ) ).

% list.inject
tff(fact_61_list_Oinject,axiom,
    ! [X21: produc6575502325842934193n_assn,X22: list_P8527749157015355191n_assn,Y21: produc6575502325842934193n_assn,Y22: list_P8527749157015355191n_assn] :
      ( ( aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X21),X22) = aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,Y21),Y22) )
    <=> ( ( X21 = Y21 )
        & ( X22 = Y22 ) ) ) ).

% list.inject
tff(fact_62_pure__assn__eq__emp__iff,axiom,
    ! [Pa: $o] :
      ( ( aa_o_assn(pure_assn,(Pa)) = one_one_assn )
    <=> (Pa) ) ).

% pure_assn_eq_emp_iff
tff(fact_63_pure__true,axiom,
    aa_o_assn(pure_assn,$true) = one_one_assn ).

% pure_true
tff(fact_64_pure__assn__eq__false__iff,axiom,
    ! [Pa: $o] :
      ( ( aa_o_assn(pure_assn,(Pa)) = bot_bot_assn )
    <=> ~ (Pa) ) ).

% pure_assn_eq_false_iff
tff(fact_65_pure__false,axiom,
    aa_o_assn(pure_assn,$false) = bot_bot_assn ).

% pure_false
tff(fact_66_assn__basic__inequalities_I3_J,axiom,
    bot_bot_assn != one_one_assn ).

% assn_basic_inequalities(3)
tff(fact_67_member__rec_I1_J,axiom,
    ! [X: b,Xs: list_b,Y: b] :
      ( aa_b_o(member_b(aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X),Xs)),Y)
    <=> ( ( X = Y )
        | aa_b_o(member_b(Xs),Y) ) ) ).

% member_rec(1)
tff(fact_68_member__rec_I1_J,axiom,
    ! [X: a,Xs: list_a,Y: a] :
      ( aa_a_o(member_a(aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X),Xs)),Y)
    <=> ( ( X = Y )
        | aa_a_o(member_a(Xs),Y) ) ) ).

% member_rec(1)
tff(fact_69_member__rec_I1_J,axiom,
    ! [X: nat,Xs: list_nat,Y: nat] :
      ( aa_nat_o(member_nat(aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X),Xs)),Y)
    <=> ( ( X = Y )
        | aa_nat_o(member_nat(Xs),Y) ) ) ).

% member_rec(1)
tff(fact_70_member__rec_I1_J,axiom,
    ! [X: int,Xs: list_int,Y: int] :
      ( aa_int_o(member_int(aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X),Xs)),Y)
    <=> ( ( X = Y )
        | aa_int_o(member_int(Xs),Y) ) ) ).

% member_rec(1)
tff(fact_71_member__rec_I1_J,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn,Y: produc6575502325842934193n_assn] :
      ( aa_Pro2491721552408928739assn_o(member743271071679453132n_assn(aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X),Xs)),Y)
    <=> ( ( X = Y )
        | aa_Pro2491721552408928739assn_o(member743271071679453132n_assn(Xs),Y) ) ) ).

% member_rec(1)
tff(fact_72_is__pure__assn__basic__simps_I2_J,axiom,
    aa_assn_o(is_pure_assn,one_one_assn) ).

% is_pure_assn_basic_simps(2)
tff(fact_73_is__pure__assn__basic__simps_I1_J,axiom,
    aa_assn_o(is_pure_assn,bot_bot_assn) ).

% is_pure_assn_basic_simps(1)
tff(fact_74_transpose_Ocases,axiom,
    ! [X: list_list_a] :
      ( ( X != nil_list_a )
     => ( ! [Xss: list_list_a] : X != cons_list_a(nil_a,Xss)
       => ~ ! [X2: a,Xs2: list_a,Xss: list_list_a] : X != cons_list_a(aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X2),Xs2),Xss) ) ) ).

% transpose.cases
tff(fact_75_transpose_Ocases,axiom,
    ! [X: list_list_b] :
      ( ( X != nil_list_b )
     => ( ! [Xss: list_list_b] : X != cons_list_b(nil_b,Xss)
       => ~ ! [X2: b,Xs2: list_b,Xss: list_list_b] : X != cons_list_b(aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X2),Xs2),Xss) ) ) ).

% transpose.cases
tff(fact_76_transpose_Ocases,axiom,
    ! [X: list_list_nat] :
      ( ( X != nil_list_nat )
     => ( ! [Xss: list_list_nat] : X != cons_list_nat(nil_nat,Xss)
       => ~ ! [X2: nat,Xs2: list_nat,Xss: list_list_nat] : X != cons_list_nat(aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X2),Xs2),Xss) ) ) ).

% transpose.cases
tff(fact_77_transpose_Ocases,axiom,
    ! [X: list_list_int] :
      ( ( X != nil_list_int )
     => ( ! [Xss: list_list_int] : X != cons_list_int(nil_int,Xss)
       => ~ ! [X2: int,Xs2: list_int,Xss: list_list_int] : X != cons_list_int(aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X2),Xs2),Xss) ) ) ).

% transpose.cases
tff(fact_78_transpose_Ocases,axiom,
    ! [X: list_l6351802567095793725n_assn] :
      ( ( X != nil_li5476096274760905021n_assn )
     => ( ! [Xss: list_l6351802567095793725n_assn] : X != cons_l2423627976422276333n_assn(nil_Pr5671120429643327159n_assn,Xss)
       => ~ ! [X2: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn,Xss: list_l6351802567095793725n_assn] : X != cons_l2423627976422276333n_assn(aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X2),Xs2),Xss) ) ) ).

% transpose.cases
tff(fact_79_not__Cons__self2,axiom,
    ! [X: nat,Xs: list_nat] : aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X),Xs) != Xs ).

% not_Cons_self2
tff(fact_80_not__Cons__self2,axiom,
    ! [X: int,Xs: list_int] : aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X),Xs) != Xs ).

% not_Cons_self2
tff(fact_81_not__Cons__self2,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] : aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X),Xs) != Xs ).

% not_Cons_self2
tff(fact_82_list__nonempty__induct,axiom,
    ! [Xs: list_a,Pa: fun_list_a_o] :
      ( ( Xs != nil_a )
     => ( ! [X2: a] : aa_list_a_o(Pa,aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X2),nil_a))
       => ( ! [X2: a,Xs2: list_a] :
              ( ( Xs2 != nil_a )
             => ( aa_list_a_o(Pa,Xs2)
               => aa_list_a_o(Pa,aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X2),Xs2)) ) )
         => aa_list_a_o(Pa,Xs) ) ) ) ).

% list_nonempty_induct
tff(fact_83_list__nonempty__induct,axiom,
    ! [Xs: list_b,Pa: fun_list_b_o] :
      ( ( Xs != nil_b )
     => ( ! [X2: b] : aa_list_b_o(Pa,aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X2),nil_b))
       => ( ! [X2: b,Xs2: list_b] :
              ( ( Xs2 != nil_b )
             => ( aa_list_b_o(Pa,Xs2)
               => aa_list_b_o(Pa,aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X2),Xs2)) ) )
         => aa_list_b_o(Pa,Xs) ) ) ) ).

% list_nonempty_induct
tff(fact_84_list__nonempty__induct,axiom,
    ! [Xs: list_nat,Pa: fun_list_nat_o] :
      ( ( Xs != nil_nat )
     => ( ! [X2: nat] : aa_list_nat_o(Pa,aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X2),nil_nat))
       => ( ! [X2: nat,Xs2: list_nat] :
              ( ( Xs2 != nil_nat )
             => ( aa_list_nat_o(Pa,Xs2)
               => aa_list_nat_o(Pa,aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X2),Xs2)) ) )
         => aa_list_nat_o(Pa,Xs) ) ) ) ).

% list_nonempty_induct
tff(fact_85_list__nonempty__induct,axiom,
    ! [Xs: list_int,Pa: fun_list_int_o] :
      ( ( Xs != nil_int )
     => ( ! [X2: int] : aa_list_int_o(Pa,aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X2),nil_int))
       => ( ! [X2: int,Xs2: list_int] :
              ( ( Xs2 != nil_int )
             => ( aa_list_int_o(Pa,Xs2)
               => aa_list_int_o(Pa,aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X2),Xs2)) ) )
         => aa_list_int_o(Pa,Xs) ) ) ) ).

% list_nonempty_induct
tff(fact_86_list__nonempty__induct,axiom,
    ! [Xs: list_P8527749157015355191n_assn,Pa: fun_li2812663585457640420assn_o] :
      ( ( Xs != nil_Pr5671120429643327159n_assn )
     => ( ! [X2: produc6575502325842934193n_assn] : aa_lis1404944545740551261assn_o(Pa,aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X2),nil_Pr5671120429643327159n_assn))
       => ( ! [X2: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn] :
              ( ( Xs2 != nil_Pr5671120429643327159n_assn )
             => ( aa_lis1404944545740551261assn_o(Pa,Xs2)
               => aa_lis1404944545740551261assn_o(Pa,aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X2),Xs2)) ) )
         => aa_lis1404944545740551261assn_o(Pa,Xs) ) ) ) ).

% list_nonempty_induct
tff(fact_87_list__induct2_H,axiom,
    ! [Pa: fun_li7061788742611047022st_a_o,Xs: list_a,Ys: list_a] :
      ( aa_list_a_o(aa_lis4963054610627903775st_a_o(Pa,nil_a),nil_a)
     => ( ! [X2: a,Xs2: list_a] : aa_list_a_o(aa_lis4963054610627903775st_a_o(Pa,aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X2),Xs2)),nil_a)
       => ( ! [Y2: a,Ys2: list_a] : aa_list_a_o(aa_lis4963054610627903775st_a_o(Pa,nil_a),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,Y2),Ys2))
         => ( ! [X2: a,Xs2: list_a,Y2: a,Ys2: list_a] :
                ( aa_list_a_o(aa_lis4963054610627903775st_a_o(Pa,Xs2),Ys2)
               => aa_list_a_o(aa_lis4963054610627903775st_a_o(Pa,aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X2),Xs2)),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,Y2),Ys2)) )
           => aa_list_a_o(aa_lis4963054610627903775st_a_o(Pa,Xs),Ys) ) ) ) ) ).

% list_induct2'
tff(fact_88_list__induct2_H,axiom,
    ! [Pa: fun_li6426139742006060333st_b_o,Xs: list_a,Ys: list_b] :
      ( aa_list_b_o(aa_lis2175599028655227742st_b_o(Pa,nil_a),nil_b)
     => ( ! [X2: a,Xs2: list_a] : aa_list_b_o(aa_lis2175599028655227742st_b_o(Pa,aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X2),Xs2)),nil_b)
       => ( ! [Y2: b,Ys2: list_b] : aa_list_b_o(aa_lis2175599028655227742st_b_o(Pa,nil_a),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,Y2),Ys2))
         => ( ! [X2: a,Xs2: list_a,Y2: b,Ys2: list_b] :
                ( aa_list_b_o(aa_lis2175599028655227742st_b_o(Pa,Xs2),Ys2)
               => aa_list_b_o(aa_lis2175599028655227742st_b_o(Pa,aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X2),Xs2)),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,Y2),Ys2)) )
           => aa_list_b_o(aa_lis2175599028655227742st_b_o(Pa,Xs),Ys) ) ) ) ) ).

% list_induct2'
tff(fact_89_list__induct2_H,axiom,
    ! [Pa: fun_li100135987217049455st_a_o,Xs: list_b,Ys: list_a] :
      ( aa_list_a_o(aa_lis7224773892088682016st_a_o(Pa,nil_b),nil_a)
     => ( ! [X2: b,Xs2: list_b] : aa_list_a_o(aa_lis7224773892088682016st_a_o(Pa,aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X2),Xs2)),nil_a)
       => ( ! [Y2: a,Ys2: list_a] : aa_list_a_o(aa_lis7224773892088682016st_a_o(Pa,nil_b),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,Y2),Ys2))
         => ( ! [X2: b,Xs2: list_b,Y2: a,Ys2: list_a] :
                ( aa_list_a_o(aa_lis7224773892088682016st_a_o(Pa,Xs2),Ys2)
               => aa_list_a_o(aa_lis7224773892088682016st_a_o(Pa,aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X2),Xs2)),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,Y2),Ys2)) )
           => aa_list_a_o(aa_lis7224773892088682016st_a_o(Pa,Xs),Ys) ) ) ) ) ).

% list_induct2'
tff(fact_90_list__induct2_H,axiom,
    ! [Pa: fun_li8687859023466838574st_b_o,Xs: list_b,Ys: list_b] :
      ( aa_list_b_o(aa_lis4437318310116005983st_b_o(Pa,nil_b),nil_b)
     => ( ! [X2: b,Xs2: list_b] : aa_list_b_o(aa_lis4437318310116005983st_b_o(Pa,aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X2),Xs2)),nil_b)
       => ( ! [Y2: b,Ys2: list_b] : aa_list_b_o(aa_lis4437318310116005983st_b_o(Pa,nil_b),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,Y2),Ys2))
         => ( ! [X2: b,Xs2: list_b,Y2: b,Ys2: list_b] :
                ( aa_list_b_o(aa_lis4437318310116005983st_b_o(Pa,Xs2),Ys2)
               => aa_list_b_o(aa_lis4437318310116005983st_b_o(Pa,aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X2),Xs2)),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,Y2),Ys2)) )
           => aa_list_b_o(aa_lis4437318310116005983st_b_o(Pa,Xs),Ys) ) ) ) ) ).

% list_induct2'
tff(fact_91_list__induct2_H,axiom,
    ! [Pa: fun_li2975960090505800144_nat_o,Xs: list_a,Ys: list_nat] :
      ( aa_list_nat_o(aa_lis6662471382306767049_nat_o(Pa,nil_a),nil_nat)
     => ( ! [X2: a,Xs2: list_a] : aa_list_nat_o(aa_lis6662471382306767049_nat_o(Pa,aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X2),Xs2)),nil_nat)
       => ( ! [Y2: nat,Ys2: list_nat] : aa_list_nat_o(aa_lis6662471382306767049_nat_o(Pa,nil_a),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Y2),Ys2))
         => ( ! [X2: a,Xs2: list_a,Y2: nat,Ys2: list_nat] :
                ( aa_list_nat_o(aa_lis6662471382306767049_nat_o(Pa,Xs2),Ys2)
               => aa_list_nat_o(aa_lis6662471382306767049_nat_o(Pa,aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X2),Xs2)),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Y2),Ys2)) )
           => aa_list_nat_o(aa_lis6662471382306767049_nat_o(Pa,Xs),Ys) ) ) ) ) ).

% list_induct2'
tff(fact_92_list__induct2_H,axiom,
    ! [Pa: fun_li9190550672693380751_nat_o,Xs: list_b,Ys: list_nat] :
      ( aa_list_nat_o(aa_lis3653689927639571848_nat_o(Pa,nil_b),nil_nat)
     => ( ! [X2: b,Xs2: list_b] : aa_list_nat_o(aa_lis3653689927639571848_nat_o(Pa,aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X2),Xs2)),nil_nat)
       => ( ! [Y2: nat,Ys2: list_nat] : aa_list_nat_o(aa_lis3653689927639571848_nat_o(Pa,nil_b),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Y2),Ys2))
         => ( ! [X2: b,Xs2: list_b,Y2: nat,Ys2: list_nat] :
                ( aa_list_nat_o(aa_lis3653689927639571848_nat_o(Pa,Xs2),Ys2)
               => aa_list_nat_o(aa_lis3653689927639571848_nat_o(Pa,aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X2),Xs2)),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Y2),Ys2)) )
           => aa_list_nat_o(aa_lis3653689927639571848_nat_o(Pa,Xs),Ys) ) ) ) ) ).

% list_induct2'
tff(fact_93_list__induct2_H,axiom,
    ! [Pa: fun_li3047910582728761460_int_o,Xs: list_a,Ys: list_int] :
      ( aa_list_int_o(aa_lis4417326414636321133_int_o(Pa,nil_a),nil_int)
     => ( ! [X2: a,Xs2: list_a] : aa_list_int_o(aa_lis4417326414636321133_int_o(Pa,aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X2),Xs2)),nil_int)
       => ( ! [Y2: int,Ys2: list_int] : aa_list_int_o(aa_lis4417326414636321133_int_o(Pa,nil_a),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Y2),Ys2))
         => ( ! [X2: a,Xs2: list_a,Y2: int,Ys2: list_int] :
                ( aa_list_int_o(aa_lis4417326414636321133_int_o(Pa,Xs2),Ys2)
               => aa_list_int_o(aa_lis4417326414636321133_int_o(Pa,aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X2),Xs2)),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Y2),Ys2)) )
           => aa_list_int_o(aa_lis4417326414636321133_int_o(Pa,Xs),Ys) ) ) ) ) ).

% list_induct2'
tff(fact_94_list__induct2_H,axiom,
    ! [Pa: fun_li39129128061566259_int_o,Xs: list_b,Ys: list_int] :
      ( aa_list_int_o(aa_lis1408544959969125932_int_o(Pa,nil_b),nil_int)
     => ( ! [X2: b,Xs2: list_b] : aa_list_int_o(aa_lis1408544959969125932_int_o(Pa,aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X2),Xs2)),nil_int)
       => ( ! [Y2: int,Ys2: list_int] : aa_list_int_o(aa_lis1408544959969125932_int_o(Pa,nil_b),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Y2),Ys2))
         => ( ! [X2: b,Xs2: list_b,Y2: int,Ys2: list_int] :
                ( aa_list_int_o(aa_lis1408544959969125932_int_o(Pa,Xs2),Ys2)
               => aa_list_int_o(aa_lis1408544959969125932_int_o(Pa,aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X2),Xs2)),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Y2),Ys2)) )
           => aa_list_int_o(aa_lis1408544959969125932_int_o(Pa,Xs),Ys) ) ) ) ) ).

% list_induct2'
tff(fact_95_list__induct2_H,axiom,
    ! [Pa: fun_li5607945090506445748st_a_o,Xs: list_nat,Ys: list_a] :
      ( aa_list_a_o(aa_lis6242202807773918837st_a_o(Pa,nil_nat),nil_a)
     => ( ! [X2: nat,Xs2: list_nat] : aa_list_a_o(aa_lis6242202807773918837st_a_o(Pa,aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X2),Xs2)),nil_a)
       => ( ! [Y2: a,Ys2: list_a] : aa_list_a_o(aa_lis6242202807773918837st_a_o(Pa,nil_nat),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,Y2),Ys2))
         => ( ! [X2: nat,Xs2: list_nat,Y2: a,Ys2: list_a] :
                ( aa_list_a_o(aa_lis6242202807773918837st_a_o(Pa,Xs2),Ys2)
               => aa_list_a_o(aa_lis6242202807773918837st_a_o(Pa,aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X2),Xs2)),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,Y2),Ys2)) )
           => aa_list_a_o(aa_lis6242202807773918837st_a_o(Pa,Xs),Ys) ) ) ) ) ).

% list_induct2'
tff(fact_96_list__induct2_H,axiom,
    ! [Pa: fun_li4972296089901459059st_b_o,Xs: list_nat,Ys: list_b] :
      ( aa_list_b_o(aa_lis3454747225801242804st_b_o(Pa,nil_nat),nil_b)
     => ( ! [X2: nat,Xs2: list_nat] : aa_list_b_o(aa_lis3454747225801242804st_b_o(Pa,aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X2),Xs2)),nil_b)
       => ( ! [Y2: b,Ys2: list_b] : aa_list_b_o(aa_lis3454747225801242804st_b_o(Pa,nil_nat),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,Y2),Ys2))
         => ( ! [X2: nat,Xs2: list_nat,Y2: b,Ys2: list_b] :
                ( aa_list_b_o(aa_lis3454747225801242804st_b_o(Pa,Xs2),Ys2)
               => aa_list_b_o(aa_lis3454747225801242804st_b_o(Pa,aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X2),Xs2)),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,Y2),Ys2)) )
           => aa_list_b_o(aa_lis3454747225801242804st_b_o(Pa,Xs),Ys) ) ) ) ) ).

% list_induct2'
tff(fact_97_neq__Nil__conv,axiom,
    ! [Xs: list_a] :
      ( ( Xs != nil_a )
    <=> ? [Y3: a,Ys3: list_a] : Xs = aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,Y3),Ys3) ) ).

% neq_Nil_conv
tff(fact_98_neq__Nil__conv,axiom,
    ! [Xs: list_b] :
      ( ( Xs != nil_b )
    <=> ? [Y3: b,Ys3: list_b] : Xs = aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,Y3),Ys3) ) ).

% neq_Nil_conv
tff(fact_99_neq__Nil__conv,axiom,
    ! [Xs: list_nat] :
      ( ( Xs != nil_nat )
    <=> ? [Y3: nat,Ys3: list_nat] : Xs = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Y3),Ys3) ) ).

% neq_Nil_conv
tff(fact_100_neq__Nil__conv,axiom,
    ! [Xs: list_int] :
      ( ( Xs != nil_int )
    <=> ? [Y3: int,Ys3: list_int] : Xs = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Y3),Ys3) ) ).

% neq_Nil_conv
tff(fact_101_neq__Nil__conv,axiom,
    ! [Xs: list_P8527749157015355191n_assn] :
      ( ( Xs != nil_Pr5671120429643327159n_assn )
    <=> ? [Y3: produc6575502325842934193n_assn,Ys3: list_P8527749157015355191n_assn] : Xs = aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,Y3),Ys3) ) ).

% neq_Nil_conv
tff(fact_102_remdups__adj_Ocases,axiom,
    ! [X: list_a] :
      ( ( X != nil_a )
     => ( ! [X2: a] : X != aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X2),nil_a)
       => ~ ! [X2: a,Y2: a,Xs2: list_a] : X != aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X2),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,Y2),Xs2)) ) ) ).

% remdups_adj.cases
tff(fact_103_remdups__adj_Ocases,axiom,
    ! [X: list_b] :
      ( ( X != nil_b )
     => ( ! [X2: b] : X != aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X2),nil_b)
       => ~ ! [X2: b,Y2: b,Xs2: list_b] : X != aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X2),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,Y2),Xs2)) ) ) ).

% remdups_adj.cases
tff(fact_104_remdups__adj_Ocases,axiom,
    ! [X: list_nat] :
      ( ( X != nil_nat )
     => ( ! [X2: nat] : X != aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X2),nil_nat)
       => ~ ! [X2: nat,Y2: nat,Xs2: list_nat] : X != aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X2),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Y2),Xs2)) ) ) ).

% remdups_adj.cases
tff(fact_105_remdups__adj_Ocases,axiom,
    ! [X: list_int] :
      ( ( X != nil_int )
     => ( ! [X2: int] : X != aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X2),nil_int)
       => ~ ! [X2: int,Y2: int,Xs2: list_int] : X != aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X2),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Y2),Xs2)) ) ) ).

% remdups_adj.cases
tff(fact_106_remdups__adj_Ocases,axiom,
    ! [X: list_P8527749157015355191n_assn] :
      ( ( X != nil_Pr5671120429643327159n_assn )
     => ( ! [X2: produc6575502325842934193n_assn] : X != aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X2),nil_Pr5671120429643327159n_assn)
       => ~ ! [X2: produc6575502325842934193n_assn,Y2: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn] : X != aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X2),aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,Y2),Xs2)) ) ) ).

% remdups_adj.cases
tff(fact_107_min__list_Ocases,axiom,
    ! [X: list_nat] :
      ( ! [X2: nat,Xs2: list_nat] : X != aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X2),Xs2)
     => ( X = nil_nat ) ) ).

% min_list.cases
tff(fact_108_min__list_Ocases,axiom,
    ! [X: list_int] :
      ( ! [X2: int,Xs2: list_int] : X != aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X2),Xs2)
     => ( X = nil_int ) ) ).

% min_list.cases
tff(fact_109_list_Oexhaust,axiom,
    ! [Y: list_a] :
      ( ( Y != nil_a )
     => ~ ! [X212: a,X222: list_a] : Y != aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X212),X222) ) ).

% list.exhaust
tff(fact_110_list_Oexhaust,axiom,
    ! [Y: list_b] :
      ( ( Y != nil_b )
     => ~ ! [X212: b,X222: list_b] : Y != aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X212),X222) ) ).

% list.exhaust
tff(fact_111_list_Oexhaust,axiom,
    ! [Y: list_nat] :
      ( ( Y != nil_nat )
     => ~ ! [X212: nat,X222: list_nat] : Y != aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X212),X222) ) ).

% list.exhaust
tff(fact_112_list_Oexhaust,axiom,
    ! [Y: list_int] :
      ( ( Y != nil_int )
     => ~ ! [X212: int,X222: list_int] : Y != aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X212),X222) ) ).

% list.exhaust
tff(fact_113_list_Oexhaust,axiom,
    ! [Y: list_P8527749157015355191n_assn] :
      ( ( Y != nil_Pr5671120429643327159n_assn )
     => ~ ! [X212: produc6575502325842934193n_assn,X222: list_P8527749157015355191n_assn] : Y != aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X212),X222) ) ).

% list.exhaust
tff(fact_114_list_OdiscI,axiom,
    ! [List: list_a,X21: a,X22: list_a] :
      ( ( List = aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X21),X22) )
     => ( List != nil_a ) ) ).

% list.discI
tff(fact_115_list_OdiscI,axiom,
    ! [List: list_b,X21: b,X22: list_b] :
      ( ( List = aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X21),X22) )
     => ( List != nil_b ) ) ).

% list.discI
tff(fact_116_list_OdiscI,axiom,
    ! [List: list_nat,X21: nat,X22: list_nat] :
      ( ( List = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X21),X22) )
     => ( List != nil_nat ) ) ).

% list.discI
tff(fact_117_list_OdiscI,axiom,
    ! [List: list_int,X21: int,X22: list_int] :
      ( ( List = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X21),X22) )
     => ( List != nil_int ) ) ).

% list.discI
tff(fact_118_list_OdiscI,axiom,
    ! [List: list_P8527749157015355191n_assn,X21: produc6575502325842934193n_assn,X22: list_P8527749157015355191n_assn] :
      ( ( List = aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X21),X22) )
     => ( List != nil_Pr5671120429643327159n_assn ) ) ).

% list.discI
tff(fact_119_list_Odistinct_I1_J,axiom,
    ! [X21: a,X22: list_a] : nil_a != aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X21),X22) ).

% list.distinct(1)
tff(fact_120_list_Odistinct_I1_J,axiom,
    ! [X21: b,X22: list_b] : nil_b != aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X21),X22) ).

% list.distinct(1)
tff(fact_121_list_Odistinct_I1_J,axiom,
    ! [X21: nat,X22: list_nat] : nil_nat != aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X21),X22) ).

% list.distinct(1)
tff(fact_122_list_Odistinct_I1_J,axiom,
    ! [X21: int,X22: list_int] : nil_int != aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X21),X22) ).

% list.distinct(1)
tff(fact_123_list_Odistinct_I1_J,axiom,
    ! [X21: produc6575502325842934193n_assn,X22: list_P8527749157015355191n_assn] : nil_Pr5671120429643327159n_assn != aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X21),X22) ).

% list.distinct(1)
tff(fact_124_mergesort__by__rel__merge__induct,axiom,
    ! [Pa: fun_li7061788742611047022st_a_o,R: fun_a_fun_a_o,Xs: list_a,Ys: list_a] :
      ( ! [Xs2: list_a] : aa_list_a_o(aa_lis4963054610627903775st_a_o(Pa,Xs2),nil_a)
     => ( ! [Ys2: list_a] : aa_list_a_o(aa_lis4963054610627903775st_a_o(Pa,nil_a),Ys2)
       => ( ! [X2: a,Xs2: list_a,Y2: a,Ys2: list_a] :
              ( aa_a_o(aa_a_fun_a_o(R,X2),Y2)
             => ( aa_list_a_o(aa_lis4963054610627903775st_a_o(Pa,Xs2),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,Y2),Ys2))
               => aa_list_a_o(aa_lis4963054610627903775st_a_o(Pa,aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X2),Xs2)),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,Y2),Ys2)) ) )
         => ( ! [X2: a,Xs2: list_a,Y2: a,Ys2: list_a] :
                ( ~ aa_a_o(aa_a_fun_a_o(R,X2),Y2)
               => ( aa_list_a_o(aa_lis4963054610627903775st_a_o(Pa,aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X2),Xs2)),Ys2)
                 => aa_list_a_o(aa_lis4963054610627903775st_a_o(Pa,aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X2),Xs2)),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,Y2),Ys2)) ) )
           => aa_list_a_o(aa_lis4963054610627903775st_a_o(Pa,Xs),Ys) ) ) ) ) ).

% mergesort_by_rel_merge_induct
tff(fact_125_mergesort__by__rel__merge__induct,axiom,
    ! [Pa: fun_li100135987217049455st_a_o,R: fun_b_fun_a_o,Xs: list_b,Ys: list_a] :
      ( ! [Xs2: list_b] : aa_list_a_o(aa_lis7224773892088682016st_a_o(Pa,Xs2),nil_a)
     => ( ! [Ys2: list_a] : aa_list_a_o(aa_lis7224773892088682016st_a_o(Pa,nil_b),Ys2)
       => ( ! [X2: b,Xs2: list_b,Y2: a,Ys2: list_a] :
              ( aa_a_o(aa_b_fun_a_o(R,X2),Y2)
             => ( aa_list_a_o(aa_lis7224773892088682016st_a_o(Pa,Xs2),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,Y2),Ys2))
               => aa_list_a_o(aa_lis7224773892088682016st_a_o(Pa,aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X2),Xs2)),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,Y2),Ys2)) ) )
         => ( ! [X2: b,Xs2: list_b,Y2: a,Ys2: list_a] :
                ( ~ aa_a_o(aa_b_fun_a_o(R,X2),Y2)
               => ( aa_list_a_o(aa_lis7224773892088682016st_a_o(Pa,aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X2),Xs2)),Ys2)
                 => aa_list_a_o(aa_lis7224773892088682016st_a_o(Pa,aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X2),Xs2)),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,Y2),Ys2)) ) )
           => aa_list_a_o(aa_lis7224773892088682016st_a_o(Pa,Xs),Ys) ) ) ) ) ).

% mergesort_by_rel_merge_induct
tff(fact_126_mergesort__by__rel__merge__induct,axiom,
    ! [Pa: fun_li6426139742006060333st_b_o,R: fun_a_fun_b_o,Xs: list_a,Ys: list_b] :
      ( ! [Xs2: list_a] : aa_list_b_o(aa_lis2175599028655227742st_b_o(Pa,Xs2),nil_b)
     => ( ! [Ys2: list_b] : aa_list_b_o(aa_lis2175599028655227742st_b_o(Pa,nil_a),Ys2)
       => ( ! [X2: a,Xs2: list_a,Y2: b,Ys2: list_b] :
              ( aa_b_o(aa_a_fun_b_o(R,X2),Y2)
             => ( aa_list_b_o(aa_lis2175599028655227742st_b_o(Pa,Xs2),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,Y2),Ys2))
               => aa_list_b_o(aa_lis2175599028655227742st_b_o(Pa,aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X2),Xs2)),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,Y2),Ys2)) ) )
         => ( ! [X2: a,Xs2: list_a,Y2: b,Ys2: list_b] :
                ( ~ aa_b_o(aa_a_fun_b_o(R,X2),Y2)
               => ( aa_list_b_o(aa_lis2175599028655227742st_b_o(Pa,aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X2),Xs2)),Ys2)
                 => aa_list_b_o(aa_lis2175599028655227742st_b_o(Pa,aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X2),Xs2)),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,Y2),Ys2)) ) )
           => aa_list_b_o(aa_lis2175599028655227742st_b_o(Pa,Xs),Ys) ) ) ) ) ).

% mergesort_by_rel_merge_induct
tff(fact_127_mergesort__by__rel__merge__induct,axiom,
    ! [Pa: fun_li8687859023466838574st_b_o,R: fun_b_fun_b_o,Xs: list_b,Ys: list_b] :
      ( ! [Xs2: list_b] : aa_list_b_o(aa_lis4437318310116005983st_b_o(Pa,Xs2),nil_b)
     => ( ! [Ys2: list_b] : aa_list_b_o(aa_lis4437318310116005983st_b_o(Pa,nil_b),Ys2)
       => ( ! [X2: b,Xs2: list_b,Y2: b,Ys2: list_b] :
              ( aa_b_o(aa_b_fun_b_o(R,X2),Y2)
             => ( aa_list_b_o(aa_lis4437318310116005983st_b_o(Pa,Xs2),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,Y2),Ys2))
               => aa_list_b_o(aa_lis4437318310116005983st_b_o(Pa,aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X2),Xs2)),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,Y2),Ys2)) ) )
         => ( ! [X2: b,Xs2: list_b,Y2: b,Ys2: list_b] :
                ( ~ aa_b_o(aa_b_fun_b_o(R,X2),Y2)
               => ( aa_list_b_o(aa_lis4437318310116005983st_b_o(Pa,aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X2),Xs2)),Ys2)
                 => aa_list_b_o(aa_lis4437318310116005983st_b_o(Pa,aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X2),Xs2)),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,Y2),Ys2)) ) )
           => aa_list_b_o(aa_lis4437318310116005983st_b_o(Pa,Xs),Ys) ) ) ) ) ).

% mergesort_by_rel_merge_induct
tff(fact_128_mergesort__by__rel__merge__induct,axiom,
    ! [Pa: fun_li5607945090506445748st_a_o,R: fun_nat_fun_a_o,Xs: list_nat,Ys: list_a] :
      ( ! [Xs2: list_nat] : aa_list_a_o(aa_lis6242202807773918837st_a_o(Pa,Xs2),nil_a)
     => ( ! [Ys2: list_a] : aa_list_a_o(aa_lis6242202807773918837st_a_o(Pa,nil_nat),Ys2)
       => ( ! [X2: nat,Xs2: list_nat,Y2: a,Ys2: list_a] :
              ( aa_a_o(aa_nat_fun_a_o(R,X2),Y2)
             => ( aa_list_a_o(aa_lis6242202807773918837st_a_o(Pa,Xs2),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,Y2),Ys2))
               => aa_list_a_o(aa_lis6242202807773918837st_a_o(Pa,aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X2),Xs2)),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,Y2),Ys2)) ) )
         => ( ! [X2: nat,Xs2: list_nat,Y2: a,Ys2: list_a] :
                ( ~ aa_a_o(aa_nat_fun_a_o(R,X2),Y2)
               => ( aa_list_a_o(aa_lis6242202807773918837st_a_o(Pa,aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X2),Xs2)),Ys2)
                 => aa_list_a_o(aa_lis6242202807773918837st_a_o(Pa,aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X2),Xs2)),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,Y2),Ys2)) ) )
           => aa_list_a_o(aa_lis6242202807773918837st_a_o(Pa,Xs),Ys) ) ) ) ) ).

% mergesort_by_rel_merge_induct
tff(fact_129_mergesort__by__rel__merge__induct,axiom,
    ! [Pa: fun_li4972296089901459059st_b_o,R: fun_nat_fun_b_o,Xs: list_nat,Ys: list_b] :
      ( ! [Xs2: list_nat] : aa_list_b_o(aa_lis3454747225801242804st_b_o(Pa,Xs2),nil_b)
     => ( ! [Ys2: list_b] : aa_list_b_o(aa_lis3454747225801242804st_b_o(Pa,nil_nat),Ys2)
       => ( ! [X2: nat,Xs2: list_nat,Y2: b,Ys2: list_b] :
              ( aa_b_o(aa_nat_fun_b_o(R,X2),Y2)
             => ( aa_list_b_o(aa_lis3454747225801242804st_b_o(Pa,Xs2),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,Y2),Ys2))
               => aa_list_b_o(aa_lis3454747225801242804st_b_o(Pa,aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X2),Xs2)),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,Y2),Ys2)) ) )
         => ( ! [X2: nat,Xs2: list_nat,Y2: b,Ys2: list_b] :
                ( ~ aa_b_o(aa_nat_fun_b_o(R,X2),Y2)
               => ( aa_list_b_o(aa_lis3454747225801242804st_b_o(Pa,aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X2),Xs2)),Ys2)
                 => aa_list_b_o(aa_lis3454747225801242804st_b_o(Pa,aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X2),Xs2)),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,Y2),Ys2)) ) )
           => aa_list_b_o(aa_lis3454747225801242804st_b_o(Pa,Xs),Ys) ) ) ) ) ).

% mergesort_by_rel_merge_induct
tff(fact_130_mergesort__by__rel__merge__induct,axiom,
    ! [Pa: fun_li8621252006174909968st_a_o,R: fun_int_fun_a_o,Xs: list_int,Ys: list_a] :
      ( ! [Xs2: list_int] : aa_list_a_o(aa_lis32137686587607249st_a_o(Pa,Xs2),nil_a)
     => ( ! [Ys2: list_a] : aa_list_a_o(aa_lis32137686587607249st_a_o(Pa,nil_int),Ys2)
       => ( ! [X2: int,Xs2: list_int,Y2: a,Ys2: list_a] :
              ( aa_a_o(aa_int_fun_a_o(R,X2),Y2)
             => ( aa_list_a_o(aa_lis32137686587607249st_a_o(Pa,Xs2),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,Y2),Ys2))
               => aa_list_a_o(aa_lis32137686587607249st_a_o(Pa,aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X2),Xs2)),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,Y2),Ys2)) ) )
         => ( ! [X2: int,Xs2: list_int,Y2: a,Ys2: list_a] :
                ( ~ aa_a_o(aa_int_fun_a_o(R,X2),Y2)
               => ( aa_list_a_o(aa_lis32137686587607249st_a_o(Pa,aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X2),Xs2)),Ys2)
                 => aa_list_a_o(aa_lis32137686587607249st_a_o(Pa,aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X2),Xs2)),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,Y2),Ys2)) ) )
           => aa_list_a_o(aa_lis32137686587607249st_a_o(Pa,Xs),Ys) ) ) ) ) ).

% mergesort_by_rel_merge_induct
tff(fact_131_mergesort__by__rel__merge__induct,axiom,
    ! [Pa: fun_li7985603005569923279st_b_o,R: fun_int_fun_b_o,Xs: list_int,Ys: list_b] :
      ( ! [Xs2: list_int] : aa_list_b_o(aa_lis6468054141469707024st_b_o(Pa,Xs2),nil_b)
     => ( ! [Ys2: list_b] : aa_list_b_o(aa_lis6468054141469707024st_b_o(Pa,nil_int),Ys2)
       => ( ! [X2: int,Xs2: list_int,Y2: b,Ys2: list_b] :
              ( aa_b_o(aa_int_fun_b_o(R,X2),Y2)
             => ( aa_list_b_o(aa_lis6468054141469707024st_b_o(Pa,Xs2),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,Y2),Ys2))
               => aa_list_b_o(aa_lis6468054141469707024st_b_o(Pa,aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X2),Xs2)),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,Y2),Ys2)) ) )
         => ( ! [X2: int,Xs2: list_int,Y2: b,Ys2: list_b] :
                ( ~ aa_b_o(aa_int_fun_b_o(R,X2),Y2)
               => ( aa_list_b_o(aa_lis6468054141469707024st_b_o(Pa,aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X2),Xs2)),Ys2)
                 => aa_list_b_o(aa_lis6468054141469707024st_b_o(Pa,aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X2),Xs2)),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,Y2),Ys2)) ) )
           => aa_list_b_o(aa_lis6468054141469707024st_b_o(Pa,Xs),Ys) ) ) ) ) ).

% mergesort_by_rel_merge_induct
tff(fact_132_mergesort__by__rel__merge__induct,axiom,
    ! [Pa: fun_li2975960090505800144_nat_o,R: fun_a_fun_nat_o,Xs: list_a,Ys: list_nat] :
      ( ! [Xs2: list_a] : aa_list_nat_o(aa_lis6662471382306767049_nat_o(Pa,Xs2),nil_nat)
     => ( ! [Ys2: list_nat] : aa_list_nat_o(aa_lis6662471382306767049_nat_o(Pa,nil_a),Ys2)
       => ( ! [X2: a,Xs2: list_a,Y2: nat,Ys2: list_nat] :
              ( aa_nat_o(aa_a_fun_nat_o(R,X2),Y2)
             => ( aa_list_nat_o(aa_lis6662471382306767049_nat_o(Pa,Xs2),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Y2),Ys2))
               => aa_list_nat_o(aa_lis6662471382306767049_nat_o(Pa,aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X2),Xs2)),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Y2),Ys2)) ) )
         => ( ! [X2: a,Xs2: list_a,Y2: nat,Ys2: list_nat] :
                ( ~ aa_nat_o(aa_a_fun_nat_o(R,X2),Y2)
               => ( aa_list_nat_o(aa_lis6662471382306767049_nat_o(Pa,aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X2),Xs2)),Ys2)
                 => aa_list_nat_o(aa_lis6662471382306767049_nat_o(Pa,aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X2),Xs2)),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Y2),Ys2)) ) )
           => aa_list_nat_o(aa_lis6662471382306767049_nat_o(Pa,Xs),Ys) ) ) ) ) ).

% mergesort_by_rel_merge_induct
tff(fact_133_mergesort__by__rel__merge__induct,axiom,
    ! [Pa: fun_li9190550672693380751_nat_o,R: fun_b_fun_nat_o,Xs: list_b,Ys: list_nat] :
      ( ! [Xs2: list_b] : aa_list_nat_o(aa_lis3653689927639571848_nat_o(Pa,Xs2),nil_nat)
     => ( ! [Ys2: list_nat] : aa_list_nat_o(aa_lis3653689927639571848_nat_o(Pa,nil_b),Ys2)
       => ( ! [X2: b,Xs2: list_b,Y2: nat,Ys2: list_nat] :
              ( aa_nat_o(aa_b_fun_nat_o(R,X2),Y2)
             => ( aa_list_nat_o(aa_lis3653689927639571848_nat_o(Pa,Xs2),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Y2),Ys2))
               => aa_list_nat_o(aa_lis3653689927639571848_nat_o(Pa,aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X2),Xs2)),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Y2),Ys2)) ) )
         => ( ! [X2: b,Xs2: list_b,Y2: nat,Ys2: list_nat] :
                ( ~ aa_nat_o(aa_b_fun_nat_o(R,X2),Y2)
               => ( aa_list_nat_o(aa_lis3653689927639571848_nat_o(Pa,aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X2),Xs2)),Ys2)
                 => aa_list_nat_o(aa_lis3653689927639571848_nat_o(Pa,aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X2),Xs2)),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Y2),Ys2)) ) )
           => aa_list_nat_o(aa_lis3653689927639571848_nat_o(Pa,Xs),Ys) ) ) ) ) ).

% mergesort_by_rel_merge_induct
tff(fact_134_mem__Collect__eq,axiom,
    ! [A: fun_Pr6460944699146269935_nat_o,Pa: fun_fu5135969500543783852at_o_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A),collec6407654713642994138_nat_o(Pa))
    <=> aa_fun1675921005893230629at_o_o(Pa,A) ) ).

% mem_Collect_eq
tff(fact_135_mem__Collect__eq,axiom,
    ! [A: nat,Pa: fun_nat_o] :
      ( aa_set_nat_o(member_nat2(A),collect_nat(Pa))
    <=> aa_nat_o(Pa,A) ) ).

% mem_Collect_eq
tff(fact_136_mem__Collect__eq,axiom,
    ! [A: product_prod_nat_nat,Pa: fun_Pr4796943543331071514_nat_o] :
      ( member8440522571783428010at_nat(A,collec3392354462482085612at_nat(Pa))
    <=> aa_Pro6150637604445964947_nat_o(Pa,A) ) ).

% mem_Collect_eq
tff(fact_137_Collect__mem__eq,axiom,
    ! [A2: set_fu3555266933746557007_nat_o] : collec6407654713642994138_nat_o(aTP_Lamm_a(A2)) = A2 ).

% Collect_mem_eq
tff(fact_138_Collect__mem__eq,axiom,
    ! [A2: set_nat] : collect_nat(aTP_Lamm_aa(A2)) = A2 ).

% Collect_mem_eq
tff(fact_139_Collect__mem__eq,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] : collec3392354462482085612at_nat(aTP_Lamm_ab(A2)) = A2 ).

% Collect_mem_eq
tff(fact_140_Collect__cong,axiom,
    ! [Pa: fun_fu5135969500543783852at_o_o,Q: fun_fu5135969500543783852at_o_o] :
      ( ! [X2: fun_Pr6460944699146269935_nat_o] :
          ( aa_fun1675921005893230629at_o_o(Pa,X2)
        <=> aa_fun1675921005893230629at_o_o(Q,X2) )
     => ( collec6407654713642994138_nat_o(Pa) = collec6407654713642994138_nat_o(Q) ) ) ).

% Collect_cong
tff(fact_141_Collect__cong,axiom,
    ! [Pa: fun_nat_o,Q: fun_nat_o] :
      ( ! [X2: nat] :
          ( aa_nat_o(Pa,X2)
        <=> aa_nat_o(Q,X2) )
     => ( collect_nat(Pa) = collect_nat(Q) ) ) ).

% Collect_cong
tff(fact_142_Collect__cong,axiom,
    ! [Pa: fun_Pr4796943543331071514_nat_o,Q: fun_Pr4796943543331071514_nat_o] :
      ( ! [X2: product_prod_nat_nat] :
          ( aa_Pro6150637604445964947_nat_o(Pa,X2)
        <=> aa_Pro6150637604445964947_nat_o(Q,X2) )
     => ( collec3392354462482085612at_nat(Pa) = collec3392354462482085612at_nat(Q) ) ) ).

% Collect_cong
tff(fact_143_list__induct__first2,axiom,
    ! [Pa: fun_list_a_o,Xs: list_a] :
      ( aa_list_a_o(Pa,nil_a)
     => ( ! [X2: a] : aa_list_a_o(Pa,aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X2),nil_a))
       => ( ! [X1: a,X23: a,Xs2: list_a] :
              ( aa_list_a_o(Pa,Xs2)
             => aa_list_a_o(Pa,aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X1),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X23),Xs2))) )
         => aa_list_a_o(Pa,Xs) ) ) ) ).

% list_induct_first2
tff(fact_144_list__induct__first2,axiom,
    ! [Pa: fun_list_b_o,Xs: list_b] :
      ( aa_list_b_o(Pa,nil_b)
     => ( ! [X2: b] : aa_list_b_o(Pa,aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X2),nil_b))
       => ( ! [X1: b,X23: b,Xs2: list_b] :
              ( aa_list_b_o(Pa,Xs2)
             => aa_list_b_o(Pa,aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X1),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X23),Xs2))) )
         => aa_list_b_o(Pa,Xs) ) ) ) ).

% list_induct_first2
tff(fact_145_list__induct__first2,axiom,
    ! [Pa: fun_list_nat_o,Xs: list_nat] :
      ( aa_list_nat_o(Pa,nil_nat)
     => ( ! [X2: nat] : aa_list_nat_o(Pa,aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X2),nil_nat))
       => ( ! [X1: nat,X23: nat,Xs2: list_nat] :
              ( aa_list_nat_o(Pa,Xs2)
             => aa_list_nat_o(Pa,aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X1),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X23),Xs2))) )
         => aa_list_nat_o(Pa,Xs) ) ) ) ).

% list_induct_first2
tff(fact_146_list__induct__first2,axiom,
    ! [Pa: fun_list_int_o,Xs: list_int] :
      ( aa_list_int_o(Pa,nil_int)
     => ( ! [X2: int] : aa_list_int_o(Pa,aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X2),nil_int))
       => ( ! [X1: int,X23: int,Xs2: list_int] :
              ( aa_list_int_o(Pa,Xs2)
             => aa_list_int_o(Pa,aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X1),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X23),Xs2))) )
         => aa_list_int_o(Pa,Xs) ) ) ) ).

% list_induct_first2
tff(fact_147_list__induct__first2,axiom,
    ! [Pa: fun_li2812663585457640420assn_o,Xs: list_P8527749157015355191n_assn] :
      ( aa_lis1404944545740551261assn_o(Pa,nil_Pr5671120429643327159n_assn)
     => ( ! [X2: produc6575502325842934193n_assn] : aa_lis1404944545740551261assn_o(Pa,aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X2),nil_Pr5671120429643327159n_assn))
       => ( ! [X1: produc6575502325842934193n_assn,X23: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn] :
              ( aa_lis1404944545740551261assn_o(Pa,Xs2)
             => aa_lis1404944545740551261assn_o(Pa,aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X1),aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X23),Xs2))) )
         => aa_lis1404944545740551261assn_o(Pa,Xs) ) ) ) ).

% list_induct_first2
tff(fact_148_list__2pre__induct,axiom,
    ! [Pa: fun_li7061788742611047022st_a_o,W1: list_a,W2: list_a] :
      ( aa_list_a_o(aa_lis4963054610627903775st_a_o(Pa,nil_a),nil_a)
     => ( ! [E: a,W12: list_a,W22: list_a] :
            ( aa_list_a_o(aa_lis4963054610627903775st_a_o(Pa,W12),W22)
           => aa_list_a_o(aa_lis4963054610627903775st_a_o(Pa,aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,E),W12)),W22) )
       => ( ! [E: a,W13: list_a,W23: list_a] :
              ( aa_list_a_o(aa_lis4963054610627903775st_a_o(Pa,W13),W23)
             => aa_list_a_o(aa_lis4963054610627903775st_a_o(Pa,W13),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,E),W23)) )
         => aa_list_a_o(aa_lis4963054610627903775st_a_o(Pa,W1),W2) ) ) ) ).

% list_2pre_induct
tff(fact_149_list__2pre__induct,axiom,
    ! [Pa: fun_li6426139742006060333st_b_o,W1: list_a,W2: list_b] :
      ( aa_list_b_o(aa_lis2175599028655227742st_b_o(Pa,nil_a),nil_b)
     => ( ! [E: a,W12: list_a,W22: list_b] :
            ( aa_list_b_o(aa_lis2175599028655227742st_b_o(Pa,W12),W22)
           => aa_list_b_o(aa_lis2175599028655227742st_b_o(Pa,aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,E),W12)),W22) )
       => ( ! [E: b,W13: list_a,W23: list_b] :
              ( aa_list_b_o(aa_lis2175599028655227742st_b_o(Pa,W13),W23)
             => aa_list_b_o(aa_lis2175599028655227742st_b_o(Pa,W13),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,E),W23)) )
         => aa_list_b_o(aa_lis2175599028655227742st_b_o(Pa,W1),W2) ) ) ) ).

% list_2pre_induct
tff(fact_150_list__2pre__induct,axiom,
    ! [Pa: fun_li100135987217049455st_a_o,W1: list_b,W2: list_a] :
      ( aa_list_a_o(aa_lis7224773892088682016st_a_o(Pa,nil_b),nil_a)
     => ( ! [E: b,W12: list_b,W22: list_a] :
            ( aa_list_a_o(aa_lis7224773892088682016st_a_o(Pa,W12),W22)
           => aa_list_a_o(aa_lis7224773892088682016st_a_o(Pa,aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,E),W12)),W22) )
       => ( ! [E: a,W13: list_b,W23: list_a] :
              ( aa_list_a_o(aa_lis7224773892088682016st_a_o(Pa,W13),W23)
             => aa_list_a_o(aa_lis7224773892088682016st_a_o(Pa,W13),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,E),W23)) )
         => aa_list_a_o(aa_lis7224773892088682016st_a_o(Pa,W1),W2) ) ) ) ).

% list_2pre_induct
tff(fact_151_list__2pre__induct,axiom,
    ! [Pa: fun_li8687859023466838574st_b_o,W1: list_b,W2: list_b] :
      ( aa_list_b_o(aa_lis4437318310116005983st_b_o(Pa,nil_b),nil_b)
     => ( ! [E: b,W12: list_b,W22: list_b] :
            ( aa_list_b_o(aa_lis4437318310116005983st_b_o(Pa,W12),W22)
           => aa_list_b_o(aa_lis4437318310116005983st_b_o(Pa,aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,E),W12)),W22) )
       => ( ! [E: b,W13: list_b,W23: list_b] :
              ( aa_list_b_o(aa_lis4437318310116005983st_b_o(Pa,W13),W23)
             => aa_list_b_o(aa_lis4437318310116005983st_b_o(Pa,W13),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,E),W23)) )
         => aa_list_b_o(aa_lis4437318310116005983st_b_o(Pa,W1),W2) ) ) ) ).

% list_2pre_induct
tff(fact_152_list__2pre__induct,axiom,
    ! [Pa: fun_li2975960090505800144_nat_o,W1: list_a,W2: list_nat] :
      ( aa_list_nat_o(aa_lis6662471382306767049_nat_o(Pa,nil_a),nil_nat)
     => ( ! [E: a,W12: list_a,W22: list_nat] :
            ( aa_list_nat_o(aa_lis6662471382306767049_nat_o(Pa,W12),W22)
           => aa_list_nat_o(aa_lis6662471382306767049_nat_o(Pa,aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,E),W12)),W22) )
       => ( ! [E: nat,W13: list_a,W23: list_nat] :
              ( aa_list_nat_o(aa_lis6662471382306767049_nat_o(Pa,W13),W23)
             => aa_list_nat_o(aa_lis6662471382306767049_nat_o(Pa,W13),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,E),W23)) )
         => aa_list_nat_o(aa_lis6662471382306767049_nat_o(Pa,W1),W2) ) ) ) ).

% list_2pre_induct
tff(fact_153_list__2pre__induct,axiom,
    ! [Pa: fun_li9190550672693380751_nat_o,W1: list_b,W2: list_nat] :
      ( aa_list_nat_o(aa_lis3653689927639571848_nat_o(Pa,nil_b),nil_nat)
     => ( ! [E: b,W12: list_b,W22: list_nat] :
            ( aa_list_nat_o(aa_lis3653689927639571848_nat_o(Pa,W12),W22)
           => aa_list_nat_o(aa_lis3653689927639571848_nat_o(Pa,aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,E),W12)),W22) )
       => ( ! [E: nat,W13: list_b,W23: list_nat] :
              ( aa_list_nat_o(aa_lis3653689927639571848_nat_o(Pa,W13),W23)
             => aa_list_nat_o(aa_lis3653689927639571848_nat_o(Pa,W13),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,E),W23)) )
         => aa_list_nat_o(aa_lis3653689927639571848_nat_o(Pa,W1),W2) ) ) ) ).

% list_2pre_induct
tff(fact_154_list__2pre__induct,axiom,
    ! [Pa: fun_li3047910582728761460_int_o,W1: list_a,W2: list_int] :
      ( aa_list_int_o(aa_lis4417326414636321133_int_o(Pa,nil_a),nil_int)
     => ( ! [E: a,W12: list_a,W22: list_int] :
            ( aa_list_int_o(aa_lis4417326414636321133_int_o(Pa,W12),W22)
           => aa_list_int_o(aa_lis4417326414636321133_int_o(Pa,aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,E),W12)),W22) )
       => ( ! [E: int,W13: list_a,W23: list_int] :
              ( aa_list_int_o(aa_lis4417326414636321133_int_o(Pa,W13),W23)
             => aa_list_int_o(aa_lis4417326414636321133_int_o(Pa,W13),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,E),W23)) )
         => aa_list_int_o(aa_lis4417326414636321133_int_o(Pa,W1),W2) ) ) ) ).

% list_2pre_induct
tff(fact_155_list__2pre__induct,axiom,
    ! [Pa: fun_li39129128061566259_int_o,W1: list_b,W2: list_int] :
      ( aa_list_int_o(aa_lis1408544959969125932_int_o(Pa,nil_b),nil_int)
     => ( ! [E: b,W12: list_b,W22: list_int] :
            ( aa_list_int_o(aa_lis1408544959969125932_int_o(Pa,W12),W22)
           => aa_list_int_o(aa_lis1408544959969125932_int_o(Pa,aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,E),W12)),W22) )
       => ( ! [E: int,W13: list_b,W23: list_int] :
              ( aa_list_int_o(aa_lis1408544959969125932_int_o(Pa,W13),W23)
             => aa_list_int_o(aa_lis1408544959969125932_int_o(Pa,W13),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,E),W23)) )
         => aa_list_int_o(aa_lis1408544959969125932_int_o(Pa,W1),W2) ) ) ) ).

% list_2pre_induct
tff(fact_156_list__2pre__induct,axiom,
    ! [Pa: fun_li5607945090506445748st_a_o,W1: list_nat,W2: list_a] :
      ( aa_list_a_o(aa_lis6242202807773918837st_a_o(Pa,nil_nat),nil_a)
     => ( ! [E: nat,W12: list_nat,W22: list_a] :
            ( aa_list_a_o(aa_lis6242202807773918837st_a_o(Pa,W12),W22)
           => aa_list_a_o(aa_lis6242202807773918837st_a_o(Pa,aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,E),W12)),W22) )
       => ( ! [E: a,W13: list_nat,W23: list_a] :
              ( aa_list_a_o(aa_lis6242202807773918837st_a_o(Pa,W13),W23)
             => aa_list_a_o(aa_lis6242202807773918837st_a_o(Pa,W13),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,E),W23)) )
         => aa_list_a_o(aa_lis6242202807773918837st_a_o(Pa,W1),W2) ) ) ) ).

% list_2pre_induct
tff(fact_157_list__2pre__induct,axiom,
    ! [Pa: fun_li4972296089901459059st_b_o,W1: list_nat,W2: list_b] :
      ( aa_list_b_o(aa_lis3454747225801242804st_b_o(Pa,nil_nat),nil_b)
     => ( ! [E: nat,W12: list_nat,W22: list_b] :
            ( aa_list_b_o(aa_lis3454747225801242804st_b_o(Pa,W12),W22)
           => aa_list_b_o(aa_lis3454747225801242804st_b_o(Pa,aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,E),W12)),W22) )
       => ( ! [E: b,W13: list_nat,W23: list_b] :
              ( aa_list_b_o(aa_lis3454747225801242804st_b_o(Pa,W13),W23)
             => aa_list_b_o(aa_lis3454747225801242804st_b_o(Pa,W13),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,E),W23)) )
         => aa_list_b_o(aa_lis3454747225801242804st_b_o(Pa,W1),W2) ) ) ) ).

% list_2pre_induct
tff(fact_158_neq__NilE,axiom,
    ! [La: list_a] :
      ( ( La != nil_a )
     => ~ ! [X2: a,Xs2: list_a] : La != aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X2),Xs2) ) ).

% neq_NilE
tff(fact_159_neq__NilE,axiom,
    ! [La: list_b] :
      ( ( La != nil_b )
     => ~ ! [X2: b,Xs2: list_b] : La != aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X2),Xs2) ) ).

% neq_NilE
tff(fact_160_neq__NilE,axiom,
    ! [La: list_nat] :
      ( ( La != nil_nat )
     => ~ ! [X2: nat,Xs2: list_nat] : La != aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X2),Xs2) ) ).

% neq_NilE
tff(fact_161_neq__NilE,axiom,
    ! [La: list_int] :
      ( ( La != nil_int )
     => ~ ! [X2: int,Xs2: list_int] : La != aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X2),Xs2) ) ).

% neq_NilE
tff(fact_162_neq__NilE,axiom,
    ! [La: list_P8527749157015355191n_assn] :
      ( ( La != nil_Pr5671120429643327159n_assn )
     => ~ ! [X2: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn] : La != aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X2),Xs2) ) ).

% neq_NilE
tff(fact_163_quicksort_Ocases,axiom,
    ! [X: list_nat] :
      ( ( X != nil_nat )
     => ~ ! [X2: nat,Xs2: list_nat] : X != aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X2),Xs2) ) ).

% quicksort.cases
tff(fact_164_quicksort_Ocases,axiom,
    ! [X: list_int] :
      ( ( X != nil_int )
     => ~ ! [X2: int,Xs2: list_int] : X != aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X2),Xs2) ) ).

% quicksort.cases
tff(fact_165_list__assn_Oelims,axiom,
    ! [X: fun_b_fun_b_assn,Xa: list_b,Xb: list_b,Y: assn] :
      ( ( vEBT_L1532435822361553411sn_b_b(X,Xa,Xb) = Y )
     => ( ( ( Xa = nil_b )
         => ( ( Xb = nil_b )
           => ( Y != one_one_assn ) ) )
       => ( ! [A3: b,As: list_b] :
              ( ( Xa = aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,A3),As) )
             => ! [C: b,Cs: list_b] :
                  ( ( Xb = aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,C),Cs) )
                 => ( Y != aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_b_assn(aa_b_fun_b_assn(X,A3),C)),vEBT_L1532435822361553411sn_b_b(X,As,Cs)) ) ) )
         => ( ( ? [V2: b,Va2: list_b] : Xa = aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,V2),Va2)
             => ( ( Xb = nil_b )
               => ( Y != bot_bot_assn ) ) )
           => ~ ( ( Xa = nil_b )
               => ( ? [V2: b,Va2: list_b] : Xb = aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,V2),Va2)
                 => ( Y != bot_bot_assn ) ) ) ) ) ) ) ).

% list_assn.elims
tff(fact_166_list__assn_Oelims,axiom,
    ! [X: fun_b_fun_nat_assn,Xa: list_b,Xb: list_nat,Y: assn] :
      ( ( vEBT_L3603864734205580236_b_nat(X,Xa,Xb) = Y )
     => ( ( ( Xa = nil_b )
         => ( ( Xb = nil_nat )
           => ( Y != one_one_assn ) ) )
       => ( ! [A3: b,As: list_b] :
              ( ( Xa = aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,A3),As) )
             => ! [C: nat,Cs: list_nat] :
                  ( ( Xb = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,C),Cs) )
                 => ( Y != aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(aa_b_fun_nat_assn(X,A3),C)),vEBT_L3603864734205580236_b_nat(X,As,Cs)) ) ) )
         => ( ( ? [V2: b,Va2: list_b] : Xa = aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,V2),Va2)
             => ( ( Xb = nil_nat )
               => ( Y != bot_bot_assn ) ) )
           => ~ ( ( Xa = nil_b )
               => ( ? [V2: nat,Va2: list_nat] : Xb = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,V2),Va2)
                 => ( Y != bot_bot_assn ) ) ) ) ) ) ) ).

% list_assn.elims
tff(fact_167_list__assn_Oelims,axiom,
    ! [X: fun_b_fun_int_assn,Xa: list_b,Xb: list_int,Y: assn] :
      ( ( vEBT_L3601374263696529960_b_int(X,Xa,Xb) = Y )
     => ( ( ( Xa = nil_b )
         => ( ( Xb = nil_int )
           => ( Y != one_one_assn ) ) )
       => ( ! [A3: b,As: list_b] :
              ( ( Xa = aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,A3),As) )
             => ! [C: int,Cs: list_int] :
                  ( ( Xb = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,C),Cs) )
                 => ( Y != aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_int_assn(aa_b_fun_int_assn(X,A3),C)),vEBT_L3601374263696529960_b_int(X,As,Cs)) ) ) )
         => ( ( ? [V2: b,Va2: list_b] : Xa = aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,V2),Va2)
             => ( ( Xb = nil_int )
               => ( Y != bot_bot_assn ) ) )
           => ~ ( ( Xa = nil_b )
               => ( ? [V2: int,Va2: list_int] : Xb = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,V2),Va2)
                 => ( Y != bot_bot_assn ) ) ) ) ) ) ) ).

% list_assn.elims
tff(fact_168_list__assn_Oelims,axiom,
    ! [X: fun_nat_fun_a_assn,Xa: list_nat,Xb: list_a,Y: assn] :
      ( ( vEBT_L6400351906427472169_nat_a(X,Xa,Xb) = Y )
     => ( ( ( Xa = nil_nat )
         => ( ( Xb = nil_a )
           => ( Y != one_one_assn ) ) )
       => ( ! [A3: nat,As: list_nat] :
              ( ( Xa = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,A3),As) )
             => ! [C: a,Cs: list_a] :
                  ( ( Xb = aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,C),Cs) )
                 => ( Y != aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_a_assn(aa_nat_fun_a_assn(X,A3),C)),vEBT_L6400351906427472169_nat_a(X,As,Cs)) ) ) )
         => ( ( ? [V2: nat,Va2: list_nat] : Xa = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,V2),Va2)
             => ( ( Xb = nil_a )
               => ( Y != bot_bot_assn ) ) )
           => ~ ( ( Xa = nil_nat )
               => ( ? [V2: a,Va2: list_a] : Xb = aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,V2),Va2)
                 => ( Y != bot_bot_assn ) ) ) ) ) ) ) ).

% list_assn.elims
tff(fact_169_list__assn_Oelims,axiom,
    ! [X: fun_nat_fun_b_assn,Xa: list_nat,Xb: list_b,Y: assn] :
      ( ( vEBT_L6400351906427472170_nat_b(X,Xa,Xb) = Y )
     => ( ( ( Xa = nil_nat )
         => ( ( Xb = nil_b )
           => ( Y != one_one_assn ) ) )
       => ( ! [A3: nat,As: list_nat] :
              ( ( Xa = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,A3),As) )
             => ! [C: b,Cs: list_b] :
                  ( ( Xb = aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,C),Cs) )
                 => ( Y != aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_b_assn(aa_nat_fun_b_assn(X,A3),C)),vEBT_L6400351906427472170_nat_b(X,As,Cs)) ) ) )
         => ( ( ? [V2: nat,Va2: list_nat] : Xa = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,V2),Va2)
             => ( ( Xb = nil_b )
               => ( Y != bot_bot_assn ) ) )
           => ~ ( ( Xa = nil_nat )
               => ( ? [V2: b,Va2: list_b] : Xb = aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,V2),Va2)
                 => ( Y != bot_bot_assn ) ) ) ) ) ) ) ).

% list_assn.elims
tff(fact_170_list__assn_Oelims,axiom,
    ! [X: fun_nat_fun_nat_assn,Xa: list_nat,Xb: list_nat,Y: assn] :
      ( ( vEBT_L8301102511889123557at_nat(X,Xa,Xb) = Y )
     => ( ( ( Xa = nil_nat )
         => ( ( Xb = nil_nat )
           => ( Y != one_one_assn ) ) )
       => ( ! [A3: nat,As: list_nat] :
              ( ( Xa = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,A3),As) )
             => ! [C: nat,Cs: list_nat] :
                  ( ( Xb = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,C),Cs) )
                 => ( Y != aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(aa_nat_fun_nat_assn(X,A3),C)),vEBT_L8301102511889123557at_nat(X,As,Cs)) ) ) )
         => ( ( ? [V2: nat,Va2: list_nat] : Xa = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,V2),Va2)
             => ( ( Xb = nil_nat )
               => ( Y != bot_bot_assn ) ) )
           => ~ ( ( Xa = nil_nat )
               => ( ? [V2: nat,Va2: list_nat] : Xb = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,V2),Va2)
                 => ( Y != bot_bot_assn ) ) ) ) ) ) ) ).

% list_assn.elims
tff(fact_171_list__assn_Oelims,axiom,
    ! [X: fun_nat_fun_int_assn,Xa: list_nat,Xb: list_int,Y: assn] :
      ( ( vEBT_L8298612041380073281at_int(X,Xa,Xb) = Y )
     => ( ( ( Xa = nil_nat )
         => ( ( Xb = nil_int )
           => ( Y != one_one_assn ) ) )
       => ( ! [A3: nat,As: list_nat] :
              ( ( Xa = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,A3),As) )
             => ! [C: int,Cs: list_int] :
                  ( ( Xb = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,C),Cs) )
                 => ( Y != aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_int_assn(aa_nat_fun_int_assn(X,A3),C)),vEBT_L8298612041380073281at_int(X,As,Cs)) ) ) )
         => ( ( ? [V2: nat,Va2: list_nat] : Xa = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,V2),Va2)
             => ( ( Xb = nil_int )
               => ( Y != bot_bot_assn ) ) )
           => ~ ( ( Xa = nil_nat )
               => ( ? [V2: int,Va2: list_int] : Xb = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,V2),Va2)
                 => ( Y != bot_bot_assn ) ) ) ) ) ) ) ).

% list_assn.elims
tff(fact_172_list__assn_Oelims,axiom,
    ! [X: fun_int_fun_a_assn,Xa: list_int,Xb: list_a,Y: assn] :
      ( ( vEBT_L4155206938757026253_int_a(X,Xa,Xb) = Y )
     => ( ( ( Xa = nil_int )
         => ( ( Xb = nil_a )
           => ( Y != one_one_assn ) ) )
       => ( ! [A3: int,As: list_int] :
              ( ( Xa = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,A3),As) )
             => ! [C: a,Cs: list_a] :
                  ( ( Xb = aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,C),Cs) )
                 => ( Y != aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_a_assn(aa_int_fun_a_assn(X,A3),C)),vEBT_L4155206938757026253_int_a(X,As,Cs)) ) ) )
         => ( ( ? [V2: int,Va2: list_int] : Xa = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,V2),Va2)
             => ( ( Xb = nil_a )
               => ( Y != bot_bot_assn ) ) )
           => ~ ( ( Xa = nil_int )
               => ( ? [V2: a,Va2: list_a] : Xb = aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,V2),Va2)
                 => ( Y != bot_bot_assn ) ) ) ) ) ) ) ).

% list_assn.elims
tff(fact_173_list__assn_Oelims,axiom,
    ! [X: fun_int_fun_b_assn,Xa: list_int,Xb: list_b,Y: assn] :
      ( ( vEBT_L4155206938757026254_int_b(X,Xa,Xb) = Y )
     => ( ( ( Xa = nil_int )
         => ( ( Xb = nil_b )
           => ( Y != one_one_assn ) ) )
       => ( ! [A3: int,As: list_int] :
              ( ( Xa = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,A3),As) )
             => ! [C: b,Cs: list_b] :
                  ( ( Xb = aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,C),Cs) )
                 => ( Y != aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_b_assn(aa_int_fun_b_assn(X,A3),C)),vEBT_L4155206938757026254_int_b(X,As,Cs)) ) ) )
         => ( ( ? [V2: int,Va2: list_int] : Xa = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,V2),Va2)
             => ( ( Xb = nil_b )
               => ( Y != bot_bot_assn ) ) )
           => ~ ( ( Xa = nil_int )
               => ( ? [V2: b,Va2: list_b] : Xb = aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,V2),Va2)
                 => ( Y != bot_bot_assn ) ) ) ) ) ) ) ).

% list_assn.elims
tff(fact_174_list__assn_Oelims,axiom,
    ! [X: fun_int_fun_nat_assn,Xa: list_int,Xb: list_nat,Y: assn] :
      ( ( vEBT_L77084186935402305nt_nat(X,Xa,Xb) = Y )
     => ( ( ( Xa = nil_int )
         => ( ( Xb = nil_nat )
           => ( Y != one_one_assn ) ) )
       => ( ! [A3: int,As: list_int] :
              ( ( Xa = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,A3),As) )
             => ! [C: nat,Cs: list_nat] :
                  ( ( Xb = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,C),Cs) )
                 => ( Y != aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(aa_int_fun_nat_assn(X,A3),C)),vEBT_L77084186935402305nt_nat(X,As,Cs)) ) ) )
         => ( ( ? [V2: int,Va2: list_int] : Xa = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,V2),Va2)
             => ( ( Xb = nil_nat )
               => ( Y != bot_bot_assn ) ) )
           => ~ ( ( Xa = nil_int )
               => ( ? [V2: nat,Va2: list_nat] : Xb = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,V2),Va2)
                 => ( Y != bot_bot_assn ) ) ) ) ) ) ) ).

% list_assn.elims
tff(fact_175_mult__1,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,one_one_int),A) = A ).

% mult_1
tff(fact_176_mult__1,axiom,
    ! [A: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,one_one_assn),A) = A ).

% mult_1
tff(fact_177_mult__1,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,one_one_nat),A) = A ).

% mult_1
tff(fact_178_mult_Oright__neutral,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),one_one_int) = A ).

% mult.right_neutral
tff(fact_179_mult_Oright__neutral,axiom,
    ! [A: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),one_one_assn) = A ).

% mult.right_neutral
tff(fact_180_mult_Oright__neutral,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),one_one_nat) = A ).

% mult.right_neutral
tff(fact_181_merge__pure__star,axiom,
    ! [A: $o,B: $o] :
      aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_o_assn(pure_assn,(A))),aa_o_assn(pure_assn,(B))) = aa_o_assn(pure_assn,
        ( (A)
        & (B) )) ).

% merge_pure_star
tff(fact_182_star__false__right,axiom,
    ! [Pa: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Pa),bot_bot_assn) = bot_bot_assn ).

% star_false_right
tff(fact_183_star__false__left,axiom,
    ! [Pa: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,bot_bot_assn),Pa) = bot_bot_assn ).

% star_false_left
tff(fact_184_is__pure__assn__starI,axiom,
    ! [A: assn,B: assn] :
      ( aa_assn_o(is_pure_assn,A)
     => ( aa_assn_o(is_pure_assn,B)
       => aa_assn_o(is_pure_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),B)) ) ) ).

% is_pure_assn_starI
tff(fact_185_assn__times__comm,axiom,
    ! [Pa: assn,Q: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Pa),Q) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Q),Pa) ).

% assn_times_comm
tff(fact_186_assn__times__assoc,axiom,
    ! [Pa: assn,Q: assn,R: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Pa),Q)),R) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Pa),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Q),R)) ).

% assn_times_assoc
tff(fact_187_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B: assn,A: assn,C2: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),C2)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B),C2)) ).

% ab_semigroup_mult_class.mult.left_commute
tff(fact_188_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B: nat,A: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C2)) ).

% ab_semigroup_mult_class.mult.left_commute
tff(fact_189_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ! [A: assn,B: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),B) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B),A) ).

% ab_semigroup_mult_class.mult.commute
tff(fact_190_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A) ).

% ab_semigroup_mult_class.mult.commute
tff(fact_191_mult_Oassoc,axiom,
    ! [A: assn,B: assn,C2: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),B)),C2) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B),C2)) ).

% mult.assoc
tff(fact_192_mult_Oassoc,axiom,
    ! [A: nat,B: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C2)) ).

% mult.assoc
tff(fact_193_mult_Ocomm__neutral,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),one_one_int) = A ).

% mult.comm_neutral
tff(fact_194_mult_Ocomm__neutral,axiom,
    ! [A: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),one_one_assn) = A ).

% mult.comm_neutral
tff(fact_195_mult_Ocomm__neutral,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),one_one_nat) = A ).

% mult.comm_neutral
tff(fact_196_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,one_one_int),A) = A ).

% comm_monoid_mult_class.mult_1
tff(fact_197_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,one_one_assn),A) = A ).

% comm_monoid_mult_class.mult_1
tff(fact_198_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,one_one_nat),A) = A ).

% comm_monoid_mult_class.mult_1
tff(fact_199_assn__one__left,axiom,
    ! [Pa: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,one_one_assn),Pa) = Pa ).

% assn_one_left
tff(fact_200_list__assn_Osimps_I2_J,axiom,
    ! [Pa: fun_nat_fun_nat_assn,A: nat,As2: list_nat,C2: nat,Cs2: list_nat] : vEBT_L8301102511889123557at_nat(Pa,aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,A),As2),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,C2),Cs2)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(aa_nat_fun_nat_assn(Pa,A),C2)),vEBT_L8301102511889123557at_nat(Pa,As2,Cs2)) ).

% list_assn.simps(2)
tff(fact_201_list__assn_Osimps_I2_J,axiom,
    ! [Pa: fun_nat_fun_int_assn,A: nat,As2: list_nat,C2: int,Cs2: list_int] : vEBT_L8298612041380073281at_int(Pa,aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,A),As2),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,C2),Cs2)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_int_assn(aa_nat_fun_int_assn(Pa,A),C2)),vEBT_L8298612041380073281at_int(Pa,As2,Cs2)) ).

% list_assn.simps(2)
tff(fact_202_list__assn_Osimps_I2_J,axiom,
    ! [Pa: fun_int_fun_nat_assn,A: int,As2: list_int,C2: nat,Cs2: list_nat] : vEBT_L77084186935402305nt_nat(Pa,aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,A),As2),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,C2),Cs2)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(aa_int_fun_nat_assn(Pa,A),C2)),vEBT_L77084186935402305nt_nat(Pa,As2,Cs2)) ).

% list_assn.simps(2)
tff(fact_203_list__assn_Osimps_I2_J,axiom,
    ! [Pa: fun_int_fun_int_assn,A: int,As2: list_int,C2: int,Cs2: list_int] : vEBT_L74593716426352029nt_int(Pa,aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,A),As2),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,C2),Cs2)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_int_assn(aa_int_fun_int_assn(Pa,A),C2)),vEBT_L74593716426352029nt_int(Pa,As2,Cs2)) ).

% list_assn.simps(2)
tff(fact_204_list__assn_Osimps_I2_J,axiom,
    ! [Pa: fun_a_fun_b_assn,A: a,As2: list_a,C2: b,Cs2: list_b] : vEBT_L4319891404334229444sn_a_b(Pa,aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,A),As2),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,C2),Cs2)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_b_assn(aa_a_fun_b_assn(Pa,A),C2)),vEBT_L4319891404334229444sn_a_b(Pa,As2,Cs2)) ).

% list_assn.simps(2)
tff(fact_205_list__assn_Osimps_I2_J,axiom,
    ! [Pa: fun_b_fun_a_assn,A: b,As2: list_b,C2: a,Cs2: list_a] : vEBT_L1532435822361553410sn_b_a(Pa,aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,A),As2),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,C2),Cs2)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_a_assn(aa_b_fun_a_assn(Pa,A),C2)),vEBT_L1532435822361553410sn_b_a(Pa,As2,Cs2)) ).

% list_assn.simps(2)
tff(fact_206_list__assn_Osimps_I2_J,axiom,
    ! [Pa: fun_a_fun_nat_assn,A: a,As2: list_a,C2: nat,Cs2: list_nat] : vEBT_L2368420405249185739_a_nat(Pa,aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,A),As2),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,C2),Cs2)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(aa_a_fun_nat_assn(Pa,A),C2)),vEBT_L2368420405249185739_a_nat(Pa,As2,Cs2)) ).

% list_assn.simps(2)
tff(fact_207_list__assn_Osimps_I2_J,axiom,
    ! [Pa: fun_a_fun_int_assn,A: a,As2: list_a,C2: int,Cs2: list_int] : vEBT_L2365929934740135463_a_int(Pa,aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,A),As2),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,C2),Cs2)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_int_assn(aa_a_fun_int_assn(Pa,A),C2)),vEBT_L2365929934740135463_a_int(Pa,As2,Cs2)) ).

% list_assn.simps(2)
tff(fact_208_list__assn_Osimps_I2_J,axiom,
    ! [Pa: fun_a_fun_a_assn,A: a,As2: list_a,C2: a,Cs2: list_a] : vEBT_L4319891404334229443sn_a_a(Pa,aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,A),As2),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,C2),Cs2)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_a_assn(aa_a_fun_a_assn(Pa,A),C2)),vEBT_L4319891404334229443sn_a_a(Pa,As2,Cs2)) ).

% list_assn.simps(2)
tff(fact_209_list__assn_Osimps_I2_J,axiom,
    ! [Pa: fun_na2667955736958379125n_assn,A: nat,As2: list_nat,C2: produc6575502325842934193n_assn,Cs2: list_P8527749157015355191n_assn] : vEBT_L9083610150846024284n_assn(Pa,aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,A),As2),aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,C2),Cs2)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_Pro2979818503705714077n_assn(aa_nat3837439340002286302n_assn(Pa,A),C2)),vEBT_L9083610150846024284n_assn(Pa,As2,Cs2)) ).

% list_assn.simps(2)
tff(fact_210_one__reorient,axiom,
    ! [X: assn] :
      ( ( one_one_assn = X )
    <=> ( X = one_one_assn ) ) ).

% one_reorient
tff(fact_211_one__reorient,axiom,
    ! [X: nat] :
      ( ( one_one_nat = X )
    <=> ( X = one_one_nat ) ) ).

% one_reorient
tff(fact_212_one__reorient,axiom,
    ! [X: int] :
      ( ( one_one_int = X )
    <=> ( X = one_one_int ) ) ).

% one_reorient
tff(fact_213_list__tail__coinc,axiom,
    ! [N1: nat,R1: list_nat,N2: nat,R2: list_nat] :
      ( ( aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,N1),R1) = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,N2),R2) )
     => ( ( N1 = N2 )
        & ( R1 = R2 ) ) ) ).

% list_tail_coinc
tff(fact_214_list__tail__coinc,axiom,
    ! [N1: int,R1: list_int,N2: int,R2: list_int] :
      ( ( aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,N1),R1) = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,N2),R2) )
     => ( ( N1 = N2 )
        & ( R1 = R2 ) ) ) ).

% list_tail_coinc
tff(fact_215_list__tail__coinc,axiom,
    ! [N1: produc6575502325842934193n_assn,R1: list_P8527749157015355191n_assn,N2: produc6575502325842934193n_assn,R2: list_P8527749157015355191n_assn] :
      ( ( aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,N1),R1) = aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,N2),R2) )
     => ( ( N1 = N2 )
        & ( R1 = R2 ) ) ) ).

% list_tail_coinc
tff(fact_216_norm__assertion__simps_I1_J,axiom,
    ! [A: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,one_one_assn),A) = A ).

% norm_assertion_simps(1)
tff(fact_217_norm__assertion__simps_I2_J,axiom,
    ! [A: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),one_one_assn) = A ).

% norm_assertion_simps(2)
tff(fact_218_product__lists_Osimps_I1_J,axiom,
    product_lists_a(nil_list_a) = cons_list_a(nil_a,nil_list_a) ).

% product_lists.simps(1)
tff(fact_219_product__lists_Osimps_I1_J,axiom,
    product_lists_b(nil_list_b) = cons_list_b(nil_b,nil_list_b) ).

% product_lists.simps(1)
tff(fact_220_product__lists_Osimps_I1_J,axiom,
    produc1566369527784499744n_assn(nil_li5476096274760905021n_assn) = cons_l2423627976422276333n_assn(nil_Pr5671120429643327159n_assn,nil_li5476096274760905021n_assn) ).

% product_lists.simps(1)
tff(fact_221_product__lists_Osimps_I1_J,axiom,
    product_lists_nat(nil_list_nat) = cons_list_nat(nil_nat,nil_list_nat) ).

% product_lists.simps(1)
tff(fact_222_product__lists_Osimps_I1_J,axiom,
    product_lists_int(nil_list_int) = cons_list_int(nil_int,nil_list_int) ).

% product_lists.simps(1)
tff(fact_223_subseqs_Osimps_I1_J,axiom,
    subseqs_a(nil_a) = cons_list_a(nil_a,nil_list_a) ).

% subseqs.simps(1)
tff(fact_224_subseqs_Osimps_I1_J,axiom,
    subseqs_b(nil_b) = cons_list_b(nil_b,nil_list_b) ).

% subseqs.simps(1)
tff(fact_225_subseqs_Osimps_I1_J,axiom,
    subseq184808802919281286n_assn(nil_Pr5671120429643327159n_assn) = cons_l2423627976422276333n_assn(nil_Pr5671120429643327159n_assn,nil_li5476096274760905021n_assn) ).

% subseqs.simps(1)
tff(fact_226_subseqs_Osimps_I1_J,axiom,
    subseqs_nat(nil_nat) = cons_list_nat(nil_nat,nil_list_nat) ).

% subseqs.simps(1)
tff(fact_227_subseqs_Osimps_I1_J,axiom,
    subseqs_int(nil_int) = cons_list_int(nil_int,nil_list_int) ).

% subseqs.simps(1)
tff(fact_228_insert__Nil,axiom,
    ! [X: a] : insert_a(X,nil_a) = aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X),nil_a) ).

% insert_Nil
tff(fact_229_insert__Nil,axiom,
    ! [X: b] : insert_b(X,nil_b) = aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X),nil_b) ).

% insert_Nil
tff(fact_230_insert__Nil,axiom,
    ! [X: nat] : insert_nat(X,nil_nat) = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X),nil_nat) ).

% insert_Nil
tff(fact_231_insert__Nil,axiom,
    ! [X: int] : insert_int(X,nil_int) = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X),nil_int) ).

% insert_Nil
tff(fact_232_insert__Nil,axiom,
    ! [X: produc6575502325842934193n_assn] : insert3246601298802261197n_assn(X,nil_Pr5671120429643327159n_assn) = aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X),nil_Pr5671120429643327159n_assn) ).

% insert_Nil
tff(fact_233_revg_Oelims,axiom,
    ! [X: list_a,Xa: list_a,Y: list_a] :
      ( ( revg_a(X,Xa) = Y )
     => ( ( ( X = nil_a )
         => ( Y != Xa ) )
       => ~ ! [A3: a,As: list_a] :
              ( ( X = aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,A3),As) )
             => ( Y != revg_a(As,aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,A3),Xa)) ) ) ) ) ).

% revg.elims
tff(fact_234_revg_Oelims,axiom,
    ! [X: list_b,Xa: list_b,Y: list_b] :
      ( ( revg_b(X,Xa) = Y )
     => ( ( ( X = nil_b )
         => ( Y != Xa ) )
       => ~ ! [A3: b,As: list_b] :
              ( ( X = aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,A3),As) )
             => ( Y != revg_b(As,aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,A3),Xa)) ) ) ) ) ).

% revg.elims
tff(fact_235_revg_Oelims,axiom,
    ! [X: list_nat,Xa: list_nat,Y: list_nat] :
      ( ( revg_nat(X,Xa) = Y )
     => ( ( ( X = nil_nat )
         => ( Y != Xa ) )
       => ~ ! [A3: nat,As: list_nat] :
              ( ( X = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,A3),As) )
             => ( Y != revg_nat(As,aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,A3),Xa)) ) ) ) ) ).

% revg.elims
tff(fact_236_revg_Oelims,axiom,
    ! [X: list_int,Xa: list_int,Y: list_int] :
      ( ( revg_int(X,Xa) = Y )
     => ( ( ( X = nil_int )
         => ( Y != Xa ) )
       => ~ ! [A3: int,As: list_int] :
              ( ( X = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,A3),As) )
             => ( Y != revg_int(As,aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,A3),Xa)) ) ) ) ) ).

% revg.elims
tff(fact_237_revg_Oelims,axiom,
    ! [X: list_P8527749157015355191n_assn,Xa: list_P8527749157015355191n_assn,Y: list_P8527749157015355191n_assn] :
      ( ( revg_P8856960164974728692n_assn(X,Xa) = Y )
     => ( ( ( X = nil_Pr5671120429643327159n_assn )
         => ( Y != Xa ) )
       => ~ ! [A3: produc6575502325842934193n_assn,As: list_P8527749157015355191n_assn] :
              ( ( X = aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,A3),As) )
             => ( Y != revg_P8856960164974728692n_assn(As,aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,A3),Xa)) ) ) ) ) ).

% revg.elims
tff(fact_238_map__tailrec__rev_Oelims,axiom,
    ! [X: fun_a_nat,Xa: list_a,Xb: list_nat,Y: list_nat] :
      ( ( map_ta8710832428924958105_a_nat(X,Xa,Xb) = Y )
     => ( ( ( Xa = nil_a )
         => ( Y != Xb ) )
       => ~ ! [A3: a,As: list_a] :
              ( ( Xa = aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,A3),As) )
             => ( Y != map_ta8710832428924958105_a_nat(X,As,aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,aa_a_nat(X,A3)),Xb)) ) ) ) ) ).

% map_tailrec_rev.elims
tff(fact_239_map__tailrec__rev_Oelims,axiom,
    ! [X: fun_b_nat,Xa: list_b,Xb: list_nat,Y: list_nat] :
      ( ( map_ta722904721026576794_b_nat(X,Xa,Xb) = Y )
     => ( ( ( Xa = nil_b )
         => ( Y != Xb ) )
       => ~ ! [A3: b,As: list_b] :
              ( ( Xa = aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,A3),As) )
             => ( Y != map_ta722904721026576794_b_nat(X,As,aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,aa_b_nat(X,A3)),Xb)) ) ) ) ) ).

% map_tailrec_rev.elims
tff(fact_240_map__tailrec__rev_Oelims,axiom,
    ! [X: fun_a_int,Xa: list_a,Xb: list_int,Y: list_int] :
      ( ( map_ta8708341958415907829_a_int(X,Xa,Xb) = Y )
     => ( ( ( Xa = nil_a )
         => ( Y != Xb ) )
       => ~ ! [A3: a,As: list_a] :
              ( ( Xa = aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,A3),As) )
             => ( Y != map_ta8708341958415907829_a_int(X,As,aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,aa_a_int(X,A3)),Xb)) ) ) ) ) ).

% map_tailrec_rev.elims
tff(fact_241_map__tailrec__rev_Oelims,axiom,
    ! [X: fun_b_int,Xa: list_b,Xb: list_int,Y: list_int] :
      ( ( map_ta720414250517526518_b_int(X,Xa,Xb) = Y )
     => ( ( ( Xa = nil_b )
         => ( Y != Xb ) )
       => ~ ! [A3: b,As: list_b] :
              ( ( Xa = aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,A3),As) )
             => ( Y != map_ta720414250517526518_b_int(X,As,aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,aa_b_int(X,A3)),Xb)) ) ) ) ) ).

% map_tailrec_rev.elims
tff(fact_242_map__tailrec__rev_Oelims,axiom,
    ! [X: fun_nat_nat,Xa: list_nat,Xb: list_nat,Y: list_nat] :
      ( ( map_ta7164188454487880599at_nat(X,Xa,Xb) = Y )
     => ( ( ( Xa = nil_nat )
         => ( Y != Xb ) )
       => ~ ! [A3: nat,As: list_nat] :
              ( ( Xa = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,A3),As) )
             => ( Y != map_ta7164188454487880599at_nat(X,As,aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,aa_nat_nat(X,A3)),Xb)) ) ) ) ) ).

% map_tailrec_rev.elims
tff(fact_243_map__tailrec__rev_Oelims,axiom,
    ! [X: fun_nat_int,Xa: list_nat,Xb: list_int,Y: list_int] :
      ( ( map_ta7161697983978830323at_int(X,Xa,Xb) = Y )
     => ( ( ( Xa = nil_nat )
         => ( Y != Xb ) )
       => ~ ! [A3: nat,As: list_nat] :
              ( ( Xa = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,A3),As) )
             => ( Y != map_ta7161697983978830323at_int(X,As,aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,aa_nat_int(X,A3)),Xb)) ) ) ) ) ).

% map_tailrec_rev.elims
tff(fact_244_map__tailrec__rev_Oelims,axiom,
    ! [X: fun_int_nat,Xa: list_int,Xb: list_nat,Y: list_nat] :
      ( ( map_ta8163542166388935155nt_nat(X,Xa,Xb) = Y )
     => ( ( ( Xa = nil_int )
         => ( Y != Xb ) )
       => ~ ! [A3: int,As: list_int] :
              ( ( Xa = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,A3),As) )
             => ( Y != map_ta8163542166388935155nt_nat(X,As,aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,aa_int_nat(X,A3)),Xb)) ) ) ) ) ).

% map_tailrec_rev.elims
tff(fact_245_map__tailrec__rev_Oelims,axiom,
    ! [X: fun_int_int,Xa: list_int,Xb: list_int,Y: list_int] :
      ( ( map_ta8161051695879884879nt_int(X,Xa,Xb) = Y )
     => ( ( ( Xa = nil_int )
         => ( Y != Xb ) )
       => ~ ! [A3: int,As: list_int] :
              ( ( Xa = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,A3),As) )
             => ( Y != map_ta8161051695879884879nt_int(X,As,aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,aa_int_int(X,A3)),Xb)) ) ) ) ) ).

% map_tailrec_rev.elims
tff(fact_246_map__tailrec__rev_Oelims,axiom,
    ! [X: fun_a_7917510167652146422n_assn,Xa: list_a,Xb: list_P8527749157015355191n_assn,Y: list_P8527749157015355191n_assn] :
      ( ( map_ta4951362296667995304n_assn(X,Xa,Xb) = Y )
     => ( ( ( Xa = nil_a )
         => ( Y != Xb ) )
       => ~ ! [A3: a,As: list_a] :
              ( ( Xa = aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,A3),As) )
             => ( Y != map_ta4951362296667995304n_assn(X,As,aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,aa_a_P1113719347253493231n_assn(X,A3)),Xb)) ) ) ) ) ).

% map_tailrec_rev.elims
tff(fact_247_map__tailrec__rev_Oelims,axiom,
    ! [X: fun_b_4490291949330546613n_assn,Xa: list_b,Xb: list_P8527749157015355191n_assn,Y: list_P8527749157015355191n_assn] :
      ( ( map_ta1524144078346395495n_assn(X,Xa,Xb) = Y )
     => ( ( ( Xa = nil_b )
         => ( Y != Xb ) )
       => ~ ! [A3: b,As: list_b] :
              ( ( Xa = aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,A3),As) )
             => ( Y != map_ta1524144078346395495n_assn(X,As,aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,aa_b_P6909873165786669230n_assn(X,A3)),Xb)) ) ) ) ) ).

% map_tailrec_rev.elims
tff(fact_248_ord_Olexordp__eq__simps_I3_J,axiom,
    ! [Less: fun_a_fun_a_o,X: a,Xs: list_a] : ~ aa_list_a_o(aa_lis4963054610627903775st_a_o(lexordp_eq_a(Less),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X),Xs)),nil_a) ).

% ord.lexordp_eq_simps(3)
tff(fact_249_ord_Olexordp__eq__simps_I3_J,axiom,
    ! [Less: fun_b_fun_b_o,X: b,Xs: list_b] : ~ aa_list_b_o(aa_lis4437318310116005983st_b_o(lexordp_eq_b(Less),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X),Xs)),nil_b) ).

% ord.lexordp_eq_simps(3)
tff(fact_250_ord_Olexordp__eq__simps_I3_J,axiom,
    ! [Less: fun_nat_fun_nat_o,X: nat,Xs: list_nat] : ~ aa_list_nat_o(aa_lis557685189327716851_nat_o(lexordp_eq_nat(Less),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X),Xs)),nil_nat) ).

% ord.lexordp_eq_simps(3)
tff(fact_251_ord_Olexordp__eq__simps_I3_J,axiom,
    ! [Less: fun_int_fun_int_o,X: int,Xs: list_int] : ~ aa_list_int_o(aa_lis7773489400162621755_int_o(lexordp_eq_int(Less),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X),Xs)),nil_int) ).

% ord.lexordp_eq_simps(3)
tff(fact_252_ord_Olexordp__eq__simps_I3_J,axiom,
    ! [Less: fun_Pr5295903952624206638assn_o,X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] : ~ aa_lis1404944545740551261assn_o(lexord6224210647917505021n_assn(Less,aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X),Xs)),nil_Pr5671120429643327159n_assn) ).

% ord.lexordp_eq_simps(3)
tff(fact_253_ord_Olexordp__eq__simps_I4_J,axiom,
    ! [Less: fun_nat_fun_nat_o,X: nat,Xs: list_nat,Y: nat,Ys: list_nat] :
      ( aa_list_nat_o(aa_lis557685189327716851_nat_o(lexordp_eq_nat(Less),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X),Xs)),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Y),Ys))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(Less,X),Y)
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(Less,Y),X)
          & aa_list_nat_o(aa_lis557685189327716851_nat_o(lexordp_eq_nat(Less),Xs),Ys) ) ) ) ).

% ord.lexordp_eq_simps(4)
tff(fact_254_ord_Olexordp__eq__simps_I4_J,axiom,
    ! [Less: fun_int_fun_int_o,X: int,Xs: list_int,Y: int,Ys: list_int] :
      ( aa_list_int_o(aa_lis7773489400162621755_int_o(lexordp_eq_int(Less),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X),Xs)),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Y),Ys))
    <=> ( aa_int_o(aa_int_fun_int_o(Less,X),Y)
        | ( ~ aa_int_o(aa_int_fun_int_o(Less,Y),X)
          & aa_list_int_o(aa_lis7773489400162621755_int_o(lexordp_eq_int(Less),Xs),Ys) ) ) ) ).

% ord.lexordp_eq_simps(4)
tff(fact_255_ord_Olexordp__eq__simps_I4_J,axiom,
    ! [Less: fun_Pr5295903952624206638assn_o,X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn,Y: produc6575502325842934193n_assn,Ys: list_P8527749157015355191n_assn] :
      ( aa_lis1404944545740551261assn_o(lexord6224210647917505021n_assn(Less,aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X),Xs)),aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,Y),Ys))
    <=> ( aa_Pro2491721552408928739assn_o(aa_Pro332967637727714143assn_o(Less,X),Y)
        | ( ~ aa_Pro2491721552408928739assn_o(aa_Pro332967637727714143assn_o(Less,Y),X)
          & aa_lis1404944545740551261assn_o(lexord6224210647917505021n_assn(Less,Xs),Ys) ) ) ) ).

% ord.lexordp_eq_simps(4)
tff(fact_256_ord_Olexordp__eq__simps_I1_J,axiom,
    ! [Less: fun_a_fun_a_o,Ys: list_a] : aa_list_a_o(aa_lis4963054610627903775st_a_o(lexordp_eq_a(Less),nil_a),Ys) ).

% ord.lexordp_eq_simps(1)
tff(fact_257_ord_Olexordp__eq__simps_I1_J,axiom,
    ! [Less: fun_b_fun_b_o,Ys: list_b] : aa_list_b_o(aa_lis4437318310116005983st_b_o(lexordp_eq_b(Less),nil_b),Ys) ).

% ord.lexordp_eq_simps(1)
tff(fact_258_ord_Olexordp__eq__simps_I1_J,axiom,
    ! [Less: fun_Pr5295903952624206638assn_o,Ys: list_P8527749157015355191n_assn] : aa_lis1404944545740551261assn_o(lexord6224210647917505021n_assn(Less,nil_Pr5671120429643327159n_assn),Ys) ).

% ord.lexordp_eq_simps(1)
tff(fact_259_ord_Olexordp__eq__simps_I1_J,axiom,
    ! [Less: fun_nat_fun_nat_o,Ys: list_nat] : aa_list_nat_o(aa_lis557685189327716851_nat_o(lexordp_eq_nat(Less),nil_nat),Ys) ).

% ord.lexordp_eq_simps(1)
tff(fact_260_ord_Olexordp__eq__simps_I1_J,axiom,
    ! [Less: fun_int_fun_int_o,Ys: list_int] : aa_list_int_o(aa_lis7773489400162621755_int_o(lexordp_eq_int(Less),nil_int),Ys) ).

% ord.lexordp_eq_simps(1)
tff(fact_261_ord_Olexordp__eq__simps_I2_J,axiom,
    ! [Less: fun_a_fun_a_o,Xs: list_a] :
      ( aa_list_a_o(aa_lis4963054610627903775st_a_o(lexordp_eq_a(Less),Xs),nil_a)
    <=> ( Xs = nil_a ) ) ).

% ord.lexordp_eq_simps(2)
tff(fact_262_ord_Olexordp__eq__simps_I2_J,axiom,
    ! [Less: fun_b_fun_b_o,Xs: list_b] :
      ( aa_list_b_o(aa_lis4437318310116005983st_b_o(lexordp_eq_b(Less),Xs),nil_b)
    <=> ( Xs = nil_b ) ) ).

% ord.lexordp_eq_simps(2)
tff(fact_263_ord_Olexordp__eq__simps_I2_J,axiom,
    ! [Less: fun_Pr5295903952624206638assn_o,Xs: list_P8527749157015355191n_assn] :
      ( aa_lis1404944545740551261assn_o(lexord6224210647917505021n_assn(Less,Xs),nil_Pr5671120429643327159n_assn)
    <=> ( Xs = nil_Pr5671120429643327159n_assn ) ) ).

% ord.lexordp_eq_simps(2)
tff(fact_264_ord_Olexordp__eq__simps_I2_J,axiom,
    ! [Less: fun_nat_fun_nat_o,Xs: list_nat] :
      ( aa_list_nat_o(aa_lis557685189327716851_nat_o(lexordp_eq_nat(Less),Xs),nil_nat)
    <=> ( Xs = nil_nat ) ) ).

% ord.lexordp_eq_simps(2)
tff(fact_265_ord_Olexordp__eq__simps_I2_J,axiom,
    ! [Less: fun_int_fun_int_o,Xs: list_int] :
      ( aa_list_int_o(aa_lis7773489400162621755_int_o(lexordp_eq_int(Less),Xs),nil_int)
    <=> ( Xs = nil_int ) ) ).

% ord.lexordp_eq_simps(2)
tff(fact_266_list__collect__set__simps_I1_J,axiom,
    ! [F: fun_a_set_o] : list_collect_set_a_o(F,nil_a) = bot_bot_set_o ).

% list_collect_set_simps(1)
tff(fact_267_list__collect__set__simps_I1_J,axiom,
    ! [F: fun_b_set_o] : list_collect_set_b_o(F,nil_b) = bot_bot_set_o ).

% list_collect_set_simps(1)
tff(fact_268_list__collect__set__simps_I1_J,axiom,
    ! [F: fun_Pr4998725199732250698_set_o] : list_c5102633440097552245assn_o(F,nil_Pr5671120429643327159n_assn) = bot_bot_set_o ).

% list_collect_set_simps(1)
tff(fact_269_list__collect__set__simps_I1_J,axiom,
    ! [F: fun_nat_set_o] : list_c8047850539171819768_nat_o(F,nil_nat) = bot_bot_set_o ).

% list_collect_set_simps(1)
tff(fact_270_list__collect__set__simps_I1_J,axiom,
    ! [F: fun_int_set_o] : list_c6226808193739131804_int_o(F,nil_int) = bot_bot_set_o ).

% list_collect_set_simps(1)
tff(fact_271_list__collect__set__simps_I1_J,axiom,
    ! [F: fun_a_set_nat] : list_c5512459755930457856_a_nat(F,nil_a) = bot_bot_set_nat ).

% list_collect_set_simps(1)
tff(fact_272_list__collect__set__simps_I1_J,axiom,
    ! [F: fun_b_set_nat] : list_c6747904084886852353_b_nat(F,nil_b) = bot_bot_set_nat ).

% list_collect_set_simps(1)
tff(fact_273_list__collect__set__simps_I1_J,axiom,
    ! [F: fun_Pr5680576844133773770et_nat] : list_c6061723043370948915sn_nat(F,nil_Pr5671120429643327159n_assn) = bot_bot_set_nat ).

% list_collect_set_simps(1)
tff(fact_274_list__collect__set__simps_I1_J,axiom,
    ! [F: fun_nat_set_nat] : list_c2452340269597857392at_nat(F,nil_nat) = bot_bot_set_nat ).

% list_collect_set_simps(1)
tff(fact_275_list__collect__set__simps_I1_J,axiom,
    ! [F: fun_int_set_nat] : list_c3451693981498911948nt_nat(F,nil_int) = bot_bot_set_nat ).

% list_collect_set_simps(1)
tff(fact_276_ord_Olexordp__eq_OCons,axiom,
    ! [Less: fun_nat_fun_nat_o,X: nat,Y: nat,Xs: list_nat,Ys: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(Less,X),Y)
     => aa_list_nat_o(aa_lis557685189327716851_nat_o(lexordp_eq_nat(Less),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X),Xs)),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Y),Ys)) ) ).

% ord.lexordp_eq.Cons
tff(fact_277_ord_Olexordp__eq_OCons,axiom,
    ! [Less: fun_int_fun_int_o,X: int,Y: int,Xs: list_int,Ys: list_int] :
      ( aa_int_o(aa_int_fun_int_o(Less,X),Y)
     => aa_list_int_o(aa_lis7773489400162621755_int_o(lexordp_eq_int(Less),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X),Xs)),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Y),Ys)) ) ).

% ord.lexordp_eq.Cons
tff(fact_278_ord_Olexordp__eq_OCons,axiom,
    ! [Less: fun_Pr5295903952624206638assn_o,X: produc6575502325842934193n_assn,Y: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn,Ys: list_P8527749157015355191n_assn] :
      ( aa_Pro2491721552408928739assn_o(aa_Pro332967637727714143assn_o(Less,X),Y)
     => aa_lis1404944545740551261assn_o(lexord6224210647917505021n_assn(Less,aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X),Xs)),aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,Y),Ys)) ) ).

% ord.lexordp_eq.Cons
tff(fact_279_ord_Olexordp__eq_OCons__eq,axiom,
    ! [Less: fun_nat_fun_nat_o,X: nat,Y: nat,Xs: list_nat,Ys: list_nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(Less,X),Y)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(Less,Y),X)
       => ( aa_list_nat_o(aa_lis557685189327716851_nat_o(lexordp_eq_nat(Less),Xs),Ys)
         => aa_list_nat_o(aa_lis557685189327716851_nat_o(lexordp_eq_nat(Less),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X),Xs)),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Y),Ys)) ) ) ) ).

% ord.lexordp_eq.Cons_eq
tff(fact_280_ord_Olexordp__eq_OCons__eq,axiom,
    ! [Less: fun_int_fun_int_o,X: int,Y: int,Xs: list_int,Ys: list_int] :
      ( ~ aa_int_o(aa_int_fun_int_o(Less,X),Y)
     => ( ~ aa_int_o(aa_int_fun_int_o(Less,Y),X)
       => ( aa_list_int_o(aa_lis7773489400162621755_int_o(lexordp_eq_int(Less),Xs),Ys)
         => aa_list_int_o(aa_lis7773489400162621755_int_o(lexordp_eq_int(Less),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X),Xs)),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Y),Ys)) ) ) ) ).

% ord.lexordp_eq.Cons_eq
tff(fact_281_ord_Olexordp__eq_OCons__eq,axiom,
    ! [Less: fun_Pr5295903952624206638assn_o,X: produc6575502325842934193n_assn,Y: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn,Ys: list_P8527749157015355191n_assn] :
      ( ~ aa_Pro2491721552408928739assn_o(aa_Pro332967637727714143assn_o(Less,X),Y)
     => ( ~ aa_Pro2491721552408928739assn_o(aa_Pro332967637727714143assn_o(Less,Y),X)
       => ( aa_lis1404944545740551261assn_o(lexord6224210647917505021n_assn(Less,Xs),Ys)
         => aa_lis1404944545740551261assn_o(lexord6224210647917505021n_assn(Less,aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X),Xs)),aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,Y),Ys)) ) ) ) ).

% ord.lexordp_eq.Cons_eq
tff(fact_282_ord_Olexordp__eq_ONil,axiom,
    ! [Less: fun_a_fun_a_o,Ys: list_a] : aa_list_a_o(aa_lis4963054610627903775st_a_o(lexordp_eq_a(Less),nil_a),Ys) ).

% ord.lexordp_eq.Nil
tff(fact_283_ord_Olexordp__eq_ONil,axiom,
    ! [Less: fun_b_fun_b_o,Ys: list_b] : aa_list_b_o(aa_lis4437318310116005983st_b_o(lexordp_eq_b(Less),nil_b),Ys) ).

% ord.lexordp_eq.Nil
tff(fact_284_ord_Olexordp__eq_ONil,axiom,
    ! [Less: fun_Pr5295903952624206638assn_o,Ys: list_P8527749157015355191n_assn] : aa_lis1404944545740551261assn_o(lexord6224210647917505021n_assn(Less,nil_Pr5671120429643327159n_assn),Ys) ).

% ord.lexordp_eq.Nil
tff(fact_285_ord_Olexordp__eq_ONil,axiom,
    ! [Less: fun_nat_fun_nat_o,Ys: list_nat] : aa_list_nat_o(aa_lis557685189327716851_nat_o(lexordp_eq_nat(Less),nil_nat),Ys) ).

% ord.lexordp_eq.Nil
tff(fact_286_ord_Olexordp__eq_ONil,axiom,
    ! [Less: fun_int_fun_int_o,Ys: list_int] : aa_list_int_o(aa_lis7773489400162621755_int_o(lexordp_eq_int(Less),nil_int),Ys) ).

% ord.lexordp_eq.Nil
tff(fact_287_map__tailrec__rev_Osimps_I2_J,axiom,
    ! [F: fun_nat_nat,A: nat,As2: list_nat,Bs: list_nat] : map_ta7164188454487880599at_nat(F,aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,A),As2),Bs) = map_ta7164188454487880599at_nat(F,As2,aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,aa_nat_nat(F,A)),Bs)) ).

% map_tailrec_rev.simps(2)
tff(fact_288_map__tailrec__rev_Osimps_I2_J,axiom,
    ! [F: fun_nat_int,A: nat,As2: list_nat,Bs: list_int] : map_ta7161697983978830323at_int(F,aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,A),As2),Bs) = map_ta7161697983978830323at_int(F,As2,aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,aa_nat_int(F,A)),Bs)) ).

% map_tailrec_rev.simps(2)
tff(fact_289_map__tailrec__rev_Osimps_I2_J,axiom,
    ! [F: fun_na5138202758309126970n_assn,A: nat,As2: list_nat,Bs: list_P8527749157015355191n_assn] : map_ta8369952943552905514n_assn(F,aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,A),As2),Bs) = map_ta8369952943552905514n_assn(F,As2,aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,aa_nat3152888703114873891n_assn(F,A)),Bs)) ).

% map_tailrec_rev.simps(2)
tff(fact_290_map__tailrec__rev_Osimps_I2_J,axiom,
    ! [F: fun_int_nat,A: int,As2: list_int,Bs: list_nat] : map_ta8163542166388935155nt_nat(F,aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,A),As2),Bs) = map_ta8163542166388935155nt_nat(F,As2,aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,aa_int_nat(F,A)),Bs)) ).

% map_tailrec_rev.simps(2)
tff(fact_291_map__tailrec__rev_Osimps_I2_J,axiom,
    ! [F: fun_int_int,A: int,As2: list_int,Bs: list_int] : map_ta8161051695879884879nt_int(F,aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,A),As2),Bs) = map_ta8161051695879884879nt_int(F,As2,aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,aa_int_int(F,A)),Bs)) ).

% map_tailrec_rev.simps(2)
tff(fact_292_map__tailrec__rev_Osimps_I2_J,axiom,
    ! [F: fun_in6898239422062238174n_assn,A: int,As2: list_int,Bs: list_P8527749157015355191n_assn] : map_ta906617570451240910n_assn(F,aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,A),As2),Bs) = map_ta906617570451240910n_assn(F,As2,aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,aa_int4912925366867985095n_assn(F,A)),Bs)) ).

% map_tailrec_rev.simps(2)
tff(fact_293_map__tailrec__rev_Osimps_I2_J,axiom,
    ! [F: fun_Pr7403558292872094996sn_nat,A: produc6575502325842934193n_assn,As2: list_P8527749157015355191n_assn,Bs: list_nat] : map_ta4049178433593862988sn_nat(F,aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,A),As2),Bs) = map_ta4049178433593862988sn_nat(F,As2,aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,aa_Pro8055486230010607173sn_nat(F,A)),Bs)) ).

% map_tailrec_rev.simps(2)
tff(fact_294_map__tailrec__rev_Osimps_I2_J,axiom,
    ! [F: fun_Pr3225707273362898288sn_int,A: produc6575502325842934193n_assn,As2: list_P8527749157015355191n_assn,Bs: list_int] : map_ta4046687963084812712sn_int(F,aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,A),As2),Bs) = map_ta4046687963084812712sn_int(F,As2,aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,aa_Pro8052995759501556897sn_int(F,A)),Bs)) ).

% map_tailrec_rev.simps(2)
tff(fact_295_map__tailrec__rev_Osimps_I2_J,axiom,
    ! [F: fun_Pr5530265736160528387n_assn,A: produc6575502325842934193n_assn,As2: list_P8527749157015355191n_assn,Bs: list_P8527749157015355191n_assn] : map_ta6859916920478844725n_assn(F,aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,A),As2),Bs) = map_ta6859916920478844725n_assn(F,As2,aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,aa_Pro2851486235043456252n_assn(F,A)),Bs)) ).

% map_tailrec_rev.simps(2)
tff(fact_296_revg_Osimps_I2_J,axiom,
    ! [A: nat,As2: list_nat,B: list_nat] : revg_nat(aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,A),As2),B) = revg_nat(As2,aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,A),B)) ).

% revg.simps(2)
tff(fact_297_revg_Osimps_I2_J,axiom,
    ! [A: int,As2: list_int,B: list_int] : revg_int(aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,A),As2),B) = revg_int(As2,aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,A),B)) ).

% revg.simps(2)
tff(fact_298_revg_Osimps_I2_J,axiom,
    ! [A: produc6575502325842934193n_assn,As2: list_P8527749157015355191n_assn,B: list_P8527749157015355191n_assn] : revg_P8856960164974728692n_assn(aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,A),As2),B) = revg_P8856960164974728692n_assn(As2,aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,A),B)) ).

% revg.simps(2)
tff(fact_299_revg_Osimps_I1_J,axiom,
    ! [B: list_a] : revg_a(nil_a,B) = B ).

% revg.simps(1)
tff(fact_300_revg_Osimps_I1_J,axiom,
    ! [B: list_b] : revg_b(nil_b,B) = B ).

% revg.simps(1)
tff(fact_301_revg_Osimps_I1_J,axiom,
    ! [B: list_P8527749157015355191n_assn] : revg_P8856960164974728692n_assn(nil_Pr5671120429643327159n_assn,B) = B ).

% revg.simps(1)
tff(fact_302_revg_Osimps_I1_J,axiom,
    ! [B: list_nat] : revg_nat(nil_nat,B) = B ).

% revg.simps(1)
tff(fact_303_revg_Osimps_I1_J,axiom,
    ! [B: list_int] : revg_int(nil_int,B) = B ).

% revg.simps(1)
tff(fact_304_ord_Olexordp__eq_Ocases,axiom,
    ! [Less: fun_a_fun_a_o,A1: list_a,A22: list_a] :
      ( aa_list_a_o(aa_lis4963054610627903775st_a_o(lexordp_eq_a(Less),A1),A22)
     => ( ( A1 != nil_a )
       => ( ! [X2: a] :
              ( ? [Xs2: list_a] : A1 = aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X2),Xs2)
             => ! [Y2: a] :
                  ( ? [Ys2: list_a] : A22 = aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,Y2),Ys2)
                 => ~ aa_a_o(aa_a_fun_a_o(Less,X2),Y2) ) )
         => ~ ! [X2: a,Y2: a,Xs2: list_a] :
                ( ( A1 = aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X2),Xs2) )
               => ! [Ys2: list_a] :
                    ( ( A22 = aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,Y2),Ys2) )
                   => ( ~ aa_a_o(aa_a_fun_a_o(Less,X2),Y2)
                     => ( ~ aa_a_o(aa_a_fun_a_o(Less,Y2),X2)
                       => ~ aa_list_a_o(aa_lis4963054610627903775st_a_o(lexordp_eq_a(Less),Xs2),Ys2) ) ) ) ) ) ) ) ).

% ord.lexordp_eq.cases
tff(fact_305_ord_Olexordp__eq_Ocases,axiom,
    ! [Less: fun_b_fun_b_o,A1: list_b,A22: list_b] :
      ( aa_list_b_o(aa_lis4437318310116005983st_b_o(lexordp_eq_b(Less),A1),A22)
     => ( ( A1 != nil_b )
       => ( ! [X2: b] :
              ( ? [Xs2: list_b] : A1 = aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X2),Xs2)
             => ! [Y2: b] :
                  ( ? [Ys2: list_b] : A22 = aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,Y2),Ys2)
                 => ~ aa_b_o(aa_b_fun_b_o(Less,X2),Y2) ) )
         => ~ ! [X2: b,Y2: b,Xs2: list_b] :
                ( ( A1 = aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X2),Xs2) )
               => ! [Ys2: list_b] :
                    ( ( A22 = aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,Y2),Ys2) )
                   => ( ~ aa_b_o(aa_b_fun_b_o(Less,X2),Y2)
                     => ( ~ aa_b_o(aa_b_fun_b_o(Less,Y2),X2)
                       => ~ aa_list_b_o(aa_lis4437318310116005983st_b_o(lexordp_eq_b(Less),Xs2),Ys2) ) ) ) ) ) ) ) ).

% ord.lexordp_eq.cases
tff(fact_306_ord_Olexordp__eq_Ocases,axiom,
    ! [Less: fun_nat_fun_nat_o,A1: list_nat,A22: list_nat] :
      ( aa_list_nat_o(aa_lis557685189327716851_nat_o(lexordp_eq_nat(Less),A1),A22)
     => ( ( A1 != nil_nat )
       => ( ! [X2: nat] :
              ( ? [Xs2: list_nat] : A1 = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X2),Xs2)
             => ! [Y2: nat] :
                  ( ? [Ys2: list_nat] : A22 = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Y2),Ys2)
                 => ~ aa_nat_o(aa_nat_fun_nat_o(Less,X2),Y2) ) )
         => ~ ! [X2: nat,Y2: nat,Xs2: list_nat] :
                ( ( A1 = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X2),Xs2) )
               => ! [Ys2: list_nat] :
                    ( ( A22 = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Y2),Ys2) )
                   => ( ~ aa_nat_o(aa_nat_fun_nat_o(Less,X2),Y2)
                     => ( ~ aa_nat_o(aa_nat_fun_nat_o(Less,Y2),X2)
                       => ~ aa_list_nat_o(aa_lis557685189327716851_nat_o(lexordp_eq_nat(Less),Xs2),Ys2) ) ) ) ) ) ) ) ).

% ord.lexordp_eq.cases
tff(fact_307_ord_Olexordp__eq_Ocases,axiom,
    ! [Less: fun_int_fun_int_o,A1: list_int,A22: list_int] :
      ( aa_list_int_o(aa_lis7773489400162621755_int_o(lexordp_eq_int(Less),A1),A22)
     => ( ( A1 != nil_int )
       => ( ! [X2: int] :
              ( ? [Xs2: list_int] : A1 = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X2),Xs2)
             => ! [Y2: int] :
                  ( ? [Ys2: list_int] : A22 = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Y2),Ys2)
                 => ~ aa_int_o(aa_int_fun_int_o(Less,X2),Y2) ) )
         => ~ ! [X2: int,Y2: int,Xs2: list_int] :
                ( ( A1 = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X2),Xs2) )
               => ! [Ys2: list_int] :
                    ( ( A22 = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Y2),Ys2) )
                   => ( ~ aa_int_o(aa_int_fun_int_o(Less,X2),Y2)
                     => ( ~ aa_int_o(aa_int_fun_int_o(Less,Y2),X2)
                       => ~ aa_list_int_o(aa_lis7773489400162621755_int_o(lexordp_eq_int(Less),Xs2),Ys2) ) ) ) ) ) ) ) ).

% ord.lexordp_eq.cases
tff(fact_308_ord_Olexordp__eq_Ocases,axiom,
    ! [Less: fun_Pr5295903952624206638assn_o,A1: list_P8527749157015355191n_assn,A22: list_P8527749157015355191n_assn] :
      ( aa_lis1404944545740551261assn_o(lexord6224210647917505021n_assn(Less,A1),A22)
     => ( ( A1 != nil_Pr5671120429643327159n_assn )
       => ( ! [X2: produc6575502325842934193n_assn] :
              ( ? [Xs2: list_P8527749157015355191n_assn] : A1 = aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X2),Xs2)
             => ! [Y2: produc6575502325842934193n_assn] :
                  ( ? [Ys2: list_P8527749157015355191n_assn] : A22 = aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,Y2),Ys2)
                 => ~ aa_Pro2491721552408928739assn_o(aa_Pro332967637727714143assn_o(Less,X2),Y2) ) )
         => ~ ! [X2: produc6575502325842934193n_assn,Y2: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn] :
                ( ( A1 = aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X2),Xs2) )
               => ! [Ys2: list_P8527749157015355191n_assn] :
                    ( ( A22 = aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,Y2),Ys2) )
                   => ( ~ aa_Pro2491721552408928739assn_o(aa_Pro332967637727714143assn_o(Less,X2),Y2)
                     => ( ~ aa_Pro2491721552408928739assn_o(aa_Pro332967637727714143assn_o(Less,Y2),X2)
                       => ~ aa_lis1404944545740551261assn_o(lexord6224210647917505021n_assn(Less,Xs2),Ys2) ) ) ) ) ) ) ) ).

% ord.lexordp_eq.cases
tff(fact_309_ord_Olexordp__eq_Osimps,axiom,
    ! [Less: fun_a_fun_a_o,A1: list_a,A22: list_a] :
      ( aa_list_a_o(aa_lis4963054610627903775st_a_o(lexordp_eq_a(Less),A1),A22)
    <=> ( ? [Ys3: list_a] :
            ( ( A1 = nil_a )
            & ( A22 = Ys3 ) )
        | ? [X3: a,Y3: a,Xs3: list_a,Ys3: list_a] :
            ( ( A1 = aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X3),Xs3) )
            & ( A22 = aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,Y3),Ys3) )
            & aa_a_o(aa_a_fun_a_o(Less,X3),Y3) )
        | ? [X3: a,Y3: a,Xs3: list_a,Ys3: list_a] :
            ( ( A1 = aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X3),Xs3) )
            & ( A22 = aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,Y3),Ys3) )
            & ~ aa_a_o(aa_a_fun_a_o(Less,X3),Y3)
            & ~ aa_a_o(aa_a_fun_a_o(Less,Y3),X3)
            & aa_list_a_o(aa_lis4963054610627903775st_a_o(lexordp_eq_a(Less),Xs3),Ys3) ) ) ) ).

% ord.lexordp_eq.simps
tff(fact_310_ord_Olexordp__eq_Osimps,axiom,
    ! [Less: fun_b_fun_b_o,A1: list_b,A22: list_b] :
      ( aa_list_b_o(aa_lis4437318310116005983st_b_o(lexordp_eq_b(Less),A1),A22)
    <=> ( ? [Ys3: list_b] :
            ( ( A1 = nil_b )
            & ( A22 = Ys3 ) )
        | ? [X3: b,Y3: b,Xs3: list_b,Ys3: list_b] :
            ( ( A1 = aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X3),Xs3) )
            & ( A22 = aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,Y3),Ys3) )
            & aa_b_o(aa_b_fun_b_o(Less,X3),Y3) )
        | ? [X3: b,Y3: b,Xs3: list_b,Ys3: list_b] :
            ( ( A1 = aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X3),Xs3) )
            & ( A22 = aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,Y3),Ys3) )
            & ~ aa_b_o(aa_b_fun_b_o(Less,X3),Y3)
            & ~ aa_b_o(aa_b_fun_b_o(Less,Y3),X3)
            & aa_list_b_o(aa_lis4437318310116005983st_b_o(lexordp_eq_b(Less),Xs3),Ys3) ) ) ) ).

% ord.lexordp_eq.simps
tff(fact_311_ord_Olexordp__eq_Osimps,axiom,
    ! [Less: fun_nat_fun_nat_o,A1: list_nat,A22: list_nat] :
      ( aa_list_nat_o(aa_lis557685189327716851_nat_o(lexordp_eq_nat(Less),A1),A22)
    <=> ( ? [Ys3: list_nat] :
            ( ( A1 = nil_nat )
            & ( A22 = Ys3 ) )
        | ? [X3: nat,Y3: nat,Xs3: list_nat,Ys3: list_nat] :
            ( ( A1 = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X3),Xs3) )
            & ( A22 = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Y3),Ys3) )
            & aa_nat_o(aa_nat_fun_nat_o(Less,X3),Y3) )
        | ? [X3: nat,Y3: nat,Xs3: list_nat,Ys3: list_nat] :
            ( ( A1 = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X3),Xs3) )
            & ( A22 = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Y3),Ys3) )
            & ~ aa_nat_o(aa_nat_fun_nat_o(Less,X3),Y3)
            & ~ aa_nat_o(aa_nat_fun_nat_o(Less,Y3),X3)
            & aa_list_nat_o(aa_lis557685189327716851_nat_o(lexordp_eq_nat(Less),Xs3),Ys3) ) ) ) ).

% ord.lexordp_eq.simps
tff(fact_312_ord_Olexordp__eq_Osimps,axiom,
    ! [Less: fun_int_fun_int_o,A1: list_int,A22: list_int] :
      ( aa_list_int_o(aa_lis7773489400162621755_int_o(lexordp_eq_int(Less),A1),A22)
    <=> ( ? [Ys3: list_int] :
            ( ( A1 = nil_int )
            & ( A22 = Ys3 ) )
        | ? [X3: int,Y3: int,Xs3: list_int,Ys3: list_int] :
            ( ( A1 = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X3),Xs3) )
            & ( A22 = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Y3),Ys3) )
            & aa_int_o(aa_int_fun_int_o(Less,X3),Y3) )
        | ? [X3: int,Y3: int,Xs3: list_int,Ys3: list_int] :
            ( ( A1 = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X3),Xs3) )
            & ( A22 = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Y3),Ys3) )
            & ~ aa_int_o(aa_int_fun_int_o(Less,X3),Y3)
            & ~ aa_int_o(aa_int_fun_int_o(Less,Y3),X3)
            & aa_list_int_o(aa_lis7773489400162621755_int_o(lexordp_eq_int(Less),Xs3),Ys3) ) ) ) ).

% ord.lexordp_eq.simps
tff(fact_313_ord_Olexordp__eq_Osimps,axiom,
    ! [Less: fun_Pr5295903952624206638assn_o,A1: list_P8527749157015355191n_assn,A22: list_P8527749157015355191n_assn] :
      ( aa_lis1404944545740551261assn_o(lexord6224210647917505021n_assn(Less,A1),A22)
    <=> ( ? [Ys3: list_P8527749157015355191n_assn] :
            ( ( A1 = nil_Pr5671120429643327159n_assn )
            & ( A22 = Ys3 ) )
        | ? [X3: produc6575502325842934193n_assn,Y3: produc6575502325842934193n_assn,Xs3: list_P8527749157015355191n_assn,Ys3: list_P8527749157015355191n_assn] :
            ( ( A1 = aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X3),Xs3) )
            & ( A22 = aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,Y3),Ys3) )
            & aa_Pro2491721552408928739assn_o(aa_Pro332967637727714143assn_o(Less,X3),Y3) )
        | ? [X3: produc6575502325842934193n_assn,Y3: produc6575502325842934193n_assn,Xs3: list_P8527749157015355191n_assn,Ys3: list_P8527749157015355191n_assn] :
            ( ( A1 = aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X3),Xs3) )
            & ( A22 = aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,Y3),Ys3) )
            & ~ aa_Pro2491721552408928739assn_o(aa_Pro332967637727714143assn_o(Less,X3),Y3)
            & ~ aa_Pro2491721552408928739assn_o(aa_Pro332967637727714143assn_o(Less,Y3),X3)
            & aa_lis1404944545740551261assn_o(lexord6224210647917505021n_assn(Less,Xs3),Ys3) ) ) ) ).

% ord.lexordp_eq.simps
tff(fact_314_assn__aci_I10_J,axiom,
    ! [A: assn,B: assn,C2: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),B)),C2) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),C2)),B) ).

% assn_aci(10)
tff(fact_315_star__aci_I3_J,axiom,
    ! [A: assn,B: assn,C2: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B),C2)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),C2)) ).

% star_aci(3)
tff(fact_316_star__aci_I2_J,axiom,
    ! [A: assn,B: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),B) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B),A) ).

% star_aci(2)
tff(fact_317_star__assoc,axiom,
    ! [A: assn,B: assn,C2: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),B)),C2) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B),C2)) ).

% star_assoc
tff(fact_318_and__extract__pure__left__iff,axiom,
    ! [B: $o,Q: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_o_assn(pure_assn,(B))),Q) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,one_one_assn),Q)),aa_o_assn(pure_assn,(B))) ).

% and_extract_pure_left_iff
tff(fact_319_and__extract__pure__right__iff,axiom,
    ! [Pa: assn,B: $o] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Pa),aa_o_assn(pure_assn,(B))) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,one_one_assn),Pa)),aa_o_assn(pure_assn,(B))) ).

% and_extract_pure_right_iff
tff(fact_320_listrelp_Ocases,axiom,
    ! [R3: fun_a_fun_a_o,A1: list_a,A22: list_a] :
      ( aa_list_a_o(aa_lis4963054610627903775st_a_o(listrelp_a_a(R3),A1),A22)
     => ( ( ( A1 = nil_a )
         => ( A22 != nil_a ) )
       => ~ ! [X2: a,Y2: a,Xs2: list_a] :
              ( ( A1 = aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X2),Xs2) )
             => ! [Ys2: list_a] :
                  ( ( A22 = aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,Y2),Ys2) )
                 => ( aa_a_o(aa_a_fun_a_o(R3,X2),Y2)
                   => ~ aa_list_a_o(aa_lis4963054610627903775st_a_o(listrelp_a_a(R3),Xs2),Ys2) ) ) ) ) ) ).

% listrelp.cases
tff(fact_321_listrelp_Ocases,axiom,
    ! [R3: fun_a_fun_b_o,A1: list_a,A22: list_b] :
      ( aa_list_b_o(aa_lis2175599028655227742st_b_o(listrelp_a_b(R3),A1),A22)
     => ( ( ( A1 = nil_a )
         => ( A22 != nil_b ) )
       => ~ ! [X2: a,Y2: b,Xs2: list_a] :
              ( ( A1 = aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X2),Xs2) )
             => ! [Ys2: list_b] :
                  ( ( A22 = aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,Y2),Ys2) )
                 => ( aa_b_o(aa_a_fun_b_o(R3,X2),Y2)
                   => ~ aa_list_b_o(aa_lis2175599028655227742st_b_o(listrelp_a_b(R3),Xs2),Ys2) ) ) ) ) ) ).

% listrelp.cases
tff(fact_322_listrelp_Ocases,axiom,
    ! [R3: fun_b_fun_a_o,A1: list_b,A22: list_a] :
      ( aa_list_a_o(aa_lis7224773892088682016st_a_o(listrelp_b_a(R3),A1),A22)
     => ( ( ( A1 = nil_b )
         => ( A22 != nil_a ) )
       => ~ ! [X2: b,Y2: a,Xs2: list_b] :
              ( ( A1 = aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X2),Xs2) )
             => ! [Ys2: list_a] :
                  ( ( A22 = aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,Y2),Ys2) )
                 => ( aa_a_o(aa_b_fun_a_o(R3,X2),Y2)
                   => ~ aa_list_a_o(aa_lis7224773892088682016st_a_o(listrelp_b_a(R3),Xs2),Ys2) ) ) ) ) ) ).

% listrelp.cases
tff(fact_323_listrelp_Ocases,axiom,
    ! [R3: fun_b_fun_b_o,A1: list_b,A22: list_b] :
      ( aa_list_b_o(aa_lis4437318310116005983st_b_o(listrelp_b_b(R3),A1),A22)
     => ( ( ( A1 = nil_b )
         => ( A22 != nil_b ) )
       => ~ ! [X2: b,Y2: b,Xs2: list_b] :
              ( ( A1 = aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X2),Xs2) )
             => ! [Ys2: list_b] :
                  ( ( A22 = aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,Y2),Ys2) )
                 => ( aa_b_o(aa_b_fun_b_o(R3,X2),Y2)
                   => ~ aa_list_b_o(aa_lis4437318310116005983st_b_o(listrelp_b_b(R3),Xs2),Ys2) ) ) ) ) ) ).

% listrelp.cases
tff(fact_324_listrelp_Ocases,axiom,
    ! [R3: fun_a_fun_nat_o,A1: list_a,A22: list_nat] :
      ( aa_list_nat_o(aa_lis6662471382306767049_nat_o(listrelp_a_nat(R3),A1),A22)
     => ( ( ( A1 = nil_a )
         => ( A22 != nil_nat ) )
       => ~ ! [X2: a,Y2: nat,Xs2: list_a] :
              ( ( A1 = aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X2),Xs2) )
             => ! [Ys2: list_nat] :
                  ( ( A22 = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Y2),Ys2) )
                 => ( aa_nat_o(aa_a_fun_nat_o(R3,X2),Y2)
                   => ~ aa_list_nat_o(aa_lis6662471382306767049_nat_o(listrelp_a_nat(R3),Xs2),Ys2) ) ) ) ) ) ).

% listrelp.cases
tff(fact_325_listrelp_Ocases,axiom,
    ! [R3: fun_b_fun_nat_o,A1: list_b,A22: list_nat] :
      ( aa_list_nat_o(aa_lis3653689927639571848_nat_o(listrelp_b_nat(R3),A1),A22)
     => ( ( ( A1 = nil_b )
         => ( A22 != nil_nat ) )
       => ~ ! [X2: b,Y2: nat,Xs2: list_b] :
              ( ( A1 = aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X2),Xs2) )
             => ! [Ys2: list_nat] :
                  ( ( A22 = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Y2),Ys2) )
                 => ( aa_nat_o(aa_b_fun_nat_o(R3,X2),Y2)
                   => ~ aa_list_nat_o(aa_lis3653689927639571848_nat_o(listrelp_b_nat(R3),Xs2),Ys2) ) ) ) ) ) ).

% listrelp.cases
tff(fact_326_listrelp_Ocases,axiom,
    ! [R3: fun_a_fun_int_o,A1: list_a,A22: list_int] :
      ( aa_list_int_o(aa_lis4417326414636321133_int_o(listrelp_a_int(R3),A1),A22)
     => ( ( ( A1 = nil_a )
         => ( A22 != nil_int ) )
       => ~ ! [X2: a,Y2: int,Xs2: list_a] :
              ( ( A1 = aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X2),Xs2) )
             => ! [Ys2: list_int] :
                  ( ( A22 = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Y2),Ys2) )
                 => ( aa_int_o(aa_a_fun_int_o(R3,X2),Y2)
                   => ~ aa_list_int_o(aa_lis4417326414636321133_int_o(listrelp_a_int(R3),Xs2),Ys2) ) ) ) ) ) ).

% listrelp.cases
tff(fact_327_listrelp_Ocases,axiom,
    ! [R3: fun_b_fun_int_o,A1: list_b,A22: list_int] :
      ( aa_list_int_o(aa_lis1408544959969125932_int_o(listrelp_b_int(R3),A1),A22)
     => ( ( ( A1 = nil_b )
         => ( A22 != nil_int ) )
       => ~ ! [X2: b,Y2: int,Xs2: list_b] :
              ( ( A1 = aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X2),Xs2) )
             => ! [Ys2: list_int] :
                  ( ( A22 = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Y2),Ys2) )
                 => ( aa_int_o(aa_b_fun_int_o(R3,X2),Y2)
                   => ~ aa_list_int_o(aa_lis1408544959969125932_int_o(listrelp_b_int(R3),Xs2),Ys2) ) ) ) ) ) ).

% listrelp.cases
tff(fact_328_listrelp_Ocases,axiom,
    ! [R3: fun_nat_fun_a_o,A1: list_nat,A22: list_a] :
      ( aa_list_a_o(aa_lis6242202807773918837st_a_o(listrelp_nat_a(R3),A1),A22)
     => ( ( ( A1 = nil_nat )
         => ( A22 != nil_a ) )
       => ~ ! [X2: nat,Y2: a,Xs2: list_nat] :
              ( ( A1 = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X2),Xs2) )
             => ! [Ys2: list_a] :
                  ( ( A22 = aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,Y2),Ys2) )
                 => ( aa_a_o(aa_nat_fun_a_o(R3,X2),Y2)
                   => ~ aa_list_a_o(aa_lis6242202807773918837st_a_o(listrelp_nat_a(R3),Xs2),Ys2) ) ) ) ) ) ).

% listrelp.cases
tff(fact_329_listrelp_Ocases,axiom,
    ! [R3: fun_nat_fun_b_o,A1: list_nat,A22: list_b] :
      ( aa_list_b_o(aa_lis3454747225801242804st_b_o(listrelp_nat_b(R3),A1),A22)
     => ( ( ( A1 = nil_nat )
         => ( A22 != nil_b ) )
       => ~ ! [X2: nat,Y2: b,Xs2: list_nat] :
              ( ( A1 = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X2),Xs2) )
             => ! [Ys2: list_b] :
                  ( ( A22 = aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,Y2),Ys2) )
                 => ( aa_b_o(aa_nat_fun_b_o(R3,X2),Y2)
                   => ~ aa_list_b_o(aa_lis3454747225801242804st_b_o(listrelp_nat_b(R3),Xs2),Ys2) ) ) ) ) ) ).

% listrelp.cases
tff(fact_330_listrelp_Osimps,axiom,
    ! [R3: fun_a_fun_a_o,A1: list_a,A22: list_a] :
      ( aa_list_a_o(aa_lis4963054610627903775st_a_o(listrelp_a_a(R3),A1),A22)
    <=> ( ( ( A1 = nil_a )
          & ( A22 = nil_a ) )
        | ? [X3: a,Y3: a,Xs3: list_a,Ys3: list_a] :
            ( ( A1 = aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X3),Xs3) )
            & ( A22 = aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,Y3),Ys3) )
            & aa_a_o(aa_a_fun_a_o(R3,X3),Y3)
            & aa_list_a_o(aa_lis4963054610627903775st_a_o(listrelp_a_a(R3),Xs3),Ys3) ) ) ) ).

% listrelp.simps
tff(fact_331_listrelp_Osimps,axiom,
    ! [R3: fun_a_fun_b_o,A1: list_a,A22: list_b] :
      ( aa_list_b_o(aa_lis2175599028655227742st_b_o(listrelp_a_b(R3),A1),A22)
    <=> ( ( ( A1 = nil_a )
          & ( A22 = nil_b ) )
        | ? [X3: a,Y3: b,Xs3: list_a,Ys3: list_b] :
            ( ( A1 = aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X3),Xs3) )
            & ( A22 = aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,Y3),Ys3) )
            & aa_b_o(aa_a_fun_b_o(R3,X3),Y3)
            & aa_list_b_o(aa_lis2175599028655227742st_b_o(listrelp_a_b(R3),Xs3),Ys3) ) ) ) ).

% listrelp.simps
tff(fact_332_listrelp_Osimps,axiom,
    ! [R3: fun_b_fun_a_o,A1: list_b,A22: list_a] :
      ( aa_list_a_o(aa_lis7224773892088682016st_a_o(listrelp_b_a(R3),A1),A22)
    <=> ( ( ( A1 = nil_b )
          & ( A22 = nil_a ) )
        | ? [X3: b,Y3: a,Xs3: list_b,Ys3: list_a] :
            ( ( A1 = aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X3),Xs3) )
            & ( A22 = aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,Y3),Ys3) )
            & aa_a_o(aa_b_fun_a_o(R3,X3),Y3)
            & aa_list_a_o(aa_lis7224773892088682016st_a_o(listrelp_b_a(R3),Xs3),Ys3) ) ) ) ).

% listrelp.simps
tff(fact_333_listrelp_Osimps,axiom,
    ! [R3: fun_b_fun_b_o,A1: list_b,A22: list_b] :
      ( aa_list_b_o(aa_lis4437318310116005983st_b_o(listrelp_b_b(R3),A1),A22)
    <=> ( ( ( A1 = nil_b )
          & ( A22 = nil_b ) )
        | ? [X3: b,Y3: b,Xs3: list_b,Ys3: list_b] :
            ( ( A1 = aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X3),Xs3) )
            & ( A22 = aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,Y3),Ys3) )
            & aa_b_o(aa_b_fun_b_o(R3,X3),Y3)
            & aa_list_b_o(aa_lis4437318310116005983st_b_o(listrelp_b_b(R3),Xs3),Ys3) ) ) ) ).

% listrelp.simps
tff(fact_334_listrelp_Osimps,axiom,
    ! [R3: fun_a_fun_nat_o,A1: list_a,A22: list_nat] :
      ( aa_list_nat_o(aa_lis6662471382306767049_nat_o(listrelp_a_nat(R3),A1),A22)
    <=> ( ( ( A1 = nil_a )
          & ( A22 = nil_nat ) )
        | ? [X3: a,Y3: nat,Xs3: list_a,Ys3: list_nat] :
            ( ( A1 = aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X3),Xs3) )
            & ( A22 = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Y3),Ys3) )
            & aa_nat_o(aa_a_fun_nat_o(R3,X3),Y3)
            & aa_list_nat_o(aa_lis6662471382306767049_nat_o(listrelp_a_nat(R3),Xs3),Ys3) ) ) ) ).

% listrelp.simps
tff(fact_335_listrelp_Osimps,axiom,
    ! [R3: fun_b_fun_nat_o,A1: list_b,A22: list_nat] :
      ( aa_list_nat_o(aa_lis3653689927639571848_nat_o(listrelp_b_nat(R3),A1),A22)
    <=> ( ( ( A1 = nil_b )
          & ( A22 = nil_nat ) )
        | ? [X3: b,Y3: nat,Xs3: list_b,Ys3: list_nat] :
            ( ( A1 = aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X3),Xs3) )
            & ( A22 = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Y3),Ys3) )
            & aa_nat_o(aa_b_fun_nat_o(R3,X3),Y3)
            & aa_list_nat_o(aa_lis3653689927639571848_nat_o(listrelp_b_nat(R3),Xs3),Ys3) ) ) ) ).

% listrelp.simps
tff(fact_336_listrelp_Osimps,axiom,
    ! [R3: fun_a_fun_int_o,A1: list_a,A22: list_int] :
      ( aa_list_int_o(aa_lis4417326414636321133_int_o(listrelp_a_int(R3),A1),A22)
    <=> ( ( ( A1 = nil_a )
          & ( A22 = nil_int ) )
        | ? [X3: a,Y3: int,Xs3: list_a,Ys3: list_int] :
            ( ( A1 = aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X3),Xs3) )
            & ( A22 = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Y3),Ys3) )
            & aa_int_o(aa_a_fun_int_o(R3,X3),Y3)
            & aa_list_int_o(aa_lis4417326414636321133_int_o(listrelp_a_int(R3),Xs3),Ys3) ) ) ) ).

% listrelp.simps
tff(fact_337_listrelp_Osimps,axiom,
    ! [R3: fun_b_fun_int_o,A1: list_b,A22: list_int] :
      ( aa_list_int_o(aa_lis1408544959969125932_int_o(listrelp_b_int(R3),A1),A22)
    <=> ( ( ( A1 = nil_b )
          & ( A22 = nil_int ) )
        | ? [X3: b,Y3: int,Xs3: list_b,Ys3: list_int] :
            ( ( A1 = aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X3),Xs3) )
            & ( A22 = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Y3),Ys3) )
            & aa_int_o(aa_b_fun_int_o(R3,X3),Y3)
            & aa_list_int_o(aa_lis1408544959969125932_int_o(listrelp_b_int(R3),Xs3),Ys3) ) ) ) ).

% listrelp.simps
tff(fact_338_listrelp_Osimps,axiom,
    ! [R3: fun_nat_fun_a_o,A1: list_nat,A22: list_a] :
      ( aa_list_a_o(aa_lis6242202807773918837st_a_o(listrelp_nat_a(R3),A1),A22)
    <=> ( ( ( A1 = nil_nat )
          & ( A22 = nil_a ) )
        | ? [X3: nat,Y3: a,Xs3: list_nat,Ys3: list_a] :
            ( ( A1 = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X3),Xs3) )
            & ( A22 = aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,Y3),Ys3) )
            & aa_a_o(aa_nat_fun_a_o(R3,X3),Y3)
            & aa_list_a_o(aa_lis6242202807773918837st_a_o(listrelp_nat_a(R3),Xs3),Ys3) ) ) ) ).

% listrelp.simps
tff(fact_339_listrelp_Osimps,axiom,
    ! [R3: fun_nat_fun_b_o,A1: list_nat,A22: list_b] :
      ( aa_list_b_o(aa_lis3454747225801242804st_b_o(listrelp_nat_b(R3),A1),A22)
    <=> ( ( ( A1 = nil_nat )
          & ( A22 = nil_b ) )
        | ? [X3: nat,Y3: b,Xs3: list_nat,Ys3: list_b] :
            ( ( A1 = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X3),Xs3) )
            & ( A22 = aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,Y3),Ys3) )
            & aa_b_o(aa_nat_fun_b_o(R3,X3),Y3)
            & aa_list_b_o(aa_lis3454747225801242804st_b_o(listrelp_nat_b(R3),Xs3),Ys3) ) ) ) ).

% listrelp.simps
tff(fact_340_mergesort__by__rel__simps_I2_J,axiom,
    ! [R: fun_a_fun_a_o,X: a] : mergesort_by_rel_a(R,aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X),nil_a)) = aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X),nil_a) ).

% mergesort_by_rel_simps(2)
tff(fact_341_mergesort__by__rel__simps_I2_J,axiom,
    ! [R: fun_b_fun_b_o,X: b] : mergesort_by_rel_b(R,aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X),nil_b)) = aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X),nil_b) ).

% mergesort_by_rel_simps(2)
tff(fact_342_mergesort__by__rel__simps_I2_J,axiom,
    ! [R: fun_nat_fun_nat_o,X: nat] : aa_list_nat_list_nat(mergesort_by_rel_nat(R),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X),nil_nat)) = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X),nil_nat) ).

% mergesort_by_rel_simps(2)
tff(fact_343_mergesort__by__rel__simps_I2_J,axiom,
    ! [R: fun_int_fun_int_o,X: int] : aa_list_int_list_int(mergesort_by_rel_int(R),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X),nil_int)) = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X),nil_int) ).

% mergesort_by_rel_simps(2)
tff(fact_344_mergesort__by__rel__simps_I2_J,axiom,
    ! [R: fun_Pr5295903952624206638assn_o,X: produc6575502325842934193n_assn] : merges5609009838848816300n_assn(R,aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X),nil_Pr5671120429643327159n_assn)) = aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X),nil_Pr5671120429643327159n_assn) ).

% mergesort_by_rel_simps(2)
tff(fact_345_lexordp__eq__simps_I3_J,axiom,
    ! [X: nat,Xs: list_nat] : ~ aa_list_nat_o(aa_lis557685189327716851_nat_o(ord_lexordp_eq_nat,aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X),Xs)),nil_nat) ).

% lexordp_eq_simps(3)
tff(fact_346_lexordp__eq__simps_I3_J,axiom,
    ! [X: int,Xs: list_int] : ~ aa_list_int_o(aa_lis7773489400162621755_int_o(ord_lexordp_eq_int,aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X),Xs)),nil_int) ).

% lexordp_eq_simps(3)
tff(fact_347_entails__solve__finalize_I2_J,axiom,
    ! [M: list_P8527749157015355191n_assn] : aa_assn_o(fI_RESULT(M,one_one_assn,one_one_assn),one_one_assn) ).

% entails_solve_finalize(2)
tff(fact_348_frame__inference__finalize,axiom,
    ! [M: list_P8527749157015355191n_assn,F2: assn] : aa_assn_o(fI_RESULT(M,F2,one_one_assn),F2) ).

% frame_inference_finalize
tff(fact_349_mult_Oright__assoc,axiom,
    ! [A: assn,B: assn,C2: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),B)),C2) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B),C2)) ).

% mult.right_assoc
tff(fact_350_mult_Oright__assoc,axiom,
    ! [A: nat,B: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C2)) ).

% mult.right_assoc
tff(fact_351_mult_Oright__commute,axiom,
    ! [A: assn,B: assn,C2: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),B)),C2) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),C2)),B) ).

% mult.right_commute
tff(fact_352_mult_Oright__commute,axiom,
    ! [A: nat,B: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)),B) ).

% mult.right_commute
tff(fact_353_list_Omap__disc__iff,axiom,
    ! [F: fun_a_a,A: list_a] :
      ( ( map_a_a(F,A) = nil_a )
    <=> ( A = nil_a ) ) ).

% list.map_disc_iff
tff(fact_354_list_Omap__disc__iff,axiom,
    ! [F: fun_b_a,A: list_b] :
      ( ( map_b_a(F,A) = nil_a )
    <=> ( A = nil_b ) ) ).

% list.map_disc_iff
tff(fact_355_list_Omap__disc__iff,axiom,
    ! [F: fun_nat_a,A: list_nat] :
      ( ( map_nat_a(F,A) = nil_a )
    <=> ( A = nil_nat ) ) ).

% list.map_disc_iff
tff(fact_356_list_Omap__disc__iff,axiom,
    ! [F: fun_int_a,A: list_int] :
      ( ( map_int_a(F,A) = nil_a )
    <=> ( A = nil_int ) ) ).

% list.map_disc_iff
tff(fact_357_list_Omap__disc__iff,axiom,
    ! [F: fun_a_b,A: list_a] :
      ( ( map_a_b(F,A) = nil_b )
    <=> ( A = nil_a ) ) ).

% list.map_disc_iff
tff(fact_358_list_Omap__disc__iff,axiom,
    ! [F: fun_b_b,A: list_b] :
      ( ( map_b_b(F,A) = nil_b )
    <=> ( A = nil_b ) ) ).

% list.map_disc_iff
tff(fact_359_list_Omap__disc__iff,axiom,
    ! [F: fun_nat_b,A: list_nat] :
      ( ( map_nat_b(F,A) = nil_b )
    <=> ( A = nil_nat ) ) ).

% list.map_disc_iff
tff(fact_360_list_Omap__disc__iff,axiom,
    ! [F: fun_int_b,A: list_int] :
      ( ( map_int_b(F,A) = nil_b )
    <=> ( A = nil_int ) ) ).

% list.map_disc_iff
tff(fact_361_list_Omap__disc__iff,axiom,
    ! [F: fun_a_nat,A: list_a] :
      ( ( map_a_nat(F,A) = nil_nat )
    <=> ( A = nil_a ) ) ).

% list.map_disc_iff
tff(fact_362_list_Omap__disc__iff,axiom,
    ! [F: fun_b_nat,A: list_b] :
      ( ( map_b_nat(F,A) = nil_nat )
    <=> ( A = nil_b ) ) ).

% list.map_disc_iff
tff(fact_363_Nil__is__map__conv,axiom,
    ! [F: fun_a_a,Xs: list_a] :
      ( ( nil_a = map_a_a(F,Xs) )
    <=> ( Xs = nil_a ) ) ).

% Nil_is_map_conv
tff(fact_364_Nil__is__map__conv,axiom,
    ! [F: fun_b_a,Xs: list_b] :
      ( ( nil_a = map_b_a(F,Xs) )
    <=> ( Xs = nil_b ) ) ).

% Nil_is_map_conv
tff(fact_365_Nil__is__map__conv,axiom,
    ! [F: fun_nat_a,Xs: list_nat] :
      ( ( nil_a = map_nat_a(F,Xs) )
    <=> ( Xs = nil_nat ) ) ).

% Nil_is_map_conv
tff(fact_366_Nil__is__map__conv,axiom,
    ! [F: fun_int_a,Xs: list_int] :
      ( ( nil_a = map_int_a(F,Xs) )
    <=> ( Xs = nil_int ) ) ).

% Nil_is_map_conv
tff(fact_367_Nil__is__map__conv,axiom,
    ! [F: fun_a_b,Xs: list_a] :
      ( ( nil_b = map_a_b(F,Xs) )
    <=> ( Xs = nil_a ) ) ).

% Nil_is_map_conv
tff(fact_368_Nil__is__map__conv,axiom,
    ! [F: fun_b_b,Xs: list_b] :
      ( ( nil_b = map_b_b(F,Xs) )
    <=> ( Xs = nil_b ) ) ).

% Nil_is_map_conv
tff(fact_369_Nil__is__map__conv,axiom,
    ! [F: fun_nat_b,Xs: list_nat] :
      ( ( nil_b = map_nat_b(F,Xs) )
    <=> ( Xs = nil_nat ) ) ).

% Nil_is_map_conv
tff(fact_370_Nil__is__map__conv,axiom,
    ! [F: fun_int_b,Xs: list_int] :
      ( ( nil_b = map_int_b(F,Xs) )
    <=> ( Xs = nil_int ) ) ).

% Nil_is_map_conv
tff(fact_371_Nil__is__map__conv,axiom,
    ! [F: fun_a_nat,Xs: list_a] :
      ( ( nil_nat = map_a_nat(F,Xs) )
    <=> ( Xs = nil_a ) ) ).

% Nil_is_map_conv
tff(fact_372_Nil__is__map__conv,axiom,
    ! [F: fun_b_nat,Xs: list_b] :
      ( ( nil_nat = map_b_nat(F,Xs) )
    <=> ( Xs = nil_b ) ) ).

% Nil_is_map_conv
tff(fact_373_map__is__Nil__conv,axiom,
    ! [F: fun_a_a,Xs: list_a] :
      ( ( map_a_a(F,Xs) = nil_a )
    <=> ( Xs = nil_a ) ) ).

% map_is_Nil_conv
tff(fact_374_map__is__Nil__conv,axiom,
    ! [F: fun_b_a,Xs: list_b] :
      ( ( map_b_a(F,Xs) = nil_a )
    <=> ( Xs = nil_b ) ) ).

% map_is_Nil_conv
tff(fact_375_map__is__Nil__conv,axiom,
    ! [F: fun_nat_a,Xs: list_nat] :
      ( ( map_nat_a(F,Xs) = nil_a )
    <=> ( Xs = nil_nat ) ) ).

% map_is_Nil_conv
tff(fact_376_map__is__Nil__conv,axiom,
    ! [F: fun_int_a,Xs: list_int] :
      ( ( map_int_a(F,Xs) = nil_a )
    <=> ( Xs = nil_int ) ) ).

% map_is_Nil_conv
tff(fact_377_map__is__Nil__conv,axiom,
    ! [F: fun_a_b,Xs: list_a] :
      ( ( map_a_b(F,Xs) = nil_b )
    <=> ( Xs = nil_a ) ) ).

% map_is_Nil_conv
tff(fact_378_map__is__Nil__conv,axiom,
    ! [F: fun_b_b,Xs: list_b] :
      ( ( map_b_b(F,Xs) = nil_b )
    <=> ( Xs = nil_b ) ) ).

% map_is_Nil_conv
tff(fact_379_map__is__Nil__conv,axiom,
    ! [F: fun_nat_b,Xs: list_nat] :
      ( ( map_nat_b(F,Xs) = nil_b )
    <=> ( Xs = nil_nat ) ) ).

% map_is_Nil_conv
tff(fact_380_map__is__Nil__conv,axiom,
    ! [F: fun_int_b,Xs: list_int] :
      ( ( map_int_b(F,Xs) = nil_b )
    <=> ( Xs = nil_int ) ) ).

% map_is_Nil_conv
tff(fact_381_map__is__Nil__conv,axiom,
    ! [F: fun_a_nat,Xs: list_a] :
      ( ( map_a_nat(F,Xs) = nil_nat )
    <=> ( Xs = nil_a ) ) ).

% map_is_Nil_conv
tff(fact_382_map__is__Nil__conv,axiom,
    ! [F: fun_b_nat,Xs: list_b] :
      ( ( map_b_nat(F,Xs) = nil_nat )
    <=> ( Xs = nil_b ) ) ).

% map_is_Nil_conv
tff(fact_383_lexordp__eq__simps_I2_J,axiom,
    ! [Xs: list_nat] :
      ( aa_list_nat_o(aa_lis557685189327716851_nat_o(ord_lexordp_eq_nat,Xs),nil_nat)
    <=> ( Xs = nil_nat ) ) ).

% lexordp_eq_simps(2)
tff(fact_384_lexordp__eq__simps_I2_J,axiom,
    ! [Xs: list_int] :
      ( aa_list_int_o(aa_lis7773489400162621755_int_o(ord_lexordp_eq_int,Xs),nil_int)
    <=> ( Xs = nil_int ) ) ).

% lexordp_eq_simps(2)
tff(fact_385_lexordp__eq__simps_I1_J,axiom,
    ! [Ys: list_nat] : aa_list_nat_o(aa_lis557685189327716851_nat_o(ord_lexordp_eq_nat,nil_nat),Ys) ).

% lexordp_eq_simps(1)
tff(fact_386_lexordp__eq__simps_I1_J,axiom,
    ! [Ys: list_int] : aa_list_int_o(aa_lis7773489400162621755_int_o(ord_lexordp_eq_int,nil_int),Ys) ).

% lexordp_eq_simps(1)
tff(fact_387_mergesort__by__rel__simps_I1_J,axiom,
    ! [R: fun_a_fun_a_o] : mergesort_by_rel_a(R,nil_a) = nil_a ).

% mergesort_by_rel_simps(1)
tff(fact_388_mergesort__by__rel__simps_I1_J,axiom,
    ! [R: fun_b_fun_b_o] : mergesort_by_rel_b(R,nil_b) = nil_b ).

% mergesort_by_rel_simps(1)
tff(fact_389_mergesort__by__rel__simps_I1_J,axiom,
    ! [R: fun_Pr5295903952624206638assn_o] : merges5609009838848816300n_assn(R,nil_Pr5671120429643327159n_assn) = nil_Pr5671120429643327159n_assn ).

% mergesort_by_rel_simps(1)
tff(fact_390_mergesort__by__rel__simps_I1_J,axiom,
    ! [R: fun_nat_fun_nat_o] : aa_list_nat_list_nat(mergesort_by_rel_nat(R),nil_nat) = nil_nat ).

% mergesort_by_rel_simps(1)
tff(fact_391_mergesort__by__rel__simps_I1_J,axiom,
    ! [R: fun_int_fun_int_o] : aa_list_int_list_int(mergesort_by_rel_int(R),nil_int) = nil_int ).

% mergesort_by_rel_simps(1)
tff(fact_392_merge__pure__and,axiom,
    ! [A: $o,B: $o] :
      aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_o_assn(pure_assn,(A))),aa_o_assn(pure_assn,(B))) = aa_o_assn(pure_assn,
        ( (A)
        & (B) )) ).

% merge_pure_and
tff(fact_393_and__extract__pure__right__ctx__iff,axiom,
    ! [Pa: assn,Q: assn,B: $o] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Pa),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Q),aa_o_assn(pure_assn,(B)))) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Pa),Q)),aa_o_assn(pure_assn,(B))) ).

% and_extract_pure_right_ctx_iff
tff(fact_394_and__extract__pure__left__ctx__iff,axiom,
    ! [Pa: assn,B: $o,Q: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Pa),aa_o_assn(pure_assn,(B)))),Q) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Pa),Q)),aa_o_assn(pure_assn,(B))) ).

% and_extract_pure_left_ctx_iff
tff(fact_395_list__collect__set__map__simps_I1_J,axiom,
    ! [F: fun_nat_set_o,X: fun_nat_nat] : list_c8047850539171819768_nat_o(F,aa_list_nat_list_nat(aa_fun4736978287202700568st_nat(map_nat_nat,X),nil_nat)) = bot_bot_set_o ).

% list_collect_set_map_simps(1)
tff(fact_396_list__collect__set__map__simps_I1_J,axiom,
    ! [F: fun_assn_set_o,X: fun_Pr3816910165196019820n_assn] : list_c312183563312650144assn_o(F,aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,X),nil_Pr5671120429643327159n_assn)) = bot_bot_set_o ).

% list_collect_set_map_simps(1)
tff(fact_397_list__collect__set__map__simps_I1_J,axiom,
    ! [F: fun_nat_set_nat,X: fun_nat_nat] : list_c2452340269597857392at_nat(F,aa_list_nat_list_nat(aa_fun4736978287202700568st_nat(map_nat_nat,X),nil_nat)) = bot_bot_set_nat ).

% list_collect_set_map_simps(1)
tff(fact_398_list__collect__set__map__simps_I1_J,axiom,
    ! [F: fun_assn_set_nat,X: fun_Pr3816910165196019820n_assn] : list_c1844713377658005960sn_nat(F,aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,X),nil_Pr5671120429643327159n_assn)) = bot_bot_set_nat ).

% list_collect_set_map_simps(1)
tff(fact_399_assn__aci_I4_J,axiom,
    ! [X: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y) ).

% assn_aci(4)
tff(fact_400_assn__aci_I3_J,axiom,
    ! [X: assn,Y: assn,Z: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),Z)) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Z)) ).

% assn_aci(3)
tff(fact_401_assn__aci_I1_J,axiom,
    ! [X: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),X) ).

% assn_aci(1)
tff(fact_402_norm__assertion__simps_I31_J,axiom,
    ! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),X) = X ).

% norm_assertion_simps(31)
tff(fact_403_norm__assertion__simps_I14_J,axiom,
    ! [X: assn,Y: assn,Z: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)),Z) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),Z)) ).

% norm_assertion_simps(14)
tff(fact_404_memb__imp__not__empty,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,S: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),S)
     => ( S != bot_bo8632255151191677115_nat_o ) ) ).

% memb_imp_not_empty
tff(fact_405_memb__imp__not__empty,axiom,
    ! [X: $o,S: set_o] :
      ( aa_set_o_o(member_o((X)),S)
     => ( S != bot_bot_set_o ) ) ).

% memb_imp_not_empty
tff(fact_406_memb__imp__not__empty,axiom,
    ! [X: nat,S: set_nat] :
      ( aa_set_nat_o(member_nat2(X),S)
     => ( S != bot_bot_set_nat ) ) ).

% memb_imp_not_empty
tff(fact_407_set__notEmptyE,axiom,
    ! [S: set_fu3555266933746557007_nat_o] :
      ( ( S != bot_bo8632255151191677115_nat_o )
     => ~ ! [X2: fun_Pr6460944699146269935_nat_o] : ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X2),S) ) ).

% set_notEmptyE
tff(fact_408_set__notEmptyE,axiom,
    ! [S: set_o] :
      ( ( S != bot_bot_set_o )
     => ~ ! [X2: $o] : ~ aa_set_o_o(member_o((X2)),S) ) ).

% set_notEmptyE
tff(fact_409_set__notEmptyE,axiom,
    ! [S: set_nat] :
      ( ( S != bot_bot_set_nat )
     => ~ ! [X2: nat] : ~ aa_set_nat_o(member_nat2(X2),S) ) ).

% set_notEmptyE
tff(fact_410_map__eq__consE,axiom,
    ! [F: fun_Pr3816910165196019820n_assn,Ls: list_P8527749157015355191n_assn,Fa: assn,Fl: list_assn] :
      ( ( aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,F),Ls) = cons_assn(Fa,Fl) )
     => ~ ! [A3: produc6575502325842934193n_assn,L: list_P8527749157015355191n_assn] :
            ( ( Ls = aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,A3),L) )
           => ( ( aa_Pro2979818503705714077n_assn(F,A3) = Fa )
             => ( aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,F),L) != Fl ) ) ) ) ).

% map_eq_consE
tff(fact_411_map__eq__consE,axiom,
    ! [F: fun_nat_nat,Ls: list_nat,Fa: nat,Fl: list_nat] :
      ( ( aa_list_nat_list_nat(aa_fun4736978287202700568st_nat(map_nat_nat,F),Ls) = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Fa),Fl) )
     => ~ ! [A3: nat,L: list_nat] :
            ( ( Ls = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,A3),L) )
           => ( ( aa_nat_nat(F,A3) = Fa )
             => ( aa_list_nat_list_nat(aa_fun4736978287202700568st_nat(map_nat_nat,F),L) != Fl ) ) ) ) ).

% map_eq_consE
tff(fact_412_map__eq__consE,axiom,
    ! [F: fun_int_nat,Ls: list_int,Fa: nat,Fl: list_nat] :
      ( ( map_int_nat(F,Ls) = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Fa),Fl) )
     => ~ ! [A3: int,L: list_int] :
            ( ( Ls = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,A3),L) )
           => ( ( aa_int_nat(F,A3) = Fa )
             => ( map_int_nat(F,L) != Fl ) ) ) ) ).

% map_eq_consE
tff(fact_413_map__eq__consE,axiom,
    ! [F: fun_Pr7403558292872094996sn_nat,Ls: list_P8527749157015355191n_assn,Fa: nat,Fl: list_nat] :
      ( ( map_Pr7570552894071451325sn_nat(F,Ls) = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Fa),Fl) )
     => ~ ! [A3: produc6575502325842934193n_assn,L: list_P8527749157015355191n_assn] :
            ( ( Ls = aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,A3),L) )
           => ( ( aa_Pro8055486230010607173sn_nat(F,A3) = Fa )
             => ( map_Pr7570552894071451325sn_nat(F,L) != Fl ) ) ) ) ).

% map_eq_consE
tff(fact_414_map__eq__consE,axiom,
    ! [F: fun_nat_int,Ls: list_nat,Fa: int,Fl: list_int] :
      ( ( map_nat_int(F,Ls) = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Fa),Fl) )
     => ~ ! [A3: nat,L: list_nat] :
            ( ( Ls = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,A3),L) )
           => ( ( aa_nat_int(F,A3) = Fa )
             => ( map_nat_int(F,L) != Fl ) ) ) ) ).

% map_eq_consE
tff(fact_415_map__eq__consE,axiom,
    ! [F: fun_int_int,Ls: list_int,Fa: int,Fl: list_int] :
      ( ( map_int_int(F,Ls) = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Fa),Fl) )
     => ~ ! [A3: int,L: list_int] :
            ( ( Ls = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,A3),L) )
           => ( ( aa_int_int(F,A3) = Fa )
             => ( map_int_int(F,L) != Fl ) ) ) ) ).

% map_eq_consE
tff(fact_416_map__eq__consE,axiom,
    ! [F: fun_Pr3225707273362898288sn_int,Ls: list_P8527749157015355191n_assn,Fa: int,Fl: list_int] :
      ( ( map_Pr7568062423562401049sn_int(F,Ls) = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Fa),Fl) )
     => ~ ! [A3: produc6575502325842934193n_assn,L: list_P8527749157015355191n_assn] :
            ( ( Ls = aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,A3),L) )
           => ( ( aa_Pro8052995759501556897sn_int(F,A3) = Fa )
             => ( map_Pr7568062423562401049sn_int(F,L) != Fl ) ) ) ) ).

% map_eq_consE
tff(fact_417_map__eq__consE,axiom,
    ! [F: fun_na5138202758309126970n_assn,Ls: list_nat,Fa: produc6575502325842934193n_assn,Fl: list_P8527749157015355191n_assn] :
      ( ( map_na2667955367175718043n_assn(F,Ls) = aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,Fa),Fl) )
     => ~ ! [A3: nat,L: list_nat] :
            ( ( Ls = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,A3),L) )
           => ( ( aa_nat3152888703114873891n_assn(F,A3) = Fa )
             => ( map_na2667955367175718043n_assn(F,L) != Fl ) ) ) ) ).

% map_eq_consE
tff(fact_418_map__eq__consE,axiom,
    ! [F: fun_in6898239422062238174n_assn,Ls: list_int,Fa: produc6575502325842934193n_assn,Fl: list_P8527749157015355191n_assn] :
      ( ( map_in4427992030928829247n_assn(F,Ls) = aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,Fa),Fl) )
     => ~ ! [A3: int,L: list_int] :
            ( ( Ls = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,A3),L) )
           => ( ( aa_int4912925366867985095n_assn(F,A3) = Fa )
             => ( map_in4427992030928829247n_assn(F,L) != Fl ) ) ) ) ).

% map_eq_consE
tff(fact_419_map__eq__consE,axiom,
    ! [F: fun_Pr5530265736160528387n_assn,Ls: list_P8527749157015355191n_assn,Fa: produc6575502325842934193n_assn,Fl: list_P8527749157015355191n_assn] :
      ( ( map_Pr7925354932063753860n_assn(F,Ls) = aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,Fa),Fl) )
     => ~ ! [A3: produc6575502325842934193n_assn,L: list_P8527749157015355191n_assn] :
            ( ( Ls = aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,A3),L) )
           => ( ( aa_Pro2851486235043456252n_assn(F,A3) = Fa )
             => ( map_Pr7925354932063753860n_assn(F,L) != Fl ) ) ) ) ).

% map_eq_consE
tff(fact_420_map__consI_I1_J,axiom,
    ! [W: list_assn,F: fun_Pr3816910165196019820n_assn,Ww: list_P8527749157015355191n_assn,A: produc6575502325842934193n_assn] :
      ( ( W = aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,F),Ww) )
     => ( cons_assn(aa_Pro2979818503705714077n_assn(F,A),W) = aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,F),aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,A),Ww)) ) ) ).

% map_consI(1)
tff(fact_421_map__consI_I1_J,axiom,
    ! [W: list_nat,F: fun_nat_nat,Ww: list_nat,A: nat] :
      ( ( W = aa_list_nat_list_nat(aa_fun4736978287202700568st_nat(map_nat_nat,F),Ww) )
     => ( aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,aa_nat_nat(F,A)),W) = aa_list_nat_list_nat(aa_fun4736978287202700568st_nat(map_nat_nat,F),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,A),Ww)) ) ) ).

% map_consI(1)
tff(fact_422_map__consI_I1_J,axiom,
    ! [W: list_nat,F: fun_int_nat,Ww: list_int,A: int] :
      ( ( W = map_int_nat(F,Ww) )
     => ( aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,aa_int_nat(F,A)),W) = map_int_nat(F,aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,A),Ww)) ) ) ).

% map_consI(1)
tff(fact_423_map__consI_I1_J,axiom,
    ! [W: list_nat,F: fun_Pr7403558292872094996sn_nat,Ww: list_P8527749157015355191n_assn,A: produc6575502325842934193n_assn] :
      ( ( W = map_Pr7570552894071451325sn_nat(F,Ww) )
     => ( aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,aa_Pro8055486230010607173sn_nat(F,A)),W) = map_Pr7570552894071451325sn_nat(F,aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,A),Ww)) ) ) ).

% map_consI(1)
tff(fact_424_map__consI_I1_J,axiom,
    ! [W: list_int,F: fun_nat_int,Ww: list_nat,A: nat] :
      ( ( W = map_nat_int(F,Ww) )
     => ( aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,aa_nat_int(F,A)),W) = map_nat_int(F,aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,A),Ww)) ) ) ).

% map_consI(1)
tff(fact_425_map__consI_I1_J,axiom,
    ! [W: list_int,F: fun_int_int,Ww: list_int,A: int] :
      ( ( W = map_int_int(F,Ww) )
     => ( aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,aa_int_int(F,A)),W) = map_int_int(F,aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,A),Ww)) ) ) ).

% map_consI(1)
tff(fact_426_map__consI_I1_J,axiom,
    ! [W: list_int,F: fun_Pr3225707273362898288sn_int,Ww: list_P8527749157015355191n_assn,A: produc6575502325842934193n_assn] :
      ( ( W = map_Pr7568062423562401049sn_int(F,Ww) )
     => ( aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,aa_Pro8052995759501556897sn_int(F,A)),W) = map_Pr7568062423562401049sn_int(F,aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,A),Ww)) ) ) ).

% map_consI(1)
tff(fact_427_map__consI_I1_J,axiom,
    ! [W: list_P8527749157015355191n_assn,F: fun_na5138202758309126970n_assn,Ww: list_nat,A: nat] :
      ( ( W = map_na2667955367175718043n_assn(F,Ww) )
     => ( aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,aa_nat3152888703114873891n_assn(F,A)),W) = map_na2667955367175718043n_assn(F,aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,A),Ww)) ) ) ).

% map_consI(1)
tff(fact_428_map__consI_I1_J,axiom,
    ! [W: list_P8527749157015355191n_assn,F: fun_in6898239422062238174n_assn,Ww: list_int,A: int] :
      ( ( W = map_in4427992030928829247n_assn(F,Ww) )
     => ( aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,aa_int4912925366867985095n_assn(F,A)),W) = map_in4427992030928829247n_assn(F,aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,A),Ww)) ) ) ).

% map_consI(1)
tff(fact_429_map__consI_I1_J,axiom,
    ! [W: list_P8527749157015355191n_assn,F: fun_Pr5530265736160528387n_assn,Ww: list_P8527749157015355191n_assn,A: produc6575502325842934193n_assn] :
      ( ( W = map_Pr7925354932063753860n_assn(F,Ww) )
     => ( aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,aa_Pro2851486235043456252n_assn(F,A)),W) = map_Pr7925354932063753860n_assn(F,aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,A),Ww)) ) ) ).

% map_consI(1)
tff(fact_430_map__eq__Cons__conv,axiom,
    ! [F: fun_Pr3816910165196019820n_assn,Xs: list_P8527749157015355191n_assn,Y: assn,Ys: list_assn] :
      ( ( aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,F),Xs) = cons_assn(Y,Ys) )
    <=> ? [Z2: produc6575502325842934193n_assn,Zs: list_P8527749157015355191n_assn] :
          ( ( Xs = aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,Z2),Zs) )
          & ( aa_Pro2979818503705714077n_assn(F,Z2) = Y )
          & ( aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,F),Zs) = Ys ) ) ) ).

% map_eq_Cons_conv
tff(fact_431_map__eq__Cons__conv,axiom,
    ! [F: fun_nat_nat,Xs: list_nat,Y: nat,Ys: list_nat] :
      ( ( aa_list_nat_list_nat(aa_fun4736978287202700568st_nat(map_nat_nat,F),Xs) = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Y),Ys) )
    <=> ? [Z2: nat,Zs: list_nat] :
          ( ( Xs = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Z2),Zs) )
          & ( aa_nat_nat(F,Z2) = Y )
          & ( aa_list_nat_list_nat(aa_fun4736978287202700568st_nat(map_nat_nat,F),Zs) = Ys ) ) ) ).

% map_eq_Cons_conv
tff(fact_432_map__eq__Cons__conv,axiom,
    ! [F: fun_int_nat,Xs: list_int,Y: nat,Ys: list_nat] :
      ( ( map_int_nat(F,Xs) = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Y),Ys) )
    <=> ? [Z2: int,Zs: list_int] :
          ( ( Xs = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Z2),Zs) )
          & ( aa_int_nat(F,Z2) = Y )
          & ( map_int_nat(F,Zs) = Ys ) ) ) ).

% map_eq_Cons_conv
tff(fact_433_map__eq__Cons__conv,axiom,
    ! [F: fun_Pr7403558292872094996sn_nat,Xs: list_P8527749157015355191n_assn,Y: nat,Ys: list_nat] :
      ( ( map_Pr7570552894071451325sn_nat(F,Xs) = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Y),Ys) )
    <=> ? [Z2: produc6575502325842934193n_assn,Zs: list_P8527749157015355191n_assn] :
          ( ( Xs = aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,Z2),Zs) )
          & ( aa_Pro8055486230010607173sn_nat(F,Z2) = Y )
          & ( map_Pr7570552894071451325sn_nat(F,Zs) = Ys ) ) ) ).

% map_eq_Cons_conv
tff(fact_434_map__eq__Cons__conv,axiom,
    ! [F: fun_nat_int,Xs: list_nat,Y: int,Ys: list_int] :
      ( ( map_nat_int(F,Xs) = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Y),Ys) )
    <=> ? [Z2: nat,Zs: list_nat] :
          ( ( Xs = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Z2),Zs) )
          & ( aa_nat_int(F,Z2) = Y )
          & ( map_nat_int(F,Zs) = Ys ) ) ) ).

% map_eq_Cons_conv
tff(fact_435_map__eq__Cons__conv,axiom,
    ! [F: fun_int_int,Xs: list_int,Y: int,Ys: list_int] :
      ( ( map_int_int(F,Xs) = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Y),Ys) )
    <=> ? [Z2: int,Zs: list_int] :
          ( ( Xs = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Z2),Zs) )
          & ( aa_int_int(F,Z2) = Y )
          & ( map_int_int(F,Zs) = Ys ) ) ) ).

% map_eq_Cons_conv
tff(fact_436_map__eq__Cons__conv,axiom,
    ! [F: fun_Pr3225707273362898288sn_int,Xs: list_P8527749157015355191n_assn,Y: int,Ys: list_int] :
      ( ( map_Pr7568062423562401049sn_int(F,Xs) = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Y),Ys) )
    <=> ? [Z2: produc6575502325842934193n_assn,Zs: list_P8527749157015355191n_assn] :
          ( ( Xs = aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,Z2),Zs) )
          & ( aa_Pro8052995759501556897sn_int(F,Z2) = Y )
          & ( map_Pr7568062423562401049sn_int(F,Zs) = Ys ) ) ) ).

% map_eq_Cons_conv
tff(fact_437_map__eq__Cons__conv,axiom,
    ! [F: fun_na5138202758309126970n_assn,Xs: list_nat,Y: produc6575502325842934193n_assn,Ys: list_P8527749157015355191n_assn] :
      ( ( map_na2667955367175718043n_assn(F,Xs) = aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,Y),Ys) )
    <=> ? [Z2: nat,Zs: list_nat] :
          ( ( Xs = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Z2),Zs) )
          & ( aa_nat3152888703114873891n_assn(F,Z2) = Y )
          & ( map_na2667955367175718043n_assn(F,Zs) = Ys ) ) ) ).

% map_eq_Cons_conv
tff(fact_438_map__eq__Cons__conv,axiom,
    ! [F: fun_in6898239422062238174n_assn,Xs: list_int,Y: produc6575502325842934193n_assn,Ys: list_P8527749157015355191n_assn] :
      ( ( map_in4427992030928829247n_assn(F,Xs) = aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,Y),Ys) )
    <=> ? [Z2: int,Zs: list_int] :
          ( ( Xs = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Z2),Zs) )
          & ( aa_int4912925366867985095n_assn(F,Z2) = Y )
          & ( map_in4427992030928829247n_assn(F,Zs) = Ys ) ) ) ).

% map_eq_Cons_conv
tff(fact_439_map__eq__Cons__conv,axiom,
    ! [F: fun_Pr5530265736160528387n_assn,Xs: list_P8527749157015355191n_assn,Y: produc6575502325842934193n_assn,Ys: list_P8527749157015355191n_assn] :
      ( ( map_Pr7925354932063753860n_assn(F,Xs) = aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,Y),Ys) )
    <=> ? [Z2: produc6575502325842934193n_assn,Zs: list_P8527749157015355191n_assn] :
          ( ( Xs = aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,Z2),Zs) )
          & ( aa_Pro2851486235043456252n_assn(F,Z2) = Y )
          & ( map_Pr7925354932063753860n_assn(F,Zs) = Ys ) ) ) ).

% map_eq_Cons_conv
tff(fact_440_Cons__eq__map__conv,axiom,
    ! [X: assn,Xs: list_assn,F: fun_Pr3816910165196019820n_assn,Ys: list_P8527749157015355191n_assn] :
      ( ( cons_assn(X,Xs) = aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,F),Ys) )
    <=> ? [Z2: produc6575502325842934193n_assn,Zs: list_P8527749157015355191n_assn] :
          ( ( Ys = aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,Z2),Zs) )
          & ( X = aa_Pro2979818503705714077n_assn(F,Z2) )
          & ( Xs = aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,F),Zs) ) ) ) ).

% Cons_eq_map_conv
tff(fact_441_Cons__eq__map__conv,axiom,
    ! [X: nat,Xs: list_nat,F: fun_nat_nat,Ys: list_nat] :
      ( ( aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X),Xs) = aa_list_nat_list_nat(aa_fun4736978287202700568st_nat(map_nat_nat,F),Ys) )
    <=> ? [Z2: nat,Zs: list_nat] :
          ( ( Ys = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Z2),Zs) )
          & ( X = aa_nat_nat(F,Z2) )
          & ( Xs = aa_list_nat_list_nat(aa_fun4736978287202700568st_nat(map_nat_nat,F),Zs) ) ) ) ).

% Cons_eq_map_conv
tff(fact_442_Cons__eq__map__conv,axiom,
    ! [X: nat,Xs: list_nat,F: fun_int_nat,Ys: list_int] :
      ( ( aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X),Xs) = map_int_nat(F,Ys) )
    <=> ? [Z2: int,Zs: list_int] :
          ( ( Ys = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Z2),Zs) )
          & ( X = aa_int_nat(F,Z2) )
          & ( Xs = map_int_nat(F,Zs) ) ) ) ).

% Cons_eq_map_conv
tff(fact_443_Cons__eq__map__conv,axiom,
    ! [X: nat,Xs: list_nat,F: fun_Pr7403558292872094996sn_nat,Ys: list_P8527749157015355191n_assn] :
      ( ( aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X),Xs) = map_Pr7570552894071451325sn_nat(F,Ys) )
    <=> ? [Z2: produc6575502325842934193n_assn,Zs: list_P8527749157015355191n_assn] :
          ( ( Ys = aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,Z2),Zs) )
          & ( X = aa_Pro8055486230010607173sn_nat(F,Z2) )
          & ( Xs = map_Pr7570552894071451325sn_nat(F,Zs) ) ) ) ).

% Cons_eq_map_conv
tff(fact_444_Cons__eq__map__conv,axiom,
    ! [X: int,Xs: list_int,F: fun_nat_int,Ys: list_nat] :
      ( ( aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X),Xs) = map_nat_int(F,Ys) )
    <=> ? [Z2: nat,Zs: list_nat] :
          ( ( Ys = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Z2),Zs) )
          & ( X = aa_nat_int(F,Z2) )
          & ( Xs = map_nat_int(F,Zs) ) ) ) ).

% Cons_eq_map_conv
tff(fact_445_Cons__eq__map__conv,axiom,
    ! [X: int,Xs: list_int,F: fun_int_int,Ys: list_int] :
      ( ( aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X),Xs) = map_int_int(F,Ys) )
    <=> ? [Z2: int,Zs: list_int] :
          ( ( Ys = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Z2),Zs) )
          & ( X = aa_int_int(F,Z2) )
          & ( Xs = map_int_int(F,Zs) ) ) ) ).

% Cons_eq_map_conv
tff(fact_446_Cons__eq__map__conv,axiom,
    ! [X: int,Xs: list_int,F: fun_Pr3225707273362898288sn_int,Ys: list_P8527749157015355191n_assn] :
      ( ( aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X),Xs) = map_Pr7568062423562401049sn_int(F,Ys) )
    <=> ? [Z2: produc6575502325842934193n_assn,Zs: list_P8527749157015355191n_assn] :
          ( ( Ys = aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,Z2),Zs) )
          & ( X = aa_Pro8052995759501556897sn_int(F,Z2) )
          & ( Xs = map_Pr7568062423562401049sn_int(F,Zs) ) ) ) ).

% Cons_eq_map_conv
tff(fact_447_Cons__eq__map__conv,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn,F: fun_na5138202758309126970n_assn,Ys: list_nat] :
      ( ( aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X),Xs) = map_na2667955367175718043n_assn(F,Ys) )
    <=> ? [Z2: nat,Zs: list_nat] :
          ( ( Ys = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Z2),Zs) )
          & ( X = aa_nat3152888703114873891n_assn(F,Z2) )
          & ( Xs = map_na2667955367175718043n_assn(F,Zs) ) ) ) ).

% Cons_eq_map_conv
tff(fact_448_Cons__eq__map__conv,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn,F: fun_in6898239422062238174n_assn,Ys: list_int] :
      ( ( aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X),Xs) = map_in4427992030928829247n_assn(F,Ys) )
    <=> ? [Z2: int,Zs: list_int] :
          ( ( Ys = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Z2),Zs) )
          & ( X = aa_int4912925366867985095n_assn(F,Z2) )
          & ( Xs = map_in4427992030928829247n_assn(F,Zs) ) ) ) ).

% Cons_eq_map_conv
tff(fact_449_Cons__eq__map__conv,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn,F: fun_Pr5530265736160528387n_assn,Ys: list_P8527749157015355191n_assn] :
      ( ( aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X),Xs) = map_Pr7925354932063753860n_assn(F,Ys) )
    <=> ? [Z2: produc6575502325842934193n_assn,Zs: list_P8527749157015355191n_assn] :
          ( ( Ys = aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,Z2),Zs) )
          & ( X = aa_Pro2851486235043456252n_assn(F,Z2) )
          & ( Xs = map_Pr7925354932063753860n_assn(F,Zs) ) ) ) ).

% Cons_eq_map_conv
tff(fact_450_map__eq__Cons__D,axiom,
    ! [F: fun_Pr3816910165196019820n_assn,Xs: list_P8527749157015355191n_assn,Y: assn,Ys: list_assn] :
      ( ( aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,F),Xs) = cons_assn(Y,Ys) )
     => ? [Z3: produc6575502325842934193n_assn,Zs2: list_P8527749157015355191n_assn] :
          ( ( Xs = aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,Z3),Zs2) )
          & ( aa_Pro2979818503705714077n_assn(F,Z3) = Y )
          & ( aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,F),Zs2) = Ys ) ) ) ).

% map_eq_Cons_D
tff(fact_451_map__eq__Cons__D,axiom,
    ! [F: fun_nat_nat,Xs: list_nat,Y: nat,Ys: list_nat] :
      ( ( aa_list_nat_list_nat(aa_fun4736978287202700568st_nat(map_nat_nat,F),Xs) = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Y),Ys) )
     => ? [Z3: nat,Zs2: list_nat] :
          ( ( Xs = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Z3),Zs2) )
          & ( aa_nat_nat(F,Z3) = Y )
          & ( aa_list_nat_list_nat(aa_fun4736978287202700568st_nat(map_nat_nat,F),Zs2) = Ys ) ) ) ).

% map_eq_Cons_D
tff(fact_452_map__eq__Cons__D,axiom,
    ! [F: fun_int_nat,Xs: list_int,Y: nat,Ys: list_nat] :
      ( ( map_int_nat(F,Xs) = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Y),Ys) )
     => ? [Z3: int,Zs2: list_int] :
          ( ( Xs = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Z3),Zs2) )
          & ( aa_int_nat(F,Z3) = Y )
          & ( map_int_nat(F,Zs2) = Ys ) ) ) ).

% map_eq_Cons_D
tff(fact_453_map__eq__Cons__D,axiom,
    ! [F: fun_Pr7403558292872094996sn_nat,Xs: list_P8527749157015355191n_assn,Y: nat,Ys: list_nat] :
      ( ( map_Pr7570552894071451325sn_nat(F,Xs) = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Y),Ys) )
     => ? [Z3: produc6575502325842934193n_assn,Zs2: list_P8527749157015355191n_assn] :
          ( ( Xs = aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,Z3),Zs2) )
          & ( aa_Pro8055486230010607173sn_nat(F,Z3) = Y )
          & ( map_Pr7570552894071451325sn_nat(F,Zs2) = Ys ) ) ) ).

% map_eq_Cons_D
tff(fact_454_map__eq__Cons__D,axiom,
    ! [F: fun_nat_int,Xs: list_nat,Y: int,Ys: list_int] :
      ( ( map_nat_int(F,Xs) = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Y),Ys) )
     => ? [Z3: nat,Zs2: list_nat] :
          ( ( Xs = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Z3),Zs2) )
          & ( aa_nat_int(F,Z3) = Y )
          & ( map_nat_int(F,Zs2) = Ys ) ) ) ).

% map_eq_Cons_D
tff(fact_455_map__eq__Cons__D,axiom,
    ! [F: fun_int_int,Xs: list_int,Y: int,Ys: list_int] :
      ( ( map_int_int(F,Xs) = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Y),Ys) )
     => ? [Z3: int,Zs2: list_int] :
          ( ( Xs = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Z3),Zs2) )
          & ( aa_int_int(F,Z3) = Y )
          & ( map_int_int(F,Zs2) = Ys ) ) ) ).

% map_eq_Cons_D
tff(fact_456_map__eq__Cons__D,axiom,
    ! [F: fun_Pr3225707273362898288sn_int,Xs: list_P8527749157015355191n_assn,Y: int,Ys: list_int] :
      ( ( map_Pr7568062423562401049sn_int(F,Xs) = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Y),Ys) )
     => ? [Z3: produc6575502325842934193n_assn,Zs2: list_P8527749157015355191n_assn] :
          ( ( Xs = aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,Z3),Zs2) )
          & ( aa_Pro8052995759501556897sn_int(F,Z3) = Y )
          & ( map_Pr7568062423562401049sn_int(F,Zs2) = Ys ) ) ) ).

% map_eq_Cons_D
tff(fact_457_map__eq__Cons__D,axiom,
    ! [F: fun_na5138202758309126970n_assn,Xs: list_nat,Y: produc6575502325842934193n_assn,Ys: list_P8527749157015355191n_assn] :
      ( ( map_na2667955367175718043n_assn(F,Xs) = aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,Y),Ys) )
     => ? [Z3: nat,Zs2: list_nat] :
          ( ( Xs = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Z3),Zs2) )
          & ( aa_nat3152888703114873891n_assn(F,Z3) = Y )
          & ( map_na2667955367175718043n_assn(F,Zs2) = Ys ) ) ) ).

% map_eq_Cons_D
tff(fact_458_map__eq__Cons__D,axiom,
    ! [F: fun_in6898239422062238174n_assn,Xs: list_int,Y: produc6575502325842934193n_assn,Ys: list_P8527749157015355191n_assn] :
      ( ( map_in4427992030928829247n_assn(F,Xs) = aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,Y),Ys) )
     => ? [Z3: int,Zs2: list_int] :
          ( ( Xs = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Z3),Zs2) )
          & ( aa_int4912925366867985095n_assn(F,Z3) = Y )
          & ( map_in4427992030928829247n_assn(F,Zs2) = Ys ) ) ) ).

% map_eq_Cons_D
tff(fact_459_map__eq__Cons__D,axiom,
    ! [F: fun_Pr5530265736160528387n_assn,Xs: list_P8527749157015355191n_assn,Y: produc6575502325842934193n_assn,Ys: list_P8527749157015355191n_assn] :
      ( ( map_Pr7925354932063753860n_assn(F,Xs) = aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,Y),Ys) )
     => ? [Z3: produc6575502325842934193n_assn,Zs2: list_P8527749157015355191n_assn] :
          ( ( Xs = aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,Z3),Zs2) )
          & ( aa_Pro2851486235043456252n_assn(F,Z3) = Y )
          & ( map_Pr7925354932063753860n_assn(F,Zs2) = Ys ) ) ) ).

% map_eq_Cons_D
tff(fact_460_Cons__eq__map__D,axiom,
    ! [X: assn,Xs: list_assn,F: fun_Pr3816910165196019820n_assn,Ys: list_P8527749157015355191n_assn] :
      ( ( cons_assn(X,Xs) = aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,F),Ys) )
     => ? [Z3: produc6575502325842934193n_assn,Zs2: list_P8527749157015355191n_assn] :
          ( ( Ys = aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,Z3),Zs2) )
          & ( X = aa_Pro2979818503705714077n_assn(F,Z3) )
          & ( Xs = aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,F),Zs2) ) ) ) ).

% Cons_eq_map_D
tff(fact_461_Cons__eq__map__D,axiom,
    ! [X: nat,Xs: list_nat,F: fun_nat_nat,Ys: list_nat] :
      ( ( aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X),Xs) = aa_list_nat_list_nat(aa_fun4736978287202700568st_nat(map_nat_nat,F),Ys) )
     => ? [Z3: nat,Zs2: list_nat] :
          ( ( Ys = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Z3),Zs2) )
          & ( X = aa_nat_nat(F,Z3) )
          & ( Xs = aa_list_nat_list_nat(aa_fun4736978287202700568st_nat(map_nat_nat,F),Zs2) ) ) ) ).

% Cons_eq_map_D
tff(fact_462_Cons__eq__map__D,axiom,
    ! [X: nat,Xs: list_nat,F: fun_int_nat,Ys: list_int] :
      ( ( aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X),Xs) = map_int_nat(F,Ys) )
     => ? [Z3: int,Zs2: list_int] :
          ( ( Ys = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Z3),Zs2) )
          & ( X = aa_int_nat(F,Z3) )
          & ( Xs = map_int_nat(F,Zs2) ) ) ) ).

% Cons_eq_map_D
tff(fact_463_Cons__eq__map__D,axiom,
    ! [X: nat,Xs: list_nat,F: fun_Pr7403558292872094996sn_nat,Ys: list_P8527749157015355191n_assn] :
      ( ( aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X),Xs) = map_Pr7570552894071451325sn_nat(F,Ys) )
     => ? [Z3: produc6575502325842934193n_assn,Zs2: list_P8527749157015355191n_assn] :
          ( ( Ys = aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,Z3),Zs2) )
          & ( X = aa_Pro8055486230010607173sn_nat(F,Z3) )
          & ( Xs = map_Pr7570552894071451325sn_nat(F,Zs2) ) ) ) ).

% Cons_eq_map_D
tff(fact_464_Cons__eq__map__D,axiom,
    ! [X: int,Xs: list_int,F: fun_nat_int,Ys: list_nat] :
      ( ( aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X),Xs) = map_nat_int(F,Ys) )
     => ? [Z3: nat,Zs2: list_nat] :
          ( ( Ys = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Z3),Zs2) )
          & ( X = aa_nat_int(F,Z3) )
          & ( Xs = map_nat_int(F,Zs2) ) ) ) ).

% Cons_eq_map_D
tff(fact_465_Cons__eq__map__D,axiom,
    ! [X: int,Xs: list_int,F: fun_int_int,Ys: list_int] :
      ( ( aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X),Xs) = map_int_int(F,Ys) )
     => ? [Z3: int,Zs2: list_int] :
          ( ( Ys = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Z3),Zs2) )
          & ( X = aa_int_int(F,Z3) )
          & ( Xs = map_int_int(F,Zs2) ) ) ) ).

% Cons_eq_map_D
tff(fact_466_Cons__eq__map__D,axiom,
    ! [X: int,Xs: list_int,F: fun_Pr3225707273362898288sn_int,Ys: list_P8527749157015355191n_assn] :
      ( ( aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X),Xs) = map_Pr7568062423562401049sn_int(F,Ys) )
     => ? [Z3: produc6575502325842934193n_assn,Zs2: list_P8527749157015355191n_assn] :
          ( ( Ys = aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,Z3),Zs2) )
          & ( X = aa_Pro8052995759501556897sn_int(F,Z3) )
          & ( Xs = map_Pr7568062423562401049sn_int(F,Zs2) ) ) ) ).

% Cons_eq_map_D
tff(fact_467_Cons__eq__map__D,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn,F: fun_na5138202758309126970n_assn,Ys: list_nat] :
      ( ( aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X),Xs) = map_na2667955367175718043n_assn(F,Ys) )
     => ? [Z3: nat,Zs2: list_nat] :
          ( ( Ys = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Z3),Zs2) )
          & ( X = aa_nat3152888703114873891n_assn(F,Z3) )
          & ( Xs = map_na2667955367175718043n_assn(F,Zs2) ) ) ) ).

% Cons_eq_map_D
tff(fact_468_Cons__eq__map__D,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn,F: fun_in6898239422062238174n_assn,Ys: list_int] :
      ( ( aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X),Xs) = map_in4427992030928829247n_assn(F,Ys) )
     => ? [Z3: int,Zs2: list_int] :
          ( ( Ys = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Z3),Zs2) )
          & ( X = aa_int4912925366867985095n_assn(F,Z3) )
          & ( Xs = map_in4427992030928829247n_assn(F,Zs2) ) ) ) ).

% Cons_eq_map_D
tff(fact_469_Cons__eq__map__D,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn,F: fun_Pr5530265736160528387n_assn,Ys: list_P8527749157015355191n_assn] :
      ( ( aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X),Xs) = map_Pr7925354932063753860n_assn(F,Ys) )
     => ? [Z3: produc6575502325842934193n_assn,Zs2: list_P8527749157015355191n_assn] :
          ( ( Ys = aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,Z3),Zs2) )
          & ( X = aa_Pro2851486235043456252n_assn(F,Z3) )
          & ( Xs = map_Pr7925354932063753860n_assn(F,Zs2) ) ) ) ).

% Cons_eq_map_D
tff(fact_470_list_Osimps_I9_J,axiom,
    ! [F: fun_nat_nat,X21: nat,X22: list_nat] : aa_list_nat_list_nat(aa_fun4736978287202700568st_nat(map_nat_nat,F),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X21),X22)) = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,aa_nat_nat(F,X21)),aa_list_nat_list_nat(aa_fun4736978287202700568st_nat(map_nat_nat,F),X22)) ).

% list.simps(9)
tff(fact_471_list_Osimps_I9_J,axiom,
    ! [F: fun_nat_int,X21: nat,X22: list_nat] : map_nat_int(F,aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X21),X22)) = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,aa_nat_int(F,X21)),map_nat_int(F,X22)) ).

% list.simps(9)
tff(fact_472_list_Osimps_I9_J,axiom,
    ! [F: fun_na5138202758309126970n_assn,X21: nat,X22: list_nat] : map_na2667955367175718043n_assn(F,aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X21),X22)) = aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,aa_nat3152888703114873891n_assn(F,X21)),map_na2667955367175718043n_assn(F,X22)) ).

% list.simps(9)
tff(fact_473_list_Osimps_I9_J,axiom,
    ! [F: fun_int_nat,X21: int,X22: list_int] : map_int_nat(F,aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X21),X22)) = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,aa_int_nat(F,X21)),map_int_nat(F,X22)) ).

% list.simps(9)
tff(fact_474_list_Osimps_I9_J,axiom,
    ! [F: fun_int_int,X21: int,X22: list_int] : map_int_int(F,aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X21),X22)) = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,aa_int_int(F,X21)),map_int_int(F,X22)) ).

% list.simps(9)
tff(fact_475_list_Osimps_I9_J,axiom,
    ! [F: fun_in6898239422062238174n_assn,X21: int,X22: list_int] : map_in4427992030928829247n_assn(F,aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X21),X22)) = aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,aa_int4912925366867985095n_assn(F,X21)),map_in4427992030928829247n_assn(F,X22)) ).

% list.simps(9)
tff(fact_476_list_Osimps_I9_J,axiom,
    ! [F: fun_Pr3816910165196019820n_assn,X21: produc6575502325842934193n_assn,X22: list_P8527749157015355191n_assn] : aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,F),aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X21),X22)) = cons_assn(aa_Pro2979818503705714077n_assn(F,X21),aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,F),X22)) ).

% list.simps(9)
tff(fact_477_list_Osimps_I9_J,axiom,
    ! [F: fun_Pr7403558292872094996sn_nat,X21: produc6575502325842934193n_assn,X22: list_P8527749157015355191n_assn] : map_Pr7570552894071451325sn_nat(F,aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X21),X22)) = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,aa_Pro8055486230010607173sn_nat(F,X21)),map_Pr7570552894071451325sn_nat(F,X22)) ).

% list.simps(9)
tff(fact_478_list_Osimps_I9_J,axiom,
    ! [F: fun_Pr3225707273362898288sn_int,X21: produc6575502325842934193n_assn,X22: list_P8527749157015355191n_assn] : map_Pr7568062423562401049sn_int(F,aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X21),X22)) = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,aa_Pro8052995759501556897sn_int(F,X21)),map_Pr7568062423562401049sn_int(F,X22)) ).

% list.simps(9)
tff(fact_479_list_Osimps_I9_J,axiom,
    ! [F: fun_Pr5530265736160528387n_assn,X21: produc6575502325842934193n_assn,X22: list_P8527749157015355191n_assn] : map_Pr7925354932063753860n_assn(F,aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X21),X22)) = aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,aa_Pro2851486235043456252n_assn(F,X21)),map_Pr7925354932063753860n_assn(F,X22)) ).

% list.simps(9)
tff(fact_480_list_Osimps_I8_J,axiom,
    ! [F: fun_a_a] : map_a_a(F,nil_a) = nil_a ).

% list.simps(8)
tff(fact_481_list_Osimps_I8_J,axiom,
    ! [F: fun_a_b] : map_a_b(F,nil_a) = nil_b ).

% list.simps(8)
tff(fact_482_list_Osimps_I8_J,axiom,
    ! [F: fun_a_nat] : map_a_nat(F,nil_a) = nil_nat ).

% list.simps(8)
tff(fact_483_list_Osimps_I8_J,axiom,
    ! [F: fun_a_int] : map_a_int(F,nil_a) = nil_int ).

% list.simps(8)
tff(fact_484_list_Osimps_I8_J,axiom,
    ! [F: fun_b_a] : map_b_a(F,nil_b) = nil_a ).

% list.simps(8)
tff(fact_485_list_Osimps_I8_J,axiom,
    ! [F: fun_b_b] : map_b_b(F,nil_b) = nil_b ).

% list.simps(8)
tff(fact_486_list_Osimps_I8_J,axiom,
    ! [F: fun_b_nat] : map_b_nat(F,nil_b) = nil_nat ).

% list.simps(8)
tff(fact_487_list_Osimps_I8_J,axiom,
    ! [F: fun_b_int] : map_b_int(F,nil_b) = nil_int ).

% list.simps(8)
tff(fact_488_list_Osimps_I8_J,axiom,
    ! [F: fun_nat_a] : map_nat_a(F,nil_nat) = nil_a ).

% list.simps(8)
tff(fact_489_list_Osimps_I8_J,axiom,
    ! [F: fun_nat_b] : map_nat_b(F,nil_nat) = nil_b ).

% list.simps(8)
tff(fact_490_norm__assertion__simps_I9_J,axiom,
    ! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,bot_bot_assn),X) = bot_bot_assn ).

% norm_assertion_simps(9)
tff(fact_491_norm__assertion__simps_I10_J,axiom,
    ! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),bot_bot_assn) = bot_bot_assn ).

% norm_assertion_simps(10)
tff(fact_492_lexordp__eq_ONil,axiom,
    ! [Ys: list_nat] : aa_list_nat_o(aa_lis557685189327716851_nat_o(ord_lexordp_eq_nat,nil_nat),Ys) ).

% lexordp_eq.Nil
tff(fact_493_lexordp__eq_ONil,axiom,
    ! [Ys: list_int] : aa_list_int_o(aa_lis7773489400162621755_int_o(ord_lexordp_eq_int,nil_int),Ys) ).

% lexordp_eq.Nil
tff(fact_494_listrelp_OCons,axiom,
    ! [R3: fun_nat_fun_nat_o,X: nat,Y: nat,Xs: list_nat,Ys: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(R3,X),Y)
     => ( aa_list_nat_o(aa_lis557685189327716851_nat_o(listrelp_nat_nat(R3),Xs),Ys)
       => aa_list_nat_o(aa_lis557685189327716851_nat_o(listrelp_nat_nat(R3),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X),Xs)),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Y),Ys)) ) ) ).

% listrelp.Cons
tff(fact_495_listrelp_OCons,axiom,
    ! [R3: fun_nat_fun_int_o,X: nat,Y: int,Xs: list_nat,Ys: list_int] :
      ( aa_int_o(aa_nat_fun_int_o(R3,X),Y)
     => ( aa_list_int_o(listrelp_nat_int(R3,Xs),Ys)
       => aa_list_int_o(listrelp_nat_int(R3,aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X),Xs)),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Y),Ys)) ) ) ).

% listrelp.Cons
tff(fact_496_listrelp_OCons,axiom,
    ! [R3: fun_na1082083995841529399assn_o,X: nat,Y: produc6575502325842934193n_assn,Xs: list_nat,Ys: list_P8527749157015355191n_assn] :
      ( aa_Pro2491721552408928739assn_o(aa_nat516171522532136568assn_o(R3,X),Y)
     => ( aa_lis1404944545740551261assn_o(listre8980241315090511200n_assn(R3,Xs),Ys)
       => aa_lis1404944545740551261assn_o(listre8980241315090511200n_assn(R3,aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X),Xs)),aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,Y),Ys)) ) ) ).

% listrelp.Cons
tff(fact_497_listrelp_OCons,axiom,
    ! [R3: fun_int_fun_nat_o,X: int,Y: nat,Xs: list_int,Ys: list_nat] :
      ( aa_nat_o(aa_int_fun_nat_o(R3,X),Y)
     => ( aa_list_nat_o(listrelp_int_nat(R3,Xs),Ys)
       => aa_list_nat_o(listrelp_int_nat(R3,aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X),Xs)),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Y),Ys)) ) ) ).

% listrelp.Cons
tff(fact_498_listrelp_OCons,axiom,
    ! [R3: fun_int_fun_int_o,X: int,Y: int,Xs: list_int,Ys: list_int] :
      ( aa_int_o(aa_int_fun_int_o(R3,X),Y)
     => ( aa_list_int_o(aa_lis7773489400162621755_int_o(listrelp_int_int(R3),Xs),Ys)
       => aa_list_int_o(aa_lis7773489400162621755_int_o(listrelp_int_int(R3),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X),Xs)),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Y),Ys)) ) ) ).

% listrelp.Cons
tff(fact_499_listrelp_OCons,axiom,
    ! [R3: fun_in7130107137290581651assn_o,X: int,Y: produc6575502325842934193n_assn,Xs: list_int,Ys: list_P8527749157015355191n_assn] :
      ( aa_Pro2491721552408928739assn_o(aa_int6564194663981188820assn_o(R3,X),Y)
     => ( aa_lis1404944545740551261assn_o(listre1516905941988846596n_assn(R3,Xs),Ys)
       => aa_lis1404944545740551261assn_o(listre1516905941988846596n_assn(R3,aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X),Xs)),aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,Y),Ys)) ) ) ).

% listrelp.Cons
tff(fact_500_listrelp_OCons,axiom,
    ! [R3: fun_Pr597617920386261747_nat_o,X: produc6575502325842934193n_assn,Y: nat,Xs: list_P8527749157015355191n_assn,Ys: list_nat] :
      ( aa_nat_o(aa_Pro7240946300278440940_nat_o(R3,X),Y)
     => ( aa_list_nat_o(listre4659466805131468674sn_nat(R3,Xs),Ys)
       => aa_list_nat_o(listre4659466805131468674sn_nat(R3,aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X),Xs)),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Y),Ys)) ) ) ).

% listrelp.Cons
tff(fact_501_listrelp_OCons,axiom,
    ! [R3: fun_Pr7575844989570591639_int_o,X: produc6575502325842934193n_assn,Y: int,Xs: list_P8527749157015355191n_assn,Ys: list_int] :
      ( aa_int_o(aa_Pro5419903954845752976_int_o(R3,X),Y)
     => ( aa_list_int_o(listre4656976334622418398sn_int(R3,Xs),Ys)
       => aa_list_int_o(listre4656976334622418398sn_int(R3,aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X),Xs)),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Y),Ys)) ) ) ).

% listrelp.Cons
tff(fact_502_listrelp_OCons,axiom,
    ! [R3: fun_Pr5295903952624206638assn_o,X: produc6575502325842934193n_assn,Y: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn,Ys: list_P8527749157015355191n_assn] :
      ( aa_Pro2491721552408928739assn_o(aa_Pro332967637727714143assn_o(R3,X),Y)
     => ( aa_lis1404944545740551261assn_o(listre7738141641822031743n_assn(R3,Xs),Ys)
       => aa_lis1404944545740551261assn_o(listre7738141641822031743n_assn(R3,aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X),Xs)),aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,Y),Ys)) ) ) ).

% listrelp.Cons
tff(fact_503_listrelp_ONil,axiom,
    ! [R3: fun_a_fun_a_o] : aa_list_a_o(aa_lis4963054610627903775st_a_o(listrelp_a_a(R3),nil_a),nil_a) ).

% listrelp.Nil
tff(fact_504_listrelp_ONil,axiom,
    ! [R3: fun_a_fun_b_o] : aa_list_b_o(aa_lis2175599028655227742st_b_o(listrelp_a_b(R3),nil_a),nil_b) ).

% listrelp.Nil
tff(fact_505_listrelp_ONil,axiom,
    ! [R3: fun_a_fun_nat_o] : aa_list_nat_o(aa_lis6662471382306767049_nat_o(listrelp_a_nat(R3),nil_a),nil_nat) ).

% listrelp.Nil
tff(fact_506_listrelp_ONil,axiom,
    ! [R3: fun_a_fun_int_o] : aa_list_int_o(aa_lis4417326414636321133_int_o(listrelp_a_int(R3),nil_a),nil_int) ).

% listrelp.Nil
tff(fact_507_listrelp_ONil,axiom,
    ! [R3: fun_b_fun_a_o] : aa_list_a_o(aa_lis7224773892088682016st_a_o(listrelp_b_a(R3),nil_b),nil_a) ).

% listrelp.Nil
tff(fact_508_listrelp_ONil,axiom,
    ! [R3: fun_b_fun_b_o] : aa_list_b_o(aa_lis4437318310116005983st_b_o(listrelp_b_b(R3),nil_b),nil_b) ).

% listrelp.Nil
tff(fact_509_listrelp_ONil,axiom,
    ! [R3: fun_b_fun_nat_o] : aa_list_nat_o(aa_lis3653689927639571848_nat_o(listrelp_b_nat(R3),nil_b),nil_nat) ).

% listrelp.Nil
tff(fact_510_listrelp_ONil,axiom,
    ! [R3: fun_b_fun_int_o] : aa_list_int_o(aa_lis1408544959969125932_int_o(listrelp_b_int(R3),nil_b),nil_int) ).

% listrelp.Nil
tff(fact_511_listrelp_ONil,axiom,
    ! [R3: fun_nat_fun_a_o] : aa_list_a_o(aa_lis6242202807773918837st_a_o(listrelp_nat_a(R3),nil_nat),nil_a) ).

% listrelp.Nil
tff(fact_512_listrelp_ONil,axiom,
    ! [R3: fun_nat_fun_b_o] : aa_list_b_o(aa_lis3454747225801242804st_b_o(listrelp_nat_b(R3),nil_nat),nil_b) ).

% listrelp.Nil
tff(fact_513_boolean__algebra_Oconj__zero__right,axiom,
    ! [X: product_unit] : aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),bot_bot_Product_unit) = bot_bot_Product_unit ).

% boolean_algebra.conj_zero_right
tff(fact_514_boolean__algebra_Oconj__zero__right,axiom,
    ! [X: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),bot_bo2099793752762293965at_nat) = bot_bo2099793752762293965at_nat ).

% boolean_algebra.conj_zero_right
tff(fact_515_boolean__algebra_Oconj__zero__right,axiom,
    ! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),bot_bot_assn) = bot_bot_assn ).

% boolean_algebra.conj_zero_right
tff(fact_516_boolean__algebra_Oconj__zero__right,axiom,
    ! [X: set_o] : inf_inf_set_o(X,bot_bot_set_o) = bot_bot_set_o ).

% boolean_algebra.conj_zero_right
tff(fact_517_boolean__algebra_Oconj__zero__right,axiom,
    ! [X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),bot_bot_set_nat) = bot_bot_set_nat ).

% boolean_algebra.conj_zero_right
tff(fact_518_boolean__algebra_Oconj__zero__left,axiom,
    ! [X: product_unit] : aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,bot_bot_Product_unit),X) = bot_bot_Product_unit ).

% boolean_algebra.conj_zero_left
tff(fact_519_boolean__algebra_Oconj__zero__left,axiom,
    ! [X: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,bot_bo2099793752762293965at_nat),X) = bot_bo2099793752762293965at_nat ).

% boolean_algebra.conj_zero_left
tff(fact_520_boolean__algebra_Oconj__zero__left,axiom,
    ! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,bot_bot_assn),X) = bot_bot_assn ).

% boolean_algebra.conj_zero_left
tff(fact_521_boolean__algebra_Oconj__zero__left,axiom,
    ! [X: set_o] : inf_inf_set_o(bot_bot_set_o,X) = bot_bot_set_o ).

% boolean_algebra.conj_zero_left
tff(fact_522_boolean__algebra_Oconj__zero__left,axiom,
    ! [X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,bot_bot_set_nat),X) = bot_bot_set_nat ).

% boolean_algebra.conj_zero_left
tff(fact_523_inf__bot__right,axiom,
    ! [X: product_unit] : aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),bot_bot_Product_unit) = bot_bot_Product_unit ).

% inf_bot_right
tff(fact_524_inf__bot__right,axiom,
    ! [X: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),bot_bo2099793752762293965at_nat) = bot_bo2099793752762293965at_nat ).

% inf_bot_right
tff(fact_525_inf__bot__right,axiom,
    ! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),bot_bot_assn) = bot_bot_assn ).

% inf_bot_right
tff(fact_526_inf__bot__right,axiom,
    ! [X: set_o] : inf_inf_set_o(X,bot_bot_set_o) = bot_bot_set_o ).

% inf_bot_right
tff(fact_527_inf__bot__right,axiom,
    ! [X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),bot_bot_set_nat) = bot_bot_set_nat ).

% inf_bot_right
tff(fact_528_inf__bot__left,axiom,
    ! [X: product_unit] : aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,bot_bot_Product_unit),X) = bot_bot_Product_unit ).

% inf_bot_left
tff(fact_529_inf__bot__left,axiom,
    ! [X: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,bot_bo2099793752762293965at_nat),X) = bot_bo2099793752762293965at_nat ).

% inf_bot_left
tff(fact_530_inf__bot__left,axiom,
    ! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,bot_bot_assn),X) = bot_bot_assn ).

% inf_bot_left
tff(fact_531_inf__bot__left,axiom,
    ! [X: set_o] : inf_inf_set_o(bot_bot_set_o,X) = bot_bot_set_o ).

% inf_bot_left
tff(fact_532_inf__bot__left,axiom,
    ! [X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,bot_bot_set_nat),X) = bot_bot_set_nat ).

% inf_bot_left
tff(fact_533_inf__right__idem,axiom,
    ! [X: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)),Y) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y) ).

% inf_right_idem
tff(fact_534_inf__right__idem,axiom,
    ! [X: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y)),Y) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y) ).

% inf_right_idem
tff(fact_535_inf__right__idem,axiom,
    ! [X: set_nat,Y: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)),Y) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y) ).

% inf_right_idem
tff(fact_536_inf__right__idem,axiom,
    ! [X: product_unit,Y: product_unit] : aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),Y)),Y) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),Y) ).

% inf_right_idem
tff(fact_537_inf__right__idem,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y)),Y) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y) ).

% inf_right_idem
tff(fact_538_inf_Oright__idem,axiom,
    ! [A: assn,B: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B)),B) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B) ).

% inf.right_idem
tff(fact_539_inf_Oright__idem,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)),B) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B) ).

% inf.right_idem
tff(fact_540_inf_Oright__idem,axiom,
    ! [A: set_nat,B: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)),B) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) ).

% inf.right_idem
tff(fact_541_inf_Oright__idem,axiom,
    ! [A: product_unit,B: product_unit] : aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,A),B)),B) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,A),B) ).

% inf.right_idem
tff(fact_542_inf_Oright__idem,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)),B) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) ).

% inf.right_idem
tff(fact_543_inf__left__idem,axiom,
    ! [X: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y) ).

% inf_left_idem
tff(fact_544_inf__left__idem,axiom,
    ! [X: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y) ).

% inf_left_idem
tff(fact_545_inf__left__idem,axiom,
    ! [X: set_nat,Y: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y) ).

% inf_left_idem
tff(fact_546_inf__left__idem,axiom,
    ! [X: product_unit,Y: product_unit] : aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),Y)) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),Y) ).

% inf_left_idem
tff(fact_547_inf__left__idem,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y) ).

% inf_left_idem
tff(fact_548_inf_Oleft__idem,axiom,
    ! [A: assn,B: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B)) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B) ).

% inf.left_idem
tff(fact_549_inf_Oleft__idem,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B) ).

% inf.left_idem
tff(fact_550_inf_Oleft__idem,axiom,
    ! [A: set_nat,B: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) ).

% inf.left_idem
tff(fact_551_inf_Oleft__idem,axiom,
    ! [A: product_unit,B: product_unit] : aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,A),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,A),B)) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,A),B) ).

% inf.left_idem
tff(fact_552_inf_Oleft__idem,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) ).

% inf.left_idem
tff(fact_553_inf__idem,axiom,
    ! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),X) = X ).

% inf_idem
tff(fact_554_inf__idem,axiom,
    ! [X: nat] : aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),X) = X ).

% inf_idem
tff(fact_555_inf__idem,axiom,
    ! [X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),X) = X ).

% inf_idem
tff(fact_556_inf__idem,axiom,
    ! [X: product_unit] : aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),X) = X ).

% inf_idem
tff(fact_557_inf__idem,axiom,
    ! [X: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),X) = X ).

% inf_idem
tff(fact_558_inf_Oidem,axiom,
    ! [A: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),A) = A ).

% inf.idem
tff(fact_559_inf_Oidem,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),A) = A ).

% inf.idem
tff(fact_560_inf_Oidem,axiom,
    ! [A: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),A) = A ).

% inf.idem
tff(fact_561_inf_Oidem,axiom,
    ! [A: product_unit] : aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,A),A) = A ).

% inf.idem
tff(fact_562_inf_Oidem,axiom,
    ! [A: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),A) = A ).

% inf.idem
tff(fact_563_empty__Collect__eq,axiom,
    ! [Pa: fun_fu5135969500543783852at_o_o] :
      ( ( bot_bo8632255151191677115_nat_o = collec6407654713642994138_nat_o(Pa) )
    <=> ! [X3: fun_Pr6460944699146269935_nat_o] : ~ aa_fun1675921005893230629at_o_o(Pa,X3) ) ).

% empty_Collect_eq
tff(fact_564_empty__Collect__eq,axiom,
    ! [Pa: fun_Pr4796943543331071514_nat_o] :
      ( ( bot_bo2099793752762293965at_nat = collec3392354462482085612at_nat(Pa) )
    <=> ! [X3: product_prod_nat_nat] : ~ aa_Pro6150637604445964947_nat_o(Pa,X3) ) ).

% empty_Collect_eq
tff(fact_565_empty__Collect__eq,axiom,
    ! [Pa: fun_o_o] :
      ( ( bot_bot_set_o = collect_o(Pa) )
    <=> ! [X3: $o] : ~ aa_o_o(Pa,(X3)) ) ).

% empty_Collect_eq
tff(fact_566_empty__Collect__eq,axiom,
    ! [Pa: fun_nat_o] :
      ( ( bot_bot_set_nat = collect_nat(Pa) )
    <=> ! [X3: nat] : ~ aa_nat_o(Pa,X3) ) ).

% empty_Collect_eq
tff(fact_567_empty__iff,axiom,
    ! [C2: fun_Pr6460944699146269935_nat_o] : ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C2),bot_bo8632255151191677115_nat_o) ).

% empty_iff
tff(fact_568_empty__iff,axiom,
    ! [C2: $o] : ~ aa_set_o_o(member_o((C2)),bot_bot_set_o) ).

% empty_iff
tff(fact_569_empty__iff,axiom,
    ! [C2: nat] : ~ aa_set_nat_o(member_nat2(C2),bot_bot_set_nat) ).

% empty_iff
tff(fact_570_all__not__in__conv,axiom,
    ! [A2: set_fu3555266933746557007_nat_o] :
      ( ! [X3: fun_Pr6460944699146269935_nat_o] : ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X3),A2)
    <=> ( A2 = bot_bo8632255151191677115_nat_o ) ) ).

% all_not_in_conv
tff(fact_571_all__not__in__conv,axiom,
    ! [A2: set_o] :
      ( ! [X3: $o] : ~ aa_set_o_o(member_o((X3)),A2)
    <=> ( A2 = bot_bot_set_o ) ) ).

% all_not_in_conv
tff(fact_572_all__not__in__conv,axiom,
    ! [A2: set_nat] :
      ( ! [X3: nat] : ~ aa_set_nat_o(member_nat2(X3),A2)
    <=> ( A2 = bot_bot_set_nat ) ) ).

% all_not_in_conv
tff(fact_573_Collect__empty__eq,axiom,
    ! [Pa: fun_fu5135969500543783852at_o_o] :
      ( ( collec6407654713642994138_nat_o(Pa) = bot_bo8632255151191677115_nat_o )
    <=> ! [X3: fun_Pr6460944699146269935_nat_o] : ~ aa_fun1675921005893230629at_o_o(Pa,X3) ) ).

% Collect_empty_eq
tff(fact_574_Collect__empty__eq,axiom,
    ! [Pa: fun_Pr4796943543331071514_nat_o] :
      ( ( collec3392354462482085612at_nat(Pa) = bot_bo2099793752762293965at_nat )
    <=> ! [X3: product_prod_nat_nat] : ~ aa_Pro6150637604445964947_nat_o(Pa,X3) ) ).

% Collect_empty_eq
tff(fact_575_Collect__empty__eq,axiom,
    ! [Pa: fun_o_o] :
      ( ( collect_o(Pa) = bot_bot_set_o )
    <=> ! [X3: $o] : ~ aa_o_o(Pa,(X3)) ) ).

% Collect_empty_eq
tff(fact_576_Collect__empty__eq,axiom,
    ! [Pa: fun_nat_o] :
      ( ( collect_nat(Pa) = bot_bot_set_nat )
    <=> ! [X3: nat] : ~ aa_nat_o(Pa,X3) ) ).

% Collect_empty_eq
tff(fact_577_Int__emptyI,axiom,
    ! [A2: set_fu3555266933746557007_nat_o,B2: set_fu3555266933746557007_nat_o] :
      ( ! [X2: fun_Pr6460944699146269935_nat_o] :
          ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X2),A2)
         => ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X2),B2) )
     => ( inf_in3376754634429691197_nat_o(A2,B2) = bot_bo8632255151191677115_nat_o ) ) ).

% Int_emptyI
tff(fact_578_Int__emptyI,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ! [X2: product_prod_nat_nat] :
          ( member8440522571783428010at_nat(X2,A2)
         => ~ member8440522571783428010at_nat(X2,B2) )
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2) = bot_bo2099793752762293965at_nat ) ) ).

% Int_emptyI
tff(fact_579_Int__emptyI,axiom,
    ! [A2: set_o,B2: set_o] :
      ( ! [X2: $o] :
          ( aa_set_o_o(member_o((X2)),A2)
         => ~ aa_set_o_o(member_o((X2)),B2) )
     => ( inf_inf_set_o(A2,B2) = bot_bot_set_o ) ) ).

% Int_emptyI
tff(fact_580_Int__emptyI,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(member_nat2(X2),A2)
         => ~ aa_set_nat_o(member_nat2(X2),B2) )
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) = bot_bot_set_nat ) ) ).

% Int_emptyI
tff(fact_581_disjoint__iff,axiom,
    ! [A2: set_fu3555266933746557007_nat_o,B2: set_fu3555266933746557007_nat_o] :
      ( ( inf_in3376754634429691197_nat_o(A2,B2) = bot_bo8632255151191677115_nat_o )
    <=> ! [X3: fun_Pr6460944699146269935_nat_o] :
          ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X3),A2)
         => ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X3),B2) ) ) ).

% disjoint_iff
tff(fact_582_disjoint__iff,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2) = bot_bo2099793752762293965at_nat )
    <=> ! [X3: product_prod_nat_nat] :
          ( member8440522571783428010at_nat(X3,A2)
         => ~ member8440522571783428010at_nat(X3,B2) ) ) ).

% disjoint_iff
tff(fact_583_disjoint__iff,axiom,
    ! [A2: set_o,B2: set_o] :
      ( ( inf_inf_set_o(A2,B2) = bot_bot_set_o )
    <=> ! [X3: $o] :
          ( aa_set_o_o(member_o((X3)),A2)
         => ~ aa_set_o_o(member_o((X3)),B2) ) ) ).

% disjoint_iff
tff(fact_584_disjoint__iff,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) = bot_bot_set_nat )
    <=> ! [X3: nat] :
          ( aa_set_nat_o(member_nat2(X3),A2)
         => ~ aa_set_nat_o(member_nat2(X3),B2) ) ) ).

% disjoint_iff
tff(fact_585_Int__empty__left,axiom,
    ! [B2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,bot_bo2099793752762293965at_nat),B2) = bot_bo2099793752762293965at_nat ).

% Int_empty_left
tff(fact_586_Int__empty__left,axiom,
    ! [B2: set_o] : inf_inf_set_o(bot_bot_set_o,B2) = bot_bot_set_o ).

% Int_empty_left
tff(fact_587_Int__empty__left,axiom,
    ! [B2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,bot_bot_set_nat),B2) = bot_bot_set_nat ).

% Int_empty_left
tff(fact_588_Int__empty__right,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),bot_bo2099793752762293965at_nat) = bot_bo2099793752762293965at_nat ).

% Int_empty_right
tff(fact_589_Int__empty__right,axiom,
    ! [A2: set_o] : inf_inf_set_o(A2,bot_bot_set_o) = bot_bot_set_o ).

% Int_empty_right
tff(fact_590_Int__empty__right,axiom,
    ! [A2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),bot_bot_set_nat) = bot_bot_set_nat ).

% Int_empty_right
tff(fact_591_disjoint__iff__not__equal,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2) = bot_bo2099793752762293965at_nat )
    <=> ! [X3: product_prod_nat_nat] :
          ( member8440522571783428010at_nat(X3,A2)
         => ! [Xa2: product_prod_nat_nat] :
              ( member8440522571783428010at_nat(Xa2,B2)
             => ( X3 != Xa2 ) ) ) ) ).

% disjoint_iff_not_equal
tff(fact_592_disjoint__iff__not__equal,axiom,
    ! [A2: set_o,B2: set_o] :
      ( ( inf_inf_set_o(A2,B2) = bot_bot_set_o )
    <=> ! [X3: $o] :
          ( aa_set_o_o(member_o((X3)),A2)
         => ! [Xa2: $o] :
              ( aa_set_o_o(member_o((Xa2)),B2)
             => ( (X3)
              <=> ~ (Xa2) ) ) ) ) ).

% disjoint_iff_not_equal
tff(fact_593_disjoint__iff__not__equal,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) = bot_bot_set_nat )
    <=> ! [X3: nat] :
          ( aa_set_nat_o(member_nat2(X3),A2)
         => ! [Xa2: nat] :
              ( aa_set_nat_o(member_nat2(Xa2),B2)
             => ( X3 != Xa2 ) ) ) ) ).

% disjoint_iff_not_equal
tff(fact_594_disjointI,axiom,
    ! [A: set_fu3555266933746557007_nat_o,B: set_fu3555266933746557007_nat_o] :
      ( ! [X2: fun_Pr6460944699146269935_nat_o] :
          ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X2),A)
         => ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X2),B) )
     => ( inf_in3376754634429691197_nat_o(A,B) = bot_bo8632255151191677115_nat_o ) ) ).

% disjointI
tff(fact_595_disjointI,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ! [X2: product_prod_nat_nat] :
          ( member8440522571783428010at_nat(X2,A)
         => ~ member8440522571783428010at_nat(X2,B) )
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) = bot_bo2099793752762293965at_nat ) ) ).

% disjointI
tff(fact_596_disjointI,axiom,
    ! [A: set_o,B: set_o] :
      ( ! [X2: $o] :
          ( aa_set_o_o(member_o((X2)),A)
         => ~ aa_set_o_o(member_o((X2)),B) )
     => ( inf_inf_set_o(A,B) = bot_bot_set_o ) ) ).

% disjointI
tff(fact_597_disjointI,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(member_nat2(X2),A)
         => ~ aa_set_nat_o(member_nat2(X2),B) )
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) = bot_bot_set_nat ) ) ).

% disjointI
tff(fact_598_emptyE,axiom,
    ! [A: fun_Pr6460944699146269935_nat_o] : ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A),bot_bo8632255151191677115_nat_o) ).

% emptyE
tff(fact_599_emptyE,axiom,
    ! [A: $o] : ~ aa_set_o_o(member_o((A)),bot_bot_set_o) ).

% emptyE
tff(fact_600_emptyE,axiom,
    ! [A: nat] : ~ aa_set_nat_o(member_nat2(A),bot_bot_set_nat) ).

% emptyE
tff(fact_601_equals0D,axiom,
    ! [A2: set_fu3555266933746557007_nat_o,A: fun_Pr6460944699146269935_nat_o] :
      ( ( A2 = bot_bo8632255151191677115_nat_o )
     => ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A),A2) ) ).

% equals0D
tff(fact_602_equals0D,axiom,
    ! [A2: set_o,A: $o] :
      ( ( A2 = bot_bot_set_o )
     => ~ aa_set_o_o(member_o((A)),A2) ) ).

% equals0D
tff(fact_603_equals0D,axiom,
    ! [A2: set_nat,A: nat] :
      ( ( A2 = bot_bot_set_nat )
     => ~ aa_set_nat_o(member_nat2(A),A2) ) ).

% equals0D
tff(fact_604_equals0I,axiom,
    ! [A2: set_fu3555266933746557007_nat_o] :
      ( ! [Y2: fun_Pr6460944699146269935_nat_o] : ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Y2),A2)
     => ( A2 = bot_bo8632255151191677115_nat_o ) ) ).

% equals0I
tff(fact_605_equals0I,axiom,
    ! [A2: set_o] :
      ( ! [Y2: $o] : ~ aa_set_o_o(member_o((Y2)),A2)
     => ( A2 = bot_bot_set_o ) ) ).

% equals0I
tff(fact_606_equals0I,axiom,
    ! [A2: set_nat] :
      ( ! [Y2: nat] : ~ aa_set_nat_o(member_nat2(Y2),A2)
     => ( A2 = bot_bot_set_nat ) ) ).

% equals0I
tff(fact_607_ex__in__conv,axiom,
    ! [A2: set_fu3555266933746557007_nat_o] :
      ( ? [X3: fun_Pr6460944699146269935_nat_o] : aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X3),A2)
    <=> ( A2 != bot_bo8632255151191677115_nat_o ) ) ).

% ex_in_conv
tff(fact_608_ex__in__conv,axiom,
    ! [A2: set_o] :
      ( ? [X3: $o] : aa_set_o_o(member_o((X3)),A2)
    <=> ( A2 != bot_bot_set_o ) ) ).

% ex_in_conv
tff(fact_609_ex__in__conv,axiom,
    ! [A2: set_nat] :
      ( ? [X3: nat] : aa_set_nat_o(member_nat2(X3),A2)
    <=> ( A2 != bot_bot_set_nat ) ) ).

% ex_in_conv
tff(fact_610_bot__set__def,axiom,
    bot_bo8632255151191677115_nat_o = collec6407654713642994138_nat_o(bot_bo8793957029302321216at_o_o) ).

% bot_set_def
tff(fact_611_bot__set__def,axiom,
    bot_bo2099793752762293965at_nat = collec3392354462482085612at_nat(bot_bo5074786513356115630_nat_o) ).

% bot_set_def
tff(fact_612_bot__set__def,axiom,
    bot_bot_set_o = collect_o(bot_bot_fun_o_o) ).

% bot_set_def
tff(fact_613_bot__set__def,axiom,
    bot_bot_set_nat = collect_nat(bot_bot_fun_nat_o) ).

% bot_set_def
tff(fact_614_inf__sup__aci_I4_J,axiom,
    ! [X: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y) ).

% inf_sup_aci(4)
tff(fact_615_inf__sup__aci_I4_J,axiom,
    ! [X: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y) ).

% inf_sup_aci(4)
tff(fact_616_inf__sup__aci_I4_J,axiom,
    ! [X: set_nat,Y: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y) ).

% inf_sup_aci(4)
tff(fact_617_inf__sup__aci_I4_J,axiom,
    ! [X: product_unit,Y: product_unit] : aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),Y)) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),Y) ).

% inf_sup_aci(4)
tff(fact_618_inf__sup__aci_I4_J,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y) ).

% inf_sup_aci(4)
tff(fact_619_inf__sup__aci_I3_J,axiom,
    ! [X: assn,Y: assn,Z: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),Z)) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Z)) ).

% inf_sup_aci(3)
tff(fact_620_inf__sup__aci_I3_J,axiom,
    ! [X: nat,Y: nat,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Y),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Z)) ).

% inf_sup_aci(3)
tff(fact_621_inf__sup__aci_I3_J,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),Z)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Z)) ).

% inf_sup_aci(3)
tff(fact_622_inf__sup__aci_I3_J,axiom,
    ! [X: product_unit,Y: product_unit,Z: product_unit] : aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,Y),Z)) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,Y),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),Z)) ).

% inf_sup_aci(3)
tff(fact_623_inf__sup__aci_I3_J,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y),Z)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Z)) ).

% inf_sup_aci(3)
tff(fact_624_inf__sup__aci_I2_J,axiom,
    ! [X: assn,Y: assn,Z: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)),Z) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),Z)) ).

% inf_sup_aci(2)
tff(fact_625_inf__sup__aci_I2_J,axiom,
    ! [X: nat,Y: nat,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y)),Z) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Y),Z)) ).

% inf_sup_aci(2)
tff(fact_626_inf__sup__aci_I2_J,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)),Z) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),Z)) ).

% inf_sup_aci(2)
tff(fact_627_inf__sup__aci_I2_J,axiom,
    ! [X: product_unit,Y: product_unit,Z: product_unit] : aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),Y)),Z) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,Y),Z)) ).

% inf_sup_aci(2)
tff(fact_628_inf__sup__aci_I2_J,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y)),Z) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y),Z)) ).

% inf_sup_aci(2)
tff(fact_629_inf__sup__aci_I1_J,axiom,
    ! [X: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),X) ).

% inf_sup_aci(1)
tff(fact_630_inf__sup__aci_I1_J,axiom,
    ! [X: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Y),X) ).

% inf_sup_aci(1)
tff(fact_631_inf__sup__aci_I1_J,axiom,
    ! [X: set_nat,Y: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),X) ).

% inf_sup_aci(1)
tff(fact_632_inf__sup__aci_I1_J,axiom,
    ! [X: product_unit,Y: product_unit] : aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),Y) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,Y),X) ).

% inf_sup_aci(1)
tff(fact_633_inf__sup__aci_I1_J,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y),X) ).

% inf_sup_aci(1)
tff(fact_634_inf_Oassoc,axiom,
    ! [A: assn,B: assn,C2: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B)),C2) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,B),C2)) ).

% inf.assoc
tff(fact_635_inf_Oassoc,axiom,
    ! [A: nat,B: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,B),C2)) ).

% inf.assoc
tff(fact_636_inf_Oassoc,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)),C2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B),C2)) ).

% inf.assoc
tff(fact_637_inf_Oassoc,axiom,
    ! [A: product_unit,B: product_unit,C2: product_unit] : aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,A),B)),C2) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,A),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,B),C2)) ).

% inf.assoc
tff(fact_638_inf_Oassoc,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)),C2) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B),C2)) ).

% inf.assoc
tff(fact_639_inf__assoc,axiom,
    ! [X: assn,Y: assn,Z: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)),Z) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),Z)) ).

% inf_assoc
tff(fact_640_inf__assoc,axiom,
    ! [X: nat,Y: nat,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y)),Z) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Y),Z)) ).

% inf_assoc
tff(fact_641_inf__assoc,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)),Z) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),Z)) ).

% inf_assoc
tff(fact_642_inf__assoc,axiom,
    ! [X: product_unit,Y: product_unit,Z: product_unit] : aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),Y)),Z) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,Y),Z)) ).

% inf_assoc
tff(fact_643_inf__assoc,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y)),Z) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y),Z)) ).

% inf_assoc
tff(fact_644_inf_Ocommute,axiom,
    ! [A: assn,B: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,B),A) ).

% inf.commute
tff(fact_645_inf_Ocommute,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,B),A) ).

% inf.commute
tff(fact_646_inf_Ocommute,axiom,
    ! [A: set_nat,B: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B),A) ).

% inf.commute
tff(fact_647_inf_Ocommute,axiom,
    ! [A: product_unit,B: product_unit] : aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,A),B) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,B),A) ).

% inf.commute
tff(fact_648_inf_Ocommute,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B),A) ).

% inf.commute
tff(fact_649_inf__commute,axiom,
    ! [X: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),X) ).

% inf_commute
tff(fact_650_inf__commute,axiom,
    ! [X: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Y),X) ).

% inf_commute
tff(fact_651_inf__commute,axiom,
    ! [X: set_nat,Y: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),X) ).

% inf_commute
tff(fact_652_inf__commute,axiom,
    ! [X: product_unit,Y: product_unit] : aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),Y) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,Y),X) ).

% inf_commute
tff(fact_653_inf__commute,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y),X) ).

% inf_commute
tff(fact_654_boolean__algebra__cancel_Oinf1,axiom,
    ! [A2: assn,K: assn,A: assn,B: assn] :
      ( ( A2 = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,K),A) )
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A2),B) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,K),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B)) ) ) ).

% boolean_algebra_cancel.inf1
tff(fact_655_boolean__algebra__cancel_Oinf1,axiom,
    ! [A2: nat,K: nat,A: nat,B: nat] :
      ( ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,K),A) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A2),B) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)) ) ) ).

% boolean_algebra_cancel.inf1
tff(fact_656_boolean__algebra__cancel_Oinf1,axiom,
    ! [A2: set_nat,K: set_nat,A: set_nat,B: set_nat] :
      ( ( A2 = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,K),A) )
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,K),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)) ) ) ).

% boolean_algebra_cancel.inf1
tff(fact_657_boolean__algebra__cancel_Oinf1,axiom,
    ! [A2: product_unit,K: product_unit,A: product_unit,B: product_unit] :
      ( ( A2 = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,K),A) )
     => ( aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,A2),B) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,K),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,A),B)) ) ) ).

% boolean_algebra_cancel.inf1
tff(fact_658_boolean__algebra__cancel_Oinf1,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,K: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( A2 = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,K),A) )
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,K),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)) ) ) ).

% boolean_algebra_cancel.inf1
tff(fact_659_boolean__algebra__cancel_Oinf2,axiom,
    ! [B2: assn,K: assn,B: assn,A: assn] :
      ( ( B2 = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,K),B) )
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B2) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,K),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B)) ) ) ).

% boolean_algebra_cancel.inf2
tff(fact_660_boolean__algebra__cancel_Oinf2,axiom,
    ! [B2: nat,K: nat,B: nat,A: nat] :
      ( ( B2 = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,K),B) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B2) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)) ) ) ).

% boolean_algebra_cancel.inf2
tff(fact_661_boolean__algebra__cancel_Oinf2,axiom,
    ! [B2: set_nat,K: set_nat,B: set_nat,A: set_nat] :
      ( ( B2 = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,K),B) )
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,K),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)) ) ) ).

% boolean_algebra_cancel.inf2
tff(fact_662_boolean__algebra__cancel_Oinf2,axiom,
    ! [B2: product_unit,K: product_unit,B: product_unit,A: product_unit] :
      ( ( B2 = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,K),B) )
     => ( aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,A),B2) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,K),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,A),B)) ) ) ).

% boolean_algebra_cancel.inf2
tff(fact_663_boolean__algebra__cancel_Oinf2,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,K: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( B2 = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,K),B) )
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B2) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,K),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)) ) ) ).

% boolean_algebra_cancel.inf2
tff(fact_664_inf_Oleft__commute,axiom,
    ! [B: assn,A: assn,C2: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,B),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),C2)) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,B),C2)) ).

% inf.left_commute
tff(fact_665_inf_Oleft__commute,axiom,
    ! [B: nat,A: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,B),C2)) ).

% inf.left_commute
tff(fact_666_inf_Oleft__commute,axiom,
    ! [B: set_nat,A: set_nat,C2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),C2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B),C2)) ).

% inf.left_commute
tff(fact_667_inf_Oleft__commute,axiom,
    ! [B: product_unit,A: product_unit,C2: product_unit] : aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,B),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,A),C2)) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,A),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,B),C2)) ).

% inf.left_commute
tff(fact_668_inf_Oleft__commute,axiom,
    ! [B: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),C2)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B),C2)) ).

% inf.left_commute
tff(fact_669_inf__left__commute,axiom,
    ! [X: assn,Y: assn,Z: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),Z)) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Z)) ).

% inf_left_commute
tff(fact_670_inf__left__commute,axiom,
    ! [X: nat,Y: nat,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Y),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Z)) ).

% inf_left_commute
tff(fact_671_inf__left__commute,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),Z)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Z)) ).

% inf_left_commute
tff(fact_672_inf__left__commute,axiom,
    ! [X: product_unit,Y: product_unit,Z: product_unit] : aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,Y),Z)) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,Y),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),Z)) ).

% inf_left_commute
tff(fact_673_inf__left__commute,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y),Z)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Z)) ).

% inf_left_commute
tff(fact_674_FI__finalize,axiom,
    ! [M2: list_P8527749157015355191n_assn,P3: assn,Up: assn,Q2: assn,Uq: assn,F: assn] :
      ( aa_assn_o(fI_RESULT(M2,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P3),Up),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Q2),Uq)),F)
     => aa_assn_o(fi(M2,P3,Q2,Up,Uq),F) ) ).

% FI_finalize
tff(fact_675_Set_Ois__empty__def,axiom,
    ! [A2: set_o] :
      ( aa_set_o_o(is_empty_o,A2)
    <=> ( A2 = bot_bot_set_o ) ) ).

% Set.is_empty_def
tff(fact_676_Set_Ois__empty__def,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(is_empty_nat,A2)
    <=> ( A2 = bot_bot_set_nat ) ) ).

% Set.is_empty_def
tff(fact_677_map__eq__map__tailrec,axiom,
    map_nat_nat = map_tailrec_nat_nat ).

% map_eq_map_tailrec
tff(fact_678_map__eq__map__tailrec,axiom,
    map_Pr8991440229025900053n_assn = map_ta2194206859901583947n_assn ).

% map_eq_map_tailrec
tff(fact_679_list__collect__set__map__simps_I3_J,axiom,
    ! [F: fun_nat_set_nat,X: fun_nat_nat,A: nat,La: list_nat] : list_c2452340269597857392at_nat(F,aa_list_nat_list_nat(aa_fun4736978287202700568st_nat(map_nat_nat,X),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,A),La))) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_nat_set_nat(F,aa_nat_nat(X,A))),list_c2452340269597857392at_nat(F,aa_list_nat_list_nat(aa_fun4736978287202700568st_nat(map_nat_nat,X),La))) ).

% list_collect_set_map_simps(3)
tff(fact_680_list__collect__set__map__simps_I3_J,axiom,
    ! [F: fun_assn_set_nat,X: fun_Pr3816910165196019820n_assn,A: produc6575502325842934193n_assn,La: list_P8527749157015355191n_assn] : list_c1844713377658005960sn_nat(F,aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,X),aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,A),La))) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_assn_set_nat(F,aa_Pro2979818503705714077n_assn(X,A))),list_c1844713377658005960sn_nat(F,aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,X),La))) ).

% list_collect_set_map_simps(3)
tff(fact_681_sorted__list__of__set_Osorted__key__list__of__set__empty,axiom,
    linord2612477271533052124et_int(bot_bot_set_int) = nil_int ).

% sorted_list_of_set.sorted_key_list_of_set_empty
tff(fact_682_sorted__list__of__set_Osorted__key__list__of__set__empty,axiom,
    linord3142498349692569832_set_o(bot_bot_set_o) = nil_o ).

% sorted_list_of_set.sorted_key_list_of_set_empty
tff(fact_683_sorted__list__of__set_Osorted__key__list__of__set__empty,axiom,
    linord2614967742042102400et_nat(bot_bot_set_nat) = nil_nat ).

% sorted_list_of_set.sorted_key_list_of_set_empty
tff(fact_684_lexordp__eq_Ocases,axiom,
    ! [A1: list_assn,A22: list_assn] :
      ( ord_lexordp_eq_assn(A1,A22)
     => ( ( A1 != nil_assn )
       => ( ! [X2: assn] :
              ( ? [Xs2: list_assn] : A1 = cons_assn(X2,Xs2)
             => ! [Y2: assn] :
                  ( ? [Ys2: list_assn] : A22 = cons_assn(Y2,Ys2)
                 => ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X2),Y2) ) )
         => ~ ! [X2: assn,Y2: assn,Xs2: list_assn] :
                ( ( A1 = cons_assn(X2,Xs2) )
               => ! [Ys2: list_assn] :
                    ( ( A22 = cons_assn(Y2,Ys2) )
                   => ( ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X2),Y2)
                     => ( ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,Y2),X2)
                       => ~ ord_lexordp_eq_assn(Xs2,Ys2) ) ) ) ) ) ) ) ).

% lexordp_eq.cases
tff(fact_685_lexordp__eq_Ocases,axiom,
    ! [A1: list_nat,A22: list_nat] :
      ( aa_list_nat_o(aa_lis557685189327716851_nat_o(ord_lexordp_eq_nat,A1),A22)
     => ( ( A1 != nil_nat )
       => ( ! [X2: nat] :
              ( ? [Xs2: list_nat] : A1 = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X2),Xs2)
             => ! [Y2: nat] :
                  ( ? [Ys2: list_nat] : A22 = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Y2),Ys2)
                 => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Y2) ) )
         => ~ ! [X2: nat,Y2: nat,Xs2: list_nat] :
                ( ( A1 = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X2),Xs2) )
               => ! [Ys2: list_nat] :
                    ( ( A22 = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Y2),Ys2) )
                   => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Y2)
                     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y2),X2)
                       => ~ aa_list_nat_o(aa_lis557685189327716851_nat_o(ord_lexordp_eq_nat,Xs2),Ys2) ) ) ) ) ) ) ) ).

% lexordp_eq.cases
tff(fact_686_lexordp__eq_Ocases,axiom,
    ! [A1: list_int,A22: list_int] :
      ( aa_list_int_o(aa_lis7773489400162621755_int_o(ord_lexordp_eq_int,A1),A22)
     => ( ( A1 != nil_int )
       => ( ! [X2: int] :
              ( ? [Xs2: list_int] : A1 = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X2),Xs2)
             => ! [Y2: int] :
                  ( ? [Ys2: list_int] : A22 = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Y2),Ys2)
                 => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Y2) ) )
         => ~ ! [X2: int,Y2: int,Xs2: list_int] :
                ( ( A1 = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X2),Xs2) )
               => ! [Ys2: list_int] :
                    ( ( A22 = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Y2),Ys2) )
                   => ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Y2)
                     => ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Y2),X2)
                       => ~ aa_list_int_o(aa_lis7773489400162621755_int_o(ord_lexordp_eq_int,Xs2),Ys2) ) ) ) ) ) ) ) ).

% lexordp_eq.cases
tff(fact_687_lexordp__eq_Osimps,axiom,
    ! [A1: list_assn,A22: list_assn] :
      ( ord_lexordp_eq_assn(A1,A22)
    <=> ( ? [Ys3: list_assn] :
            ( ( A1 = nil_assn )
            & ( A22 = Ys3 ) )
        | ? [X3: assn,Y3: assn,Xs3: list_assn,Ys3: list_assn] :
            ( ( A1 = cons_assn(X3,Xs3) )
            & ( A22 = cons_assn(Y3,Ys3) )
            & aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X3),Y3) )
        | ? [X3: assn,Y3: assn,Xs3: list_assn,Ys3: list_assn] :
            ( ( A1 = cons_assn(X3,Xs3) )
            & ( A22 = cons_assn(Y3,Ys3) )
            & ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X3),Y3)
            & ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,Y3),X3)
            & ord_lexordp_eq_assn(Xs3,Ys3) ) ) ) ).

% lexordp_eq.simps
tff(fact_688_lexordp__eq_Osimps,axiom,
    ! [A1: list_nat,A22: list_nat] :
      ( aa_list_nat_o(aa_lis557685189327716851_nat_o(ord_lexordp_eq_nat,A1),A22)
    <=> ( ? [Ys3: list_nat] :
            ( ( A1 = nil_nat )
            & ( A22 = Ys3 ) )
        | ? [X3: nat,Y3: nat,Xs3: list_nat,Ys3: list_nat] :
            ( ( A1 = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X3),Xs3) )
            & ( A22 = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Y3),Ys3) )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y3) )
        | ? [X3: nat,Y3: nat,Xs3: list_nat,Ys3: list_nat] :
            ( ( A1 = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X3),Xs3) )
            & ( A22 = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Y3),Ys3) )
            & ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y3)
            & ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y3),X3)
            & aa_list_nat_o(aa_lis557685189327716851_nat_o(ord_lexordp_eq_nat,Xs3),Ys3) ) ) ) ).

% lexordp_eq.simps
tff(fact_689_lexordp__eq_Osimps,axiom,
    ! [A1: list_int,A22: list_int] :
      ( aa_list_int_o(aa_lis7773489400162621755_int_o(ord_lexordp_eq_int,A1),A22)
    <=> ( ? [Ys3: list_int] :
            ( ( A1 = nil_int )
            & ( A22 = Ys3 ) )
        | ? [X3: int,Y3: int,Xs3: list_int,Ys3: list_int] :
            ( ( A1 = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X3),Xs3) )
            & ( A22 = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Y3),Ys3) )
            & aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Y3) )
        | ? [X3: int,Y3: int,Xs3: list_int,Ys3: list_int] :
            ( ( A1 = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X3),Xs3) )
            & ( A22 = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Y3),Ys3) )
            & ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Y3)
            & ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Y3),X3)
            & aa_list_int_o(aa_lis7773489400162621755_int_o(ord_lexordp_eq_int,Xs3),Ys3) ) ) ) ).

% lexordp_eq.simps
tff(fact_690_bot__empty__eq,axiom,
    ! [X4: fun_Pr6460944699146269935_nat_o] :
      ( aa_fun1675921005893230629at_o_o(bot_bo8793957029302321216at_o_o,X4)
    <=> aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X4),bot_bo8632255151191677115_nat_o) ) ).

% bot_empty_eq
tff(fact_691_bot__empty__eq,axiom,
    ! [X4: $o] :
      ( aa_o_o(bot_bot_fun_o_o,(X4))
    <=> aa_set_o_o(member_o((X4)),bot_bot_set_o) ) ).

% bot_empty_eq
tff(fact_692_bot__empty__eq,axiom,
    ! [X4: nat] :
      ( aa_nat_o(bot_bot_fun_nat_o,X4)
    <=> aa_set_nat_o(member_nat2(X4),bot_bot_set_nat) ) ).

% bot_empty_eq
tff(fact_693_Collect__empty__eq__bot,axiom,
    ! [Pa: fun_fu5135969500543783852at_o_o] :
      ( ( collec6407654713642994138_nat_o(Pa) = bot_bo8632255151191677115_nat_o )
    <=> ( Pa = bot_bo8793957029302321216at_o_o ) ) ).

% Collect_empty_eq_bot
tff(fact_694_Collect__empty__eq__bot,axiom,
    ! [Pa: fun_Pr4796943543331071514_nat_o] :
      ( ( collec3392354462482085612at_nat(Pa) = bot_bo2099793752762293965at_nat )
    <=> ( Pa = bot_bo5074786513356115630_nat_o ) ) ).

% Collect_empty_eq_bot
tff(fact_695_Collect__empty__eq__bot,axiom,
    ! [Pa: fun_o_o] :
      ( ( collect_o(Pa) = bot_bot_set_o )
    <=> ( Pa = bot_bot_fun_o_o ) ) ).

% Collect_empty_eq_bot
tff(fact_696_Collect__empty__eq__bot,axiom,
    ! [Pa: fun_nat_o] :
      ( ( collect_nat(Pa) = bot_bot_set_nat )
    <=> ( Pa = bot_bot_fun_nat_o ) ) ).

% Collect_empty_eq_bot
tff(fact_697_mult_Osafe__commute,axiom,
    ! [X: nat,Y: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(syntax4682126007086162916at_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X),Y)),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A) ) ) ).

% mult.safe_commute
tff(fact_698_mult_Osafe__commute,axiom,
    ! [X: assn,Y: assn,A: assn,B: assn] :
      ( aa_assn_o(syntax7398250324933576852n_assn(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,X),Y)),A)
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),B) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B),A) ) ) ).

% mult.safe_commute
tff(fact_699_sup_Oidem,axiom,
    ! [A: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),A) = A ).

% sup.idem
tff(fact_700_sup_Oidem,axiom,
    ! [A: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),A) = A ).

% sup.idem
tff(fact_701_sup__idem,axiom,
    ! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),X) = X ).

% sup_idem
tff(fact_702_sup__idem,axiom,
    ! [X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),X) = X ).

% sup_idem
tff(fact_703_sup_Oleft__idem,axiom,
    ! [A: assn,B: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B)) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B) ).

% sup.left_idem
tff(fact_704_sup_Oleft__idem,axiom,
    ! [A: set_nat,B: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) ).

% sup.left_idem
tff(fact_705_sup__left__idem,axiom,
    ! [X: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y)) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y) ).

% sup_left_idem
tff(fact_706_sup__left__idem,axiom,
    ! [X: set_nat,Y: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y) ).

% sup_left_idem
tff(fact_707_sup_Oright__idem,axiom,
    ! [A: assn,B: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B)),B) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B) ).

% sup.right_idem
tff(fact_708_sup_Oright__idem,axiom,
    ! [A: set_nat,B: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)),B) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) ).

% sup.right_idem
tff(fact_709_Int__iff,axiom,
    ! [C2: fun_Pr6460944699146269935_nat_o,A2: set_fu3555266933746557007_nat_o,B2: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C2),inf_in3376754634429691197_nat_o(A2,B2))
    <=> ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C2),A2)
        & aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C2),B2) ) ) ).

% Int_iff
tff(fact_710_Int__iff,axiom,
    ! [C2: nat,A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(member_nat2(C2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2))
    <=> ( aa_set_nat_o(member_nat2(C2),A2)
        & aa_set_nat_o(member_nat2(C2),B2) ) ) ).

% Int_iff
tff(fact_711_Int__iff,axiom,
    ! [C2: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( member8440522571783428010at_nat(C2,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2))
    <=> ( member8440522571783428010at_nat(C2,A2)
        & member8440522571783428010at_nat(C2,B2) ) ) ).

% Int_iff
tff(fact_712_IntI,axiom,
    ! [C2: fun_Pr6460944699146269935_nat_o,A2: set_fu3555266933746557007_nat_o,B2: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C2),A2)
     => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C2),B2)
       => aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C2),inf_in3376754634429691197_nat_o(A2,B2)) ) ) ).

% IntI
tff(fact_713_IntI,axiom,
    ! [C2: nat,A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(member_nat2(C2),A2)
     => ( aa_set_nat_o(member_nat2(C2),B2)
       => aa_set_nat_o(member_nat2(C2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2)) ) ) ).

% IntI
tff(fact_714_IntI,axiom,
    ! [C2: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( member8440522571783428010at_nat(C2,A2)
     => ( member8440522571783428010at_nat(C2,B2)
       => member8440522571783428010at_nat(C2,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2)) ) ) ).

% IntI
tff(fact_715_UnCI,axiom,
    ! [C2: fun_Pr6460944699146269935_nat_o,B2: set_fu3555266933746557007_nat_o,A2: set_fu3555266933746557007_nat_o] :
      ( ( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C2),B2)
       => aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C2),A2) )
     => aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C2),sup_su2896084399781877795_nat_o(A2,B2)) ) ).

% UnCI
tff(fact_716_UnCI,axiom,
    ! [C2: nat,B2: set_nat,A2: set_nat] :
      ( ( ~ aa_set_nat_o(member_nat2(C2),B2)
       => aa_set_nat_o(member_nat2(C2),A2) )
     => aa_set_nat_o(member_nat2(C2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)) ) ).

% UnCI
tff(fact_717_Un__iff,axiom,
    ! [C2: fun_Pr6460944699146269935_nat_o,A2: set_fu3555266933746557007_nat_o,B2: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C2),sup_su2896084399781877795_nat_o(A2,B2))
    <=> ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C2),A2)
        | aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C2),B2) ) ) ).

% Un_iff
tff(fact_718_Un__iff,axiom,
    ! [C2: nat,A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(member_nat2(C2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2))
    <=> ( aa_set_nat_o(member_nat2(C2),A2)
        | aa_set_nat_o(member_nat2(C2),B2) ) ) ).

% Un_iff
tff(fact_719_sup__bot_Oright__neutral,axiom,
    ! [A: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),bot_bot_assn) = A ).

% sup_bot.right_neutral
tff(fact_720_sup__bot_Oright__neutral,axiom,
    ! [A: set_o] : sup_sup_set_o(A,bot_bot_set_o) = A ).

% sup_bot.right_neutral
tff(fact_721_sup__bot_Oright__neutral,axiom,
    ! [A: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),bot_bot_set_nat) = A ).

% sup_bot.right_neutral
tff(fact_722_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: assn,B: assn] :
      ( ( bot_bot_assn = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B) )
    <=> ( ( A = bot_bot_assn )
        & ( B = bot_bot_assn ) ) ) ).

% sup_bot.neutr_eq_iff
tff(fact_723_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: set_o,B: set_o] :
      ( ( bot_bot_set_o = sup_sup_set_o(A,B) )
    <=> ( ( A = bot_bot_set_o )
        & ( B = bot_bot_set_o ) ) ) ).

% sup_bot.neutr_eq_iff
tff(fact_724_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( bot_bot_set_nat = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) )
    <=> ( ( A = bot_bot_set_nat )
        & ( B = bot_bot_set_nat ) ) ) ).

% sup_bot.neutr_eq_iff
tff(fact_725_sup__bot_Oleft__neutral,axiom,
    ! [A: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,bot_bot_assn),A) = A ).

% sup_bot.left_neutral
tff(fact_726_sup__bot_Oleft__neutral,axiom,
    ! [A: set_o] : sup_sup_set_o(bot_bot_set_o,A) = A ).

% sup_bot.left_neutral
tff(fact_727_sup__bot_Oleft__neutral,axiom,
    ! [A: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,bot_bot_set_nat),A) = A ).

% sup_bot.left_neutral
tff(fact_728_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: assn,B: assn] :
      ( ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B) = bot_bot_assn )
    <=> ( ( A = bot_bot_assn )
        & ( B = bot_bot_assn ) ) ) ).

% sup_bot.eq_neutr_iff
tff(fact_729_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: set_o,B: set_o] :
      ( ( sup_sup_set_o(A,B) = bot_bot_set_o )
    <=> ( ( A = bot_bot_set_o )
        & ( B = bot_bot_set_o ) ) ) ).

% sup_bot.eq_neutr_iff
tff(fact_730_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) = bot_bot_set_nat )
    <=> ( ( A = bot_bot_set_nat )
        & ( B = bot_bot_set_nat ) ) ) ).

% sup_bot.eq_neutr_iff
tff(fact_731_sup__eq__bot__iff,axiom,
    ! [X: assn,Y: assn] :
      ( ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y) = bot_bot_assn )
    <=> ( ( X = bot_bot_assn )
        & ( Y = bot_bot_assn ) ) ) ).

% sup_eq_bot_iff
tff(fact_732_sup__eq__bot__iff,axiom,
    ! [X: set_o,Y: set_o] :
      ( ( sup_sup_set_o(X,Y) = bot_bot_set_o )
    <=> ( ( X = bot_bot_set_o )
        & ( Y = bot_bot_set_o ) ) ) ).

% sup_eq_bot_iff
tff(fact_733_sup__eq__bot__iff,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y) = bot_bot_set_nat )
    <=> ( ( X = bot_bot_set_nat )
        & ( Y = bot_bot_set_nat ) ) ) ).

% sup_eq_bot_iff
tff(fact_734_bot__eq__sup__iff,axiom,
    ! [X: assn,Y: assn] :
      ( ( bot_bot_assn = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y) )
    <=> ( ( X = bot_bot_assn )
        & ( Y = bot_bot_assn ) ) ) ).

% bot_eq_sup_iff
tff(fact_735_bot__eq__sup__iff,axiom,
    ! [X: set_o,Y: set_o] :
      ( ( bot_bot_set_o = sup_sup_set_o(X,Y) )
    <=> ( ( X = bot_bot_set_o )
        & ( Y = bot_bot_set_o ) ) ) ).

% bot_eq_sup_iff
tff(fact_736_bot__eq__sup__iff,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( bot_bot_set_nat = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y) )
    <=> ( ( X = bot_bot_set_nat )
        & ( Y = bot_bot_set_nat ) ) ) ).

% bot_eq_sup_iff
tff(fact_737_sup__bot__right,axiom,
    ! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),bot_bot_assn) = X ).

% sup_bot_right
tff(fact_738_sup__bot__right,axiom,
    ! [X: set_o] : sup_sup_set_o(X,bot_bot_set_o) = X ).

% sup_bot_right
tff(fact_739_sup__bot__right,axiom,
    ! [X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),bot_bot_set_nat) = X ).

% sup_bot_right
tff(fact_740_sup__bot__left,axiom,
    ! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,bot_bot_assn),X) = X ).

% sup_bot_left
tff(fact_741_sup__bot__left,axiom,
    ! [X: set_o] : sup_sup_set_o(bot_bot_set_o,X) = X ).

% sup_bot_left
tff(fact_742_sup__bot__left,axiom,
    ! [X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,bot_bot_set_nat),X) = X ).

% sup_bot_left
tff(fact_743_sup__inf__absorb,axiom,
    ! [X: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y)) = X ).

% sup_inf_absorb
tff(fact_744_sup__inf__absorb,axiom,
    ! [X: product_unit,Y: product_unit] : aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,X),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),Y)) = X ).

% sup_inf_absorb
tff(fact_745_sup__inf__absorb,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y)) = X ).

% sup_inf_absorb
tff(fact_746_sup__inf__absorb,axiom,
    ! [X: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)) = X ).

% sup_inf_absorb
tff(fact_747_sup__inf__absorb,axiom,
    ! [X: set_nat,Y: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)) = X ).

% sup_inf_absorb
tff(fact_748_inf__sup__absorb,axiom,
    ! [X: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y)) = X ).

% inf_sup_absorb
tff(fact_749_inf__sup__absorb,axiom,
    ! [X: product_unit,Y: product_unit] : aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,X),Y)) = X ).

% inf_sup_absorb
tff(fact_750_inf__sup__absorb,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X),Y)) = X ).

% inf_sup_absorb
tff(fact_751_inf__sup__absorb,axiom,
    ! [X: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y)) = X ).

% inf_sup_absorb
tff(fact_752_inf__sup__absorb,axiom,
    ! [X: set_nat,Y: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y)) = X ).

% inf_sup_absorb
tff(fact_753_Un__empty,axiom,
    ! [A2: set_o,B2: set_o] :
      ( ( sup_sup_set_o(A2,B2) = bot_bot_set_o )
    <=> ( ( A2 = bot_bot_set_o )
        & ( B2 = bot_bot_set_o ) ) ) ).

% Un_empty
tff(fact_754_Un__empty,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2) = bot_bot_set_nat )
    <=> ( ( A2 = bot_bot_set_nat )
        & ( B2 = bot_bot_set_nat ) ) ) ).

% Un_empty
tff(fact_755_Un__Int__eq_I1_J,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,S),T)),S) = S ).

% Un_Int_eq(1)
tff(fact_756_Un__Int__eq_I1_J,axiom,
    ! [S: set_nat,T: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,S),T)),S) = S ).

% Un_Int_eq(1)
tff(fact_757_Un__Int__eq_I2_J,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,S),T)),T) = T ).

% Un_Int_eq(2)
tff(fact_758_Un__Int__eq_I2_J,axiom,
    ! [S: set_nat,T: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,S),T)),T) = T ).

% Un_Int_eq(2)
tff(fact_759_Un__Int__eq_I3_J,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,S),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,S),T)) = S ).

% Un_Int_eq(3)
tff(fact_760_Un__Int__eq_I3_J,axiom,
    ! [S: set_nat,T: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,S),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,S),T)) = S ).

% Un_Int_eq(3)
tff(fact_761_Un__Int__eq_I4_J,axiom,
    ! [T: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,T),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,S),T)) = T ).

% Un_Int_eq(4)
tff(fact_762_Un__Int__eq_I4_J,axiom,
    ! [T: set_nat,S: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,T),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,S),T)) = T ).

% Un_Int_eq(4)
tff(fact_763_Int__Un__eq_I1_J,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,S),T)),S) = S ).

% Int_Un_eq(1)
tff(fact_764_Int__Un__eq_I1_J,axiom,
    ! [S: set_nat,T: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,S),T)),S) = S ).

% Int_Un_eq(1)
tff(fact_765_Int__Un__eq_I2_J,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,S),T)),T) = T ).

% Int_Un_eq(2)
tff(fact_766_Int__Un__eq_I2_J,axiom,
    ! [S: set_nat,T: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,S),T)),T) = T ).

% Int_Un_eq(2)
tff(fact_767_Int__Un__eq_I3_J,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,S),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,S),T)) = S ).

% Int_Un_eq(3)
tff(fact_768_Int__Un__eq_I3_J,axiom,
    ! [S: set_nat,T: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,S),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,S),T)) = S ).

% Int_Un_eq(3)
tff(fact_769_Int__Un__eq_I4_J,axiom,
    ! [T: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,T),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,S),T)) = T ).

% Int_Un_eq(4)
tff(fact_770_Int__Un__eq_I4_J,axiom,
    ! [T: set_nat,S: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,T),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,S),T)) = T ).

% Int_Un_eq(4)
tff(fact_771_lexordp__eq__simps_I4_J,axiom,
    ! [X: assn,Xs: list_assn,Y: assn,Ys: list_assn] :
      ( ord_lexordp_eq_assn(cons_assn(X,Xs),cons_assn(Y,Ys))
    <=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),Y)
        | ( ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,Y),X)
          & ord_lexordp_eq_assn(Xs,Ys) ) ) ) ).

% lexordp_eq_simps(4)
tff(fact_772_lexordp__eq__simps_I4_J,axiom,
    ! [X: nat,Xs: list_nat,Y: nat,Ys: list_nat] :
      ( aa_list_nat_o(aa_lis557685189327716851_nat_o(ord_lexordp_eq_nat,aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X),Xs)),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Y),Ys))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),X)
          & aa_list_nat_o(aa_lis557685189327716851_nat_o(ord_lexordp_eq_nat,Xs),Ys) ) ) ) ).

% lexordp_eq_simps(4)
tff(fact_773_lexordp__eq__simps_I4_J,axiom,
    ! [X: int,Xs: list_int,Y: int,Ys: list_int] :
      ( aa_list_int_o(aa_lis7773489400162621755_int_o(ord_lexordp_eq_int,aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X),Xs)),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Y),Ys))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
        | ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Y),X)
          & aa_list_int_o(aa_lis7773489400162621755_int_o(ord_lexordp_eq_int,Xs),Ys) ) ) ) ).

% lexordp_eq_simps(4)
tff(fact_774_list__collect__set__simps_I3_J,axiom,
    ! [F: fun_nat_set_nat,A: nat,La: list_nat] : list_c2452340269597857392at_nat(F,aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,A),La)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_nat_set_nat(F,A)),list_c2452340269597857392at_nat(F,La)) ).

% list_collect_set_simps(3)
tff(fact_775_list__collect__set__simps_I3_J,axiom,
    ! [F: fun_int_set_nat,A: int,La: list_int] : list_c3451693981498911948nt_nat(F,aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,A),La)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_int_set_nat(F,A)),list_c3451693981498911948nt_nat(F,La)) ).

% list_collect_set_simps(3)
tff(fact_776_list__collect__set__simps_I3_J,axiom,
    ! [F: fun_Pr5680576844133773770et_nat,A: produc6575502325842934193n_assn,La: list_P8527749157015355191n_assn] : list_c6061723043370948915sn_nat(F,aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,A),La)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_Pro3832724493296963067et_nat(F,A)),list_c6061723043370948915sn_nat(F,La)) ).

% list_collect_set_simps(3)
tff(fact_777_Int__left__commute,axiom,
    ! [A2: set_nat,B2: set_nat,C3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),C3)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),C3)) ).

% Int_left_commute
tff(fact_778_Int__left__commute,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C3: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B2),C3)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),C3)) ).

% Int_left_commute
tff(fact_779_Un__Int__distrib2,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,C3: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B2),C3)),A2) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,B2),A2)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,C3),A2)) ).

% Un_Int_distrib2
tff(fact_780_Un__Int__distrib2,axiom,
    ! [B2: set_nat,C3: set_nat,A2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),C3)),A2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B2),A2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,C3),A2)) ).

% Un_Int_distrib2
tff(fact_781_Int__left__absorb,axiom,
    ! [A2: set_nat,B2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) ).

% Int_left_absorb
tff(fact_782_Int__left__absorb,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2) ).

% Int_left_absorb
tff(fact_783_Int__Un__distrib2,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,C3: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,B2),C3)),A2) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B2),A2)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,C3),A2)) ).

% Int_Un_distrib2
tff(fact_784_Int__Un__distrib2,axiom,
    ! [B2: set_nat,C3: set_nat,A2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B2),C3)),A2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),A2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,C3),A2)) ).

% Int_Un_distrib2
tff(fact_785_Un__Int__distrib,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C3: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B2),C3)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A2),B2)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A2),C3)) ).

% Un_Int_distrib
tff(fact_786_Un__Int__distrib,axiom,
    ! [A2: set_nat,B2: set_nat,C3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),C3)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),C3)) ).

% Un_Int_distrib
tff(fact_787_Int__Un__distrib,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C3: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,B2),C3)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),C3)) ).

% Int_Un_distrib
tff(fact_788_Int__Un__distrib,axiom,
    ! [A2: set_nat,B2: set_nat,C3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B2),C3)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),C3)) ).

% Int_Un_distrib
tff(fact_789_Un__Int__crazy,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C3: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B2),C3))),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,C3),A2)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A2),B2)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,B2),C3))),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,C3),A2)) ).

% Un_Int_crazy
tff(fact_790_Un__Int__crazy,axiom,
    ! [A2: set_nat,B2: set_nat,C3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),C3))),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,C3),A2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B2),C3))),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,C3),A2)) ).

% Un_Int_crazy
tff(fact_791_Int__commute,axiom,
    ! [A2: set_nat,B2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),A2) ).

% Int_commute
tff(fact_792_Int__commute,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B2),A2) ).

% Int_commute
tff(fact_793_Int__absorb,axiom,
    ! [A2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),A2) = A2 ).

% Int_absorb
tff(fact_794_Int__absorb,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),A2) = A2 ).

% Int_absorb
tff(fact_795_Int__assoc,axiom,
    ! [A2: set_nat,B2: set_nat,C3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2)),C3) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),C3)) ).

% Int_assoc
tff(fact_796_Int__assoc,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C3: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2)),C3) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B2),C3)) ).

% Int_assoc
tff(fact_797_IntD2,axiom,
    ! [C2: fun_Pr6460944699146269935_nat_o,A2: set_fu3555266933746557007_nat_o,B2: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C2),inf_in3376754634429691197_nat_o(A2,B2))
     => aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C2),B2) ) ).

% IntD2
tff(fact_798_IntD2,axiom,
    ! [C2: nat,A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(member_nat2(C2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2))
     => aa_set_nat_o(member_nat2(C2),B2) ) ).

% IntD2
tff(fact_799_IntD2,axiom,
    ! [C2: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( member8440522571783428010at_nat(C2,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2))
     => member8440522571783428010at_nat(C2,B2) ) ).

% IntD2
tff(fact_800_IntD1,axiom,
    ! [C2: fun_Pr6460944699146269935_nat_o,A2: set_fu3555266933746557007_nat_o,B2: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C2),inf_in3376754634429691197_nat_o(A2,B2))
     => aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C2),A2) ) ).

% IntD1
tff(fact_801_IntD1,axiom,
    ! [C2: nat,A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(member_nat2(C2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2))
     => aa_set_nat_o(member_nat2(C2),A2) ) ).

% IntD1
tff(fact_802_IntD1,axiom,
    ! [C2: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( member8440522571783428010at_nat(C2,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2))
     => member8440522571783428010at_nat(C2,A2) ) ).

% IntD1
tff(fact_803_IntE,axiom,
    ! [C2: fun_Pr6460944699146269935_nat_o,A2: set_fu3555266933746557007_nat_o,B2: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C2),inf_in3376754634429691197_nat_o(A2,B2))
     => ~ ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C2),A2)
         => ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C2),B2) ) ) ).

% IntE
tff(fact_804_IntE,axiom,
    ! [C2: nat,A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(member_nat2(C2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2))
     => ~ ( aa_set_nat_o(member_nat2(C2),A2)
         => ~ aa_set_nat_o(member_nat2(C2),B2) ) ) ).

% IntE
tff(fact_805_IntE,axiom,
    ! [C2: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( member8440522571783428010at_nat(C2,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2))
     => ~ ( member8440522571783428010at_nat(C2,A2)
         => ~ member8440522571783428010at_nat(C2,B2) ) ) ).

% IntE
tff(fact_806_inf__sup__aci_I8_J,axiom,
    ! [X: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y)) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y) ).

% inf_sup_aci(8)
tff(fact_807_inf__sup__aci_I8_J,axiom,
    ! [X: set_nat,Y: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y) ).

% inf_sup_aci(8)
tff(fact_808_inf__sup__aci_I7_J,axiom,
    ! [X: assn,Y: assn,Z: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),Z)) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Z)) ).

% inf_sup_aci(7)
tff(fact_809_inf__sup__aci_I7_J,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),Z)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Z)) ).

% inf_sup_aci(7)
tff(fact_810_inf__sup__aci_I6_J,axiom,
    ! [X: assn,Y: assn,Z: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y)),Z) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),Z)) ).

% inf_sup_aci(6)
tff(fact_811_inf__sup__aci_I6_J,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y)),Z) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),Z)) ).

% inf_sup_aci(6)
tff(fact_812_inf__sup__aci_I5_J,axiom,
    ! [X: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),X) ).

% inf_sup_aci(5)
tff(fact_813_inf__sup__aci_I5_J,axiom,
    ! [X: set_nat,Y: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),X) ).

% inf_sup_aci(5)
tff(fact_814_sup_Oassoc,axiom,
    ! [A: assn,B: assn,C2: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B)),C2) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,B),C2)) ).

% sup.assoc
tff(fact_815_sup_Oassoc,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)),C2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B),C2)) ).

% sup.assoc
tff(fact_816_sup__assoc,axiom,
    ! [X: assn,Y: assn,Z: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y)),Z) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),Z)) ).

% sup_assoc
tff(fact_817_sup__assoc,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y)),Z) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),Z)) ).

% sup_assoc
tff(fact_818_less__supI1,axiom,
    ! [X: set_nat,A: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,X),A)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) ) ).

% less_supI1
tff(fact_819_less__supI1,axiom,
    ! [X: assn,A: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),A)
     => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B)) ) ).

% less_supI1
tff(fact_820_less__supI1,axiom,
    ! [X: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)) ) ).

% less_supI1
tff(fact_821_less__supI1,axiom,
    ! [X: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,X),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)) ) ).

% less_supI1
tff(fact_822_less__supI2,axiom,
    ! [X: set_nat,B: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,X),B)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) ) ).

% less_supI2
tff(fact_823_less__supI2,axiom,
    ! [X: assn,B: assn,A: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),B)
     => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B)) ) ).

% less_supI2
tff(fact_824_less__supI2,axiom,
    ! [X: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)) ) ).

% less_supI2
tff(fact_825_less__supI2,axiom,
    ! [X: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,X),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)) ) ).

% less_supI2
tff(fact_826_sup_Oabsorb3,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,B),A)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) = A ) ) ).

% sup.absorb3
tff(fact_827_sup_Oabsorb3,axiom,
    ! [B: assn,A: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),A)
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B) = A ) ) ).

% sup.absorb3
tff(fact_828_sup_Oabsorb3,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B) = A ) ) ).

% sup.absorb3
tff(fact_829_sup_Oabsorb3,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ( aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B) = A ) ) ).

% sup.absorb3
tff(fact_830_sup_Oabsorb4,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),B)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) = B ) ) ).

% sup.absorb4
tff(fact_831_sup_Oabsorb4,axiom,
    ! [A: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B) = B ) ) ).

% sup.absorb4
tff(fact_832_sup_Oabsorb4,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B) = B ) ) ).

% sup.absorb4
tff(fact_833_sup_Oabsorb4,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B) = B ) ) ).

% sup.absorb4
tff(fact_834_sup_Ocommute,axiom,
    ! [A: assn,B: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,B),A) ).

% sup.commute
tff(fact_835_sup_Ocommute,axiom,
    ! [A: set_nat,B: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B),A) ).

% sup.commute
tff(fact_836_sup__commute,axiom,
    ! [X: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),X) ).

% sup_commute
tff(fact_837_sup__commute,axiom,
    ! [X: set_nat,Y: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),X) ).

% sup_commute
tff(fact_838_boolean__algebra__cancel_Osup1,axiom,
    ! [A2: assn,K: assn,A: assn,B: assn] :
      ( ( A2 = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,K),A) )
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A2),B) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,K),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B)) ) ) ).

% boolean_algebra_cancel.sup1
tff(fact_839_boolean__algebra__cancel_Osup1,axiom,
    ! [A2: set_nat,K: set_nat,A: set_nat,B: set_nat] :
      ( ( A2 = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,K),A) )
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,K),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) ) ) ).

% boolean_algebra_cancel.sup1
tff(fact_840_boolean__algebra__cancel_Osup2,axiom,
    ! [B2: assn,K: assn,B: assn,A: assn] :
      ( ( B2 = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,K),B) )
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B2) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,K),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B)) ) ) ).

% boolean_algebra_cancel.sup2
tff(fact_841_boolean__algebra__cancel_Osup2,axiom,
    ! [B2: set_nat,K: set_nat,B: set_nat,A: set_nat] :
      ( ( B2 = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,K),B) )
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,K),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) ) ) ).

% boolean_algebra_cancel.sup2
tff(fact_842_sup_Oleft__commute,axiom,
    ! [B: assn,A: assn,C2: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,B),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),C2)) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,B),C2)) ).

% sup.left_commute
tff(fact_843_sup_Oleft__commute,axiom,
    ! [B: set_nat,A: set_nat,C2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),C2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B),C2)) ).

% sup.left_commute
tff(fact_844_sup__left__commute,axiom,
    ! [X: assn,Y: assn,Z: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),Z)) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Z)) ).

% sup_left_commute
tff(fact_845_sup__left__commute,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),Z)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Z)) ).

% sup_left_commute
tff(fact_846_sup_Ostrict__boundedE,axiom,
    ! [B: set_nat,C2: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B),C2)),A)
     => ~ ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,B),A)
         => ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,C2),A) ) ) ).

% sup.strict_boundedE
tff(fact_847_sup_Ostrict__boundedE,axiom,
    ! [B: assn,C2: assn,A: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,B),C2)),A)
     => ~ ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),A)
         => ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,C2),A) ) ) ).

% sup.strict_boundedE
tff(fact_848_sup_Ostrict__boundedE,axiom,
    ! [B: nat,C2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,B),C2)),A)
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),A) ) ) ).

% sup.strict_boundedE
tff(fact_849_sup_Ostrict__boundedE,axiom,
    ! [B: int,C2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,B),C2)),A)
     => ~ ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
         => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,C2),A) ) ) ).

% sup.strict_boundedE
tff(fact_850_sup_Ostrict__order__iff,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,B),A)
    <=> ( ( A = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) )
        & ( A != B ) ) ) ).

% sup.strict_order_iff
tff(fact_851_sup_Ostrict__order__iff,axiom,
    ! [B: assn,A: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),A)
    <=> ( ( A = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B) )
        & ( A != B ) ) ) ).

% sup.strict_order_iff
tff(fact_852_sup_Ostrict__order__iff,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
    <=> ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B) )
        & ( A != B ) ) ) ).

% sup.strict_order_iff
tff(fact_853_sup_Ostrict__order__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
    <=> ( ( A = aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B) )
        & ( A != B ) ) ) ).

% sup.strict_order_iff
tff(fact_854_sup_Ostrict__coboundedI1,axiom,
    ! [C2: set_nat,A: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,C2),A)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,C2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) ) ).

% sup.strict_coboundedI1
tff(fact_855_sup_Ostrict__coboundedI1,axiom,
    ! [C2: assn,A: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,C2),A)
     => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,C2),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B)) ) ).

% sup.strict_coboundedI1
tff(fact_856_sup_Ostrict__coboundedI1,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)) ) ).

% sup.strict_coboundedI1
tff(fact_857_sup_Ostrict__coboundedI1,axiom,
    ! [C2: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,C2),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)) ) ).

% sup.strict_coboundedI1
tff(fact_858_sup_Ostrict__coboundedI2,axiom,
    ! [C2: set_nat,B: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,C2),B)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,C2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) ) ).

% sup.strict_coboundedI2
tff(fact_859_sup_Ostrict__coboundedI2,axiom,
    ! [C2: assn,B: assn,A: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,C2),B)
     => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,C2),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B)) ) ).

% sup.strict_coboundedI2
tff(fact_860_sup_Ostrict__coboundedI2,axiom,
    ! [C2: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)) ) ).

% sup.strict_coboundedI2
tff(fact_861_sup_Ostrict__coboundedI2,axiom,
    ! [C2: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,C2),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)) ) ).

% sup.strict_coboundedI2
tff(fact_862_UnE,axiom,
    ! [C2: fun_Pr6460944699146269935_nat_o,A2: set_fu3555266933746557007_nat_o,B2: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C2),sup_su2896084399781877795_nat_o(A2,B2))
     => ( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C2),A2)
       => aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C2),B2) ) ) ).

% UnE
tff(fact_863_UnE,axiom,
    ! [C2: nat,A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(member_nat2(C2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2))
     => ( ~ aa_set_nat_o(member_nat2(C2),A2)
       => aa_set_nat_o(member_nat2(C2),B2) ) ) ).

% UnE
tff(fact_864_UnI1,axiom,
    ! [C2: fun_Pr6460944699146269935_nat_o,A2: set_fu3555266933746557007_nat_o,B2: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C2),A2)
     => aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C2),sup_su2896084399781877795_nat_o(A2,B2)) ) ).

% UnI1
tff(fact_865_UnI1,axiom,
    ! [C2: nat,A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(member_nat2(C2),A2)
     => aa_set_nat_o(member_nat2(C2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)) ) ).

% UnI1
tff(fact_866_UnI2,axiom,
    ! [C2: fun_Pr6460944699146269935_nat_o,B2: set_fu3555266933746557007_nat_o,A2: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C2),B2)
     => aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C2),sup_su2896084399781877795_nat_o(A2,B2)) ) ).

% UnI2
tff(fact_867_UnI2,axiom,
    ! [C2: nat,B2: set_nat,A2: set_nat] :
      ( aa_set_nat_o(member_nat2(C2),B2)
     => aa_set_nat_o(member_nat2(C2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)) ) ).

% UnI2
tff(fact_868_bex__Un,axiom,
    ! [A2: set_nat,B2: set_nat,Pa: fun_nat_o] :
      ( ? [X3: nat] :
          ( aa_set_nat_o(member_nat2(X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2))
          & aa_nat_o(Pa,X3) )
    <=> ( ? [X3: nat] :
            ( aa_set_nat_o(member_nat2(X3),A2)
            & aa_nat_o(Pa,X3) )
        | ? [X3: nat] :
            ( aa_set_nat_o(member_nat2(X3),B2)
            & aa_nat_o(Pa,X3) ) ) ) ).

% bex_Un
tff(fact_869_ball__Un,axiom,
    ! [A2: set_nat,B2: set_nat,Pa: fun_nat_o] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat2(X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2))
         => aa_nat_o(Pa,X3) )
    <=> ( ! [X3: nat] :
            ( aa_set_nat_o(member_nat2(X3),A2)
           => aa_nat_o(Pa,X3) )
        & ! [X3: nat] :
            ( aa_set_nat_o(member_nat2(X3),B2)
           => aa_nat_o(Pa,X3) ) ) ) ).

% ball_Un
tff(fact_870_Un__assoc,axiom,
    ! [A2: set_nat,B2: set_nat,C3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)),C3) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B2),C3)) ).

% Un_assoc
tff(fact_871_Un__absorb,axiom,
    ! [A2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),A2) = A2 ).

% Un_absorb
tff(fact_872_Un__commute,axiom,
    ! [A2: set_nat,B2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B2),A2) ).

% Un_commute
tff(fact_873_Un__left__absorb,axiom,
    ! [A2: set_nat,B2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2) ).

% Un_left_absorb
tff(fact_874_Un__left__commute,axiom,
    ! [A2: set_nat,B2: set_nat,C3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B2),C3)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),C3)) ).

% Un_left_commute
tff(fact_875_lt__ex,axiom,
    ! [X: int] :
    ? [Y2: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,Y2),X) ).

% lt_ex
tff(fact_876_gt__ex,axiom,
    ! [X: nat] :
    ? [X_1: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),X_1) ).

% gt_ex
tff(fact_877_gt__ex,axiom,
    ! [X: int] :
    ? [X_1: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,X),X_1) ).

% gt_ex
tff(fact_878_less__imp__neq,axiom,
    ! [X: assn,Y: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),Y)
     => ( X != Y ) ) ).

% less_imp_neq
tff(fact_879_less__imp__neq,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
     => ( X != Y ) ) ).

% less_imp_neq
tff(fact_880_less__imp__neq,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
     => ( X != Y ) ) ).

% less_imp_neq
tff(fact_881_order_Oasym,axiom,
    ! [A: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
     => ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),A) ) ).

% order.asym
tff(fact_882_order_Oasym,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A) ) ).

% order.asym
tff(fact_883_order_Oasym,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ).

% order.asym
tff(fact_884_ord__eq__less__trans,axiom,
    ! [A: assn,B: assn,C2: assn] :
      ( ( A = B )
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),C2)
       => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),C2) ) ) ).

% ord_eq_less_trans
tff(fact_885_ord__eq__less__trans,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( A = B )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),C2) ) ) ).

% ord_eq_less_trans
tff(fact_886_ord__eq__less__trans,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( A = B )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),C2) ) ) ).

% ord_eq_less_trans
tff(fact_887_ord__less__eq__trans,axiom,
    ! [A: assn,B: assn,C2: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
     => ( ( B = C2 )
       => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),C2) ) ) ).

% ord_less_eq_trans
tff(fact_888_ord__less__eq__trans,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( ( B = C2 )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),C2) ) ) ).

% ord_less_eq_trans
tff(fact_889_ord__less__eq__trans,axiom,
    ! [A: int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( ( B = C2 )
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),C2) ) ) ).

% ord_less_eq_trans
tff(fact_890_less__induct,axiom,
    ! [Pa: fun_nat_o,A: nat] :
      ( ! [X2: nat] :
          ( ! [Y4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y4),X2)
             => aa_nat_o(Pa,Y4) )
         => aa_nat_o(Pa,X2) )
     => aa_nat_o(Pa,A) ) ).

% less_induct
tff(fact_891_antisym__conv3,axiom,
    ! [Y: nat,X: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),X)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
      <=> ( X = Y ) ) ) ).

% antisym_conv3
tff(fact_892_antisym__conv3,axiom,
    ! [Y: int,X: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Y),X)
     => ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
      <=> ( X = Y ) ) ) ).

% antisym_conv3
tff(fact_893_linorder__cases,axiom,
    ! [X: nat,Y: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
     => ( ( X != Y )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),X) ) ) ).

% linorder_cases
tff(fact_894_linorder__cases,axiom,
    ! [X: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
     => ( ( X != Y )
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Y),X) ) ) ).

% linorder_cases
tff(fact_895_dual__order_Oasym,axiom,
    ! [B: assn,A: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),A)
     => ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B) ) ).

% dual_order.asym
tff(fact_896_dual__order_Oasym,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% dual_order.asym
tff(fact_897_dual__order_Oasym,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% dual_order.asym
tff(fact_898_dual__order_Oirrefl,axiom,
    ! [A: assn] : ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),A) ).

% dual_order.irrefl
tff(fact_899_dual__order_Oirrefl,axiom,
    ! [A: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),A) ).

% dual_order.irrefl
tff(fact_900_dual__order_Oirrefl,axiom,
    ! [A: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A),A) ).

% dual_order.irrefl
tff(fact_901_exists__least__iff,axiom,
    ! [Pa: fun_nat_o] :
      ( ? [X_12: nat] : aa_nat_o(Pa,X_12)
    <=> ? [N3: nat] :
          ( aa_nat_o(Pa,N3)
          & ! [M3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M3),N3)
             => ~ aa_nat_o(Pa,M3) ) ) ) ).

% exists_least_iff
tff(fact_902_linorder__less__wlog,axiom,
    ! [Pa: fun_nat_fun_nat_o,A: nat,B: nat] :
      ( ! [A3: nat,B3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A3),B3)
         => aa_nat_o(aa_nat_fun_nat_o(Pa,A3),B3) )
     => ( ! [A3: nat] : aa_nat_o(aa_nat_fun_nat_o(Pa,A3),A3)
       => ( ! [A3: nat,B3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(Pa,B3),A3)
             => aa_nat_o(aa_nat_fun_nat_o(Pa,A3),B3) )
         => aa_nat_o(aa_nat_fun_nat_o(Pa,A),B) ) ) ) ).

% linorder_less_wlog
tff(fact_903_linorder__less__wlog,axiom,
    ! [Pa: fun_int_fun_int_o,A: int,B: int] :
      ( ! [A3: int,B3: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,A3),B3)
         => aa_int_o(aa_int_fun_int_o(Pa,A3),B3) )
     => ( ! [A3: int] : aa_int_o(aa_int_fun_int_o(Pa,A3),A3)
       => ( ! [A3: int,B3: int] :
              ( aa_int_o(aa_int_fun_int_o(Pa,B3),A3)
             => aa_int_o(aa_int_fun_int_o(Pa,A3),B3) )
         => aa_int_o(aa_int_fun_int_o(Pa,A),B) ) ) ) ).

% linorder_less_wlog
tff(fact_904_order_Ostrict__trans,axiom,
    ! [A: assn,B: assn,C2: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),C2)
       => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),C2) ) ) ).

% order.strict_trans
tff(fact_905_order_Ostrict__trans,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),C2) ) ) ).

% order.strict_trans
tff(fact_906_order_Ostrict__trans,axiom,
    ! [A: int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),C2) ) ) ).

% order.strict_trans
tff(fact_907_not__less__iff__gr__or__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),X)
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_908_not__less__iff__gr__or__eq,axiom,
    ! [X: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),X)
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_909_dual__order_Ostrict__trans,axiom,
    ! [B: assn,A: assn,C2: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),A)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,C2),B)
       => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,C2),A) ) ) ).

% dual_order.strict_trans
tff(fact_910_dual__order_Ostrict__trans,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),A) ) ) ).

% dual_order.strict_trans
tff(fact_911_dual__order_Ostrict__trans,axiom,
    ! [B: int,A: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,C2),A) ) ) ).

% dual_order.strict_trans
tff(fact_912_order_Ostrict__implies__not__eq,axiom,
    ! [A: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
     => ( A != B ) ) ).

% order.strict_implies_not_eq
tff(fact_913_order_Ostrict__implies__not__eq,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( A != B ) ) ).

% order.strict_implies_not_eq
tff(fact_914_order_Ostrict__implies__not__eq,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( A != B ) ) ).

% order.strict_implies_not_eq
tff(fact_915_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: assn,A: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),A)
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_916_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_917_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_918_linorder__neqE,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),X) ) ) ).

% linorder_neqE
tff(fact_919_linorder__neqE,axiom,
    ! [X: int,Y: int] :
      ( ( X != Y )
     => ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Y),X) ) ) ).

% linorder_neqE
tff(fact_920_order__less__asym,axiom,
    ! [X: assn,Y: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),Y)
     => ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,Y),X) ) ).

% order_less_asym
tff(fact_921_order__less__asym,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),X) ) ).

% order_less_asym
tff(fact_922_order__less__asym,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Y),X) ) ).

% order_less_asym
tff(fact_923_linorder__neq__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),X) ) ) ).

% linorder_neq_iff
tff(fact_924_linorder__neq__iff,axiom,
    ! [X: int,Y: int] :
      ( ( X != Y )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
        | aa_int_o(aa_int_fun_int_o(ord_less_int,Y),X) ) ) ).

% linorder_neq_iff
tff(fact_925_order__less__asym_H,axiom,
    ! [A: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
     => ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),A) ) ).

% order_less_asym'
tff(fact_926_order__less__asym_H,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A) ) ).

% order_less_asym'
tff(fact_927_order__less__asym_H,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ).

% order_less_asym'
tff(fact_928_order__less__trans,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),Y)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,Y),Z)
       => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),Z) ) ) ).

% order_less_trans
tff(fact_929_order__less__trans,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Z)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z) ) ) ).

% order_less_trans
tff(fact_930_order__less__trans,axiom,
    ! [X: int,Y: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z) ) ) ).

% order_less_trans
tff(fact_931_ord__eq__less__subst,axiom,
    ! [A: assn,F: fun_assn_assn,B: assn,C2: assn] :
      ( ( A = aa_assn_assn(F,B) )
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),C2)
       => ( ! [X2: assn,Y2: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X2),Y2)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_assn_assn(F,X2)),aa_assn_assn(F,Y2)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),aa_assn_assn(F,C2)) ) ) ) ).

% ord_eq_less_subst
tff(fact_932_ord__eq__less__subst,axiom,
    ! [A: nat,F: fun_assn_nat,B: assn,C2: assn] :
      ( ( A = aa_assn_nat(F,B) )
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),C2)
       => ( ! [X2: assn,Y2: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X2),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_assn_nat(F,X2)),aa_assn_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_assn_nat(F,C2)) ) ) ) ).

% ord_eq_less_subst
tff(fact_933_ord__eq__less__subst,axiom,
    ! [A: int,F: fun_assn_int,B: assn,C2: assn] :
      ( ( A = aa_assn_int(F,B) )
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),C2)
       => ( ! [X2: assn,Y2: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X2),Y2)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_assn_int(F,X2)),aa_assn_int(F,Y2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_assn_int(F,C2)) ) ) ) ).

% ord_eq_less_subst
tff(fact_934_ord__eq__less__subst,axiom,
    ! [A: assn,F: fun_nat_assn,B: nat,C2: nat] :
      ( ( A = aa_nat_assn(F,B) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C2)
       => ( ! [X2: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Y2)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_nat_assn(F,X2)),aa_nat_assn(F,Y2)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),aa_nat_assn(F,C2)) ) ) ) ).

% ord_eq_less_subst
tff(fact_935_ord__eq__less__subst,axiom,
    ! [A: nat,F: fun_nat_nat,B: nat,C2: nat] :
      ( ( A = aa_nat_nat(F,B) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C2)
       => ( ! [X2: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,X2)),aa_nat_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_nat_nat(F,C2)) ) ) ) ).

% ord_eq_less_subst
tff(fact_936_ord__eq__less__subst,axiom,
    ! [A: int,F: fun_nat_int,B: nat,C2: nat] :
      ( ( A = aa_nat_int(F,B) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C2)
       => ( ! [X2: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Y2)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,X2)),aa_nat_int(F,Y2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(F,C2)) ) ) ) ).

% ord_eq_less_subst
tff(fact_937_ord__eq__less__subst,axiom,
    ! [A: assn,F: fun_int_assn,B: int,C2: int] :
      ( ( A = aa_int_assn(F,B) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C2)
       => ( ! [X2: int,Y2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Y2)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_int_assn(F,X2)),aa_int_assn(F,Y2)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),aa_int_assn(F,C2)) ) ) ) ).

% ord_eq_less_subst
tff(fact_938_ord__eq__less__subst,axiom,
    ! [A: nat,F: fun_int_nat,B: int,C2: int] :
      ( ( A = aa_int_nat(F,B) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C2)
       => ( ! [X2: int,Y2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(F,X2)),aa_int_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_int_nat(F,C2)) ) ) ) ).

% ord_eq_less_subst
tff(fact_939_ord__eq__less__subst,axiom,
    ! [A: int,F: fun_int_int,B: int,C2: int] :
      ( ( A = aa_int_int(F,B) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C2)
       => ( ! [X2: int,Y2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Y2)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(F,X2)),aa_int_int(F,Y2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(F,C2)) ) ) ) ).

% ord_eq_less_subst
tff(fact_940_ord__less__eq__subst,axiom,
    ! [A: assn,B: assn,F: fun_assn_assn,C2: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
     => ( ( aa_assn_assn(F,B) = C2 )
       => ( ! [X2: assn,Y2: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X2),Y2)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_assn_assn(F,X2)),aa_assn_assn(F,Y2)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_assn_assn(F,A)),C2) ) ) ) ).

% ord_less_eq_subst
tff(fact_941_ord__less__eq__subst,axiom,
    ! [A: assn,B: assn,F: fun_assn_nat,C2: nat] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
     => ( ( aa_assn_nat(F,B) = C2 )
       => ( ! [X2: assn,Y2: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X2),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_assn_nat(F,X2)),aa_assn_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_assn_nat(F,A)),C2) ) ) ) ).

% ord_less_eq_subst
tff(fact_942_ord__less__eq__subst,axiom,
    ! [A: assn,B: assn,F: fun_assn_int,C2: int] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
     => ( ( aa_assn_int(F,B) = C2 )
       => ( ! [X2: assn,Y2: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X2),Y2)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_assn_int(F,X2)),aa_assn_int(F,Y2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_assn_int(F,A)),C2) ) ) ) ).

% ord_less_eq_subst
tff(fact_943_ord__less__eq__subst,axiom,
    ! [A: nat,B: nat,F: fun_nat_assn,C2: assn] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( ( aa_nat_assn(F,B) = C2 )
       => ( ! [X2: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Y2)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_nat_assn(F,X2)),aa_nat_assn(F,Y2)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_nat_assn(F,A)),C2) ) ) ) ).

% ord_less_eq_subst
tff(fact_944_ord__less__eq__subst,axiom,
    ! [A: nat,B: nat,F: fun_nat_nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( ( aa_nat_nat(F,B) = C2 )
       => ( ! [X2: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,X2)),aa_nat_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,A)),C2) ) ) ) ).

% ord_less_eq_subst
tff(fact_945_ord__less__eq__subst,axiom,
    ! [A: nat,B: nat,F: fun_nat_int,C2: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( ( aa_nat_int(F,B) = C2 )
       => ( ! [X2: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Y2)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,X2)),aa_nat_int(F,Y2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,A)),C2) ) ) ) ).

% ord_less_eq_subst
tff(fact_946_ord__less__eq__subst,axiom,
    ! [A: int,B: int,F: fun_int_assn,C2: assn] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( ( aa_int_assn(F,B) = C2 )
       => ( ! [X2: int,Y2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Y2)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_int_assn(F,X2)),aa_int_assn(F,Y2)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_int_assn(F,A)),C2) ) ) ) ).

% ord_less_eq_subst
tff(fact_947_ord__less__eq__subst,axiom,
    ! [A: int,B: int,F: fun_int_nat,C2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( ( aa_int_nat(F,B) = C2 )
       => ( ! [X2: int,Y2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(F,X2)),aa_int_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(F,A)),C2) ) ) ) ).

% ord_less_eq_subst
tff(fact_948_ord__less__eq__subst,axiom,
    ! [A: int,B: int,F: fun_int_int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( ( aa_int_int(F,B) = C2 )
       => ( ! [X2: int,Y2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Y2)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(F,X2)),aa_int_int(F,Y2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(F,A)),C2) ) ) ) ).

% ord_less_eq_subst
tff(fact_949_order__less__irrefl,axiom,
    ! [X: assn] : ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),X) ).

% order_less_irrefl
tff(fact_950_order__less__irrefl,axiom,
    ! [X: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),X) ).

% order_less_irrefl
tff(fact_951_order__less__irrefl,axiom,
    ! [X: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X),X) ).

% order_less_irrefl
tff(fact_952_order__less__subst1,axiom,
    ! [A: assn,F: fun_assn_assn,B: assn,C2: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),aa_assn_assn(F,B))
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),C2)
       => ( ! [X2: assn,Y2: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X2),Y2)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_assn_assn(F,X2)),aa_assn_assn(F,Y2)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),aa_assn_assn(F,C2)) ) ) ) ).

% order_less_subst1
tff(fact_953_order__less__subst1,axiom,
    ! [A: assn,F: fun_nat_assn,B: nat,C2: nat] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),aa_nat_assn(F,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C2)
       => ( ! [X2: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Y2)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_nat_assn(F,X2)),aa_nat_assn(F,Y2)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),aa_nat_assn(F,C2)) ) ) ) ).

% order_less_subst1
tff(fact_954_order__less__subst1,axiom,
    ! [A: assn,F: fun_int_assn,B: int,C2: int] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),aa_int_assn(F,B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C2)
       => ( ! [X2: int,Y2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Y2)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_int_assn(F,X2)),aa_int_assn(F,Y2)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),aa_int_assn(F,C2)) ) ) ) ).

% order_less_subst1
tff(fact_955_order__less__subst1,axiom,
    ! [A: nat,F: fun_assn_nat,B: assn,C2: assn] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_assn_nat(F,B))
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),C2)
       => ( ! [X2: assn,Y2: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X2),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_assn_nat(F,X2)),aa_assn_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_assn_nat(F,C2)) ) ) ) ).

% order_less_subst1
tff(fact_956_order__less__subst1,axiom,
    ! [A: nat,F: fun_nat_nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_nat_nat(F,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C2)
       => ( ! [X2: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,X2)),aa_nat_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_nat_nat(F,C2)) ) ) ) ).

% order_less_subst1
tff(fact_957_order__less__subst1,axiom,
    ! [A: nat,F: fun_int_nat,B: int,C2: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_int_nat(F,B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C2)
       => ( ! [X2: int,Y2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(F,X2)),aa_int_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_int_nat(F,C2)) ) ) ) ).

% order_less_subst1
tff(fact_958_order__less__subst1,axiom,
    ! [A: int,F: fun_assn_int,B: assn,C2: assn] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_assn_int(F,B))
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),C2)
       => ( ! [X2: assn,Y2: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X2),Y2)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_assn_int(F,X2)),aa_assn_int(F,Y2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_assn_int(F,C2)) ) ) ) ).

% order_less_subst1
tff(fact_959_order__less__subst1,axiom,
    ! [A: int,F: fun_nat_int,B: nat,C2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(F,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C2)
       => ( ! [X2: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Y2)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,X2)),aa_nat_int(F,Y2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(F,C2)) ) ) ) ).

% order_less_subst1
tff(fact_960_order__less__subst1,axiom,
    ! [A: int,F: fun_int_int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(F,B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C2)
       => ( ! [X2: int,Y2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Y2)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(F,X2)),aa_int_int(F,Y2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(F,C2)) ) ) ) ).

% order_less_subst1
tff(fact_961_order__less__subst2,axiom,
    ! [A: assn,B: assn,F: fun_assn_assn,C2: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_assn_assn(F,B)),C2)
       => ( ! [X2: assn,Y2: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X2),Y2)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_assn_assn(F,X2)),aa_assn_assn(F,Y2)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_assn_assn(F,A)),C2) ) ) ) ).

% order_less_subst2
tff(fact_962_order__less__subst2,axiom,
    ! [A: assn,B: assn,F: fun_assn_nat,C2: nat] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_assn_nat(F,B)),C2)
       => ( ! [X2: assn,Y2: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X2),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_assn_nat(F,X2)),aa_assn_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_assn_nat(F,A)),C2) ) ) ) ).

% order_less_subst2
tff(fact_963_order__less__subst2,axiom,
    ! [A: assn,B: assn,F: fun_assn_int,C2: int] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_assn_int(F,B)),C2)
       => ( ! [X2: assn,Y2: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X2),Y2)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_assn_int(F,X2)),aa_assn_int(F,Y2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_assn_int(F,A)),C2) ) ) ) ).

% order_less_subst2
tff(fact_964_order__less__subst2,axiom,
    ! [A: nat,B: nat,F: fun_nat_assn,C2: assn] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_nat_assn(F,B)),C2)
       => ( ! [X2: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Y2)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_nat_assn(F,X2)),aa_nat_assn(F,Y2)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_nat_assn(F,A)),C2) ) ) ) ).

% order_less_subst2
tff(fact_965_order__less__subst2,axiom,
    ! [A: nat,B: nat,F: fun_nat_nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,B)),C2)
       => ( ! [X2: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,X2)),aa_nat_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,A)),C2) ) ) ) ).

% order_less_subst2
tff(fact_966_order__less__subst2,axiom,
    ! [A: nat,B: nat,F: fun_nat_int,C2: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,B)),C2)
       => ( ! [X2: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Y2)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,X2)),aa_nat_int(F,Y2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,A)),C2) ) ) ) ).

% order_less_subst2
tff(fact_967_order__less__subst2,axiom,
    ! [A: int,B: int,F: fun_int_assn,C2: assn] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_int_assn(F,B)),C2)
       => ( ! [X2: int,Y2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Y2)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_int_assn(F,X2)),aa_int_assn(F,Y2)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_int_assn(F,A)),C2) ) ) ) ).

% order_less_subst2
tff(fact_968_order__less__subst2,axiom,
    ! [A: int,B: int,F: fun_int_nat,C2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(F,B)),C2)
       => ( ! [X2: int,Y2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(F,X2)),aa_int_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(F,A)),C2) ) ) ) ).

% order_less_subst2
tff(fact_969_order__less__subst2,axiom,
    ! [A: int,B: int,F: fun_int_int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(F,B)),C2)
       => ( ! [X2: int,Y2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Y2)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(F,X2)),aa_int_int(F,Y2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(F,A)),C2) ) ) ) ).

% order_less_subst2
tff(fact_970_order__less__not__sym,axiom,
    ! [X: assn,Y: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),Y)
     => ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,Y),X) ) ).

% order_less_not_sym
tff(fact_971_order__less__not__sym,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),X) ) ).

% order_less_not_sym
tff(fact_972_order__less__not__sym,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Y),X) ) ).

% order_less_not_sym
tff(fact_973_order__less__imp__triv,axiom,
    ! [X: assn,Y: assn,Pa: $o] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),Y)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,Y),X)
       => (Pa) ) ) ).

% order_less_imp_triv
tff(fact_974_order__less__imp__triv,axiom,
    ! [X: nat,Y: nat,Pa: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),X)
       => (Pa) ) ) ).

% order_less_imp_triv
tff(fact_975_order__less__imp__triv,axiom,
    ! [X: int,Y: int,Pa: $o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),X)
       => (Pa) ) ) ).

% order_less_imp_triv
tff(fact_976_linorder__less__linear,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
      | ( X = Y )
      | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),X) ) ).

% linorder_less_linear
tff(fact_977_linorder__less__linear,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
      | ( X = Y )
      | aa_int_o(aa_int_fun_int_o(ord_less_int,Y),X) ) ).

% linorder_less_linear
tff(fact_978_order__less__imp__not__eq,axiom,
    ! [X: assn,Y: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),Y)
     => ( X != Y ) ) ).

% order_less_imp_not_eq
tff(fact_979_order__less__imp__not__eq,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
     => ( X != Y ) ) ).

% order_less_imp_not_eq
tff(fact_980_order__less__imp__not__eq,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
     => ( X != Y ) ) ).

% order_less_imp_not_eq
tff(fact_981_order__less__imp__not__eq2,axiom,
    ! [X: assn,Y: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),Y)
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
tff(fact_982_order__less__imp__not__eq2,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
tff(fact_983_order__less__imp__not__eq2,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
tff(fact_984_order__less__imp__not__less,axiom,
    ! [X: assn,Y: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),Y)
     => ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,Y),X) ) ).

% order_less_imp_not_less
tff(fact_985_order__less__imp__not__less,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),X) ) ).

% order_less_imp_not_less
tff(fact_986_order__less__imp__not__less,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Y),X) ) ).

% order_less_imp_not_less
tff(fact_987_syntax__fo__nomatch__def,axiom,
    ! [Pat: assn,Obj: assn] : aa_assn_o(syntax7398250324933576852n_assn(Pat),Obj) ).

% syntax_fo_nomatch_def
tff(fact_988_boolean__algebra_Odisj__zero__right,axiom,
    ! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),bot_bot_assn) = X ).

% boolean_algebra.disj_zero_right
tff(fact_989_boolean__algebra_Odisj__zero__right,axiom,
    ! [X: set_o] : sup_sup_set_o(X,bot_bot_set_o) = X ).

% boolean_algebra.disj_zero_right
tff(fact_990_boolean__algebra_Odisj__zero__right,axiom,
    ! [X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),bot_bot_set_nat) = X ).

% boolean_algebra.disj_zero_right
tff(fact_991_boolean__algebra_Odisj__conj__distrib2,axiom,
    ! [Y: product_unit,Z: product_unit,X: product_unit] : aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,Y),Z)),X) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,Y),X)),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,Z),X)) ).

% boolean_algebra.disj_conj_distrib2
tff(fact_992_boolean__algebra_Odisj__conj__distrib2,axiom,
    ! [Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y),Z)),X) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Y),X)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Z),X)) ).

% boolean_algebra.disj_conj_distrib2
tff(fact_993_boolean__algebra_Odisj__conj__distrib2,axiom,
    ! [Y: assn,Z: assn,X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),Z)),X) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),X)),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Z),X)) ).

% boolean_algebra.disj_conj_distrib2
tff(fact_994_boolean__algebra_Odisj__conj__distrib2,axiom,
    ! [Y: set_nat,Z: set_nat,X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),Z)),X) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),X)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Z),X)) ).

% boolean_algebra.disj_conj_distrib2
tff(fact_995_boolean__algebra_Oconj__disj__distrib2,axiom,
    ! [Y: product_unit,Z: product_unit,X: product_unit] : aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,Y),Z)),X) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,Y),X)),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,Z),X)) ).

% boolean_algebra.conj_disj_distrib2
tff(fact_996_boolean__algebra_Oconj__disj__distrib2,axiom,
    ! [Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Y),Z)),X) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y),X)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Z),X)) ).

% boolean_algebra.conj_disj_distrib2
tff(fact_997_boolean__algebra_Oconj__disj__distrib2,axiom,
    ! [Y: assn,Z: assn,X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),Z)),X) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),X)),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Z),X)) ).

% boolean_algebra.conj_disj_distrib2
tff(fact_998_boolean__algebra_Oconj__disj__distrib2,axiom,
    ! [Y: set_nat,Z: set_nat,X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),Z)),X) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),X)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Z),X)) ).

% boolean_algebra.conj_disj_distrib2
tff(fact_999_boolean__algebra_Odisj__conj__distrib,axiom,
    ! [X: product_unit,Y: product_unit,Z: product_unit] : aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,X),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,Y),Z)) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,X),Y)),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,X),Z)) ).

% boolean_algebra.disj_conj_distrib
tff(fact_1000_boolean__algebra_Odisj__conj__distrib,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y),Z)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X),Y)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X),Z)) ).

% boolean_algebra.disj_conj_distrib
tff(fact_1001_boolean__algebra_Odisj__conj__distrib,axiom,
    ! [X: assn,Y: assn,Z: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),Z)) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y)),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Z)) ).

% boolean_algebra.disj_conj_distrib
tff(fact_1002_boolean__algebra_Odisj__conj__distrib,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),Z)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Z)) ).

% boolean_algebra.disj_conj_distrib
tff(fact_1003_boolean__algebra_Oconj__disj__distrib,axiom,
    ! [X: product_unit,Y: product_unit,Z: product_unit] : aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,Y),Z)) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),Y)),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),Z)) ).

% boolean_algebra.conj_disj_distrib
tff(fact_1004_boolean__algebra_Oconj__disj__distrib,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Y),Z)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Z)) ).

% boolean_algebra.conj_disj_distrib
tff(fact_1005_boolean__algebra_Oconj__disj__distrib,axiom,
    ! [X: assn,Y: assn,Z: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),Z)) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Z)) ).

% boolean_algebra.conj_disj_distrib
tff(fact_1006_boolean__algebra_Oconj__disj__distrib,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),Z)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Z)) ).

% boolean_algebra.conj_disj_distrib
tff(fact_1007_sup__inf__distrib2,axiom,
    ! [Y: nat,Z: nat,X: nat] : aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Y),Z)),X) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Y),X)),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Z),X)) ).

% sup_inf_distrib2
tff(fact_1008_sup__inf__distrib2,axiom,
    ! [Y: product_unit,Z: product_unit,X: product_unit] : aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,Y),Z)),X) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,Y),X)),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,Z),X)) ).

% sup_inf_distrib2
tff(fact_1009_sup__inf__distrib2,axiom,
    ! [Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y),Z)),X) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Y),X)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Z),X)) ).

% sup_inf_distrib2
tff(fact_1010_sup__inf__distrib2,axiom,
    ! [Y: assn,Z: assn,X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),Z)),X) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),X)),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Z),X)) ).

% sup_inf_distrib2
tff(fact_1011_sup__inf__distrib2,axiom,
    ! [Y: set_nat,Z: set_nat,X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),Z)),X) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),X)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Z),X)) ).

% sup_inf_distrib2
tff(fact_1012_sup__inf__distrib1,axiom,
    ! [X: nat,Y: nat,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Y),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y)),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Z)) ).

% sup_inf_distrib1
tff(fact_1013_sup__inf__distrib1,axiom,
    ! [X: product_unit,Y: product_unit,Z: product_unit] : aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,X),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,Y),Z)) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,X),Y)),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,X),Z)) ).

% sup_inf_distrib1
tff(fact_1014_sup__inf__distrib1,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y),Z)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X),Y)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X),Z)) ).

% sup_inf_distrib1
tff(fact_1015_sup__inf__distrib1,axiom,
    ! [X: assn,Y: assn,Z: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),Z)) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y)),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Z)) ).

% sup_inf_distrib1
tff(fact_1016_sup__inf__distrib1,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),Z)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Z)) ).

% sup_inf_distrib1
tff(fact_1017_inf__sup__distrib2,axiom,
    ! [Y: nat,Z: nat,X: nat] : aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Y),Z)),X) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Y),X)),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Z),X)) ).

% inf_sup_distrib2
tff(fact_1018_inf__sup__distrib2,axiom,
    ! [Y: product_unit,Z: product_unit,X: product_unit] : aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,Y),Z)),X) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,Y),X)),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,Z),X)) ).

% inf_sup_distrib2
tff(fact_1019_inf__sup__distrib2,axiom,
    ! [Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Y),Z)),X) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y),X)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Z),X)) ).

% inf_sup_distrib2
tff(fact_1020_inf__sup__distrib2,axiom,
    ! [Y: assn,Z: assn,X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),Z)),X) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),X)),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Z),X)) ).

% inf_sup_distrib2
tff(fact_1021_inf__sup__distrib2,axiom,
    ! [Y: set_nat,Z: set_nat,X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),Z)),X) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),X)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Z),X)) ).

% inf_sup_distrib2
tff(fact_1022_inf__sup__distrib1,axiom,
    ! [X: nat,Y: nat,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Y),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y)),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Z)) ).

% inf_sup_distrib1
tff(fact_1023_inf__sup__distrib1,axiom,
    ! [X: product_unit,Y: product_unit,Z: product_unit] : aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,Y),Z)) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),Y)),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),Z)) ).

% inf_sup_distrib1
tff(fact_1024_inf__sup__distrib1,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Y),Z)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Z)) ).

% inf_sup_distrib1
tff(fact_1025_inf__sup__distrib1,axiom,
    ! [X: assn,Y: assn,Z: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),Z)) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Z)) ).

% inf_sup_distrib1
tff(fact_1026_inf__sup__distrib1,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),Z)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Z)) ).

% inf_sup_distrib1
tff(fact_1027_distrib__imp2,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ! [X2: nat,Y2: nat,Z3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X2),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Y2),Z3)) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X2),Y2)),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X2),Z3))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Y),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y)),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Z)) ) ) ).

% distrib_imp2
tff(fact_1028_distrib__imp2,axiom,
    ! [X: product_unit,Y: product_unit,Z: product_unit] :
      ( ! [X2: product_unit,Y2: product_unit,Z3: product_unit] : aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,X2),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,Y2),Z3)) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,X2),Y2)),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,X2),Z3))
     => ( aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,Y),Z)) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),Y)),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),Z)) ) ) ).

% distrib_imp2
tff(fact_1029_distrib__imp2,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( ! [X2: set_Pr1261947904930325089at_nat,Y2: set_Pr1261947904930325089at_nat,Z3: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y2),Z3)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X2),Y2)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X2),Z3))
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Y),Z)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Z)) ) ) ).

% distrib_imp2
tff(fact_1030_distrib__imp2,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ! [X2: assn,Y2: assn,Z3: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X2),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y2),Z3)) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X2),Y2)),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X2),Z3))
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),Z)) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Z)) ) ) ).

% distrib_imp2
tff(fact_1031_distrib__imp2,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ! [X2: set_nat,Y2: set_nat,Z3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y2),Z3)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X2),Y2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X2),Z3))
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),Z)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Z)) ) ) ).

% distrib_imp2
tff(fact_1032_distrib__imp1,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ! [X2: nat,Y2: nat,Z3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X2),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Y2),Z3)) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X2),Y2)),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X2),Z3))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Y),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y)),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Z)) ) ) ).

% distrib_imp1
tff(fact_1033_distrib__imp1,axiom,
    ! [X: product_unit,Y: product_unit,Z: product_unit] :
      ( ! [X2: product_unit,Y2: product_unit,Z3: product_unit] : aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X2),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,Y2),Z3)) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X2),Y2)),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X2),Z3))
     => ( aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,X),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,Y),Z)) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,X),Y)),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,X),Z)) ) ) ).

% distrib_imp1
tff(fact_1034_distrib__imp1,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( ! [X2: set_Pr1261947904930325089at_nat,Y2: set_Pr1261947904930325089at_nat,Z3: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Y2),Z3)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X2),Y2)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X2),Z3))
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y),Z)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X),Y)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X),Z)) ) ) ).

% distrib_imp1
tff(fact_1035_distrib__imp1,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ! [X2: assn,Y2: assn,Z3: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X2),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y2),Z3)) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X2),Y2)),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X2),Z3))
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),Z)) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y)),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Z)) ) ) ).

% distrib_imp1
tff(fact_1036_distrib__imp1,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ! [X2: set_nat,Y2: set_nat,Z3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y2),Z3)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X2),Y2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X2),Z3))
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),Z)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Z)) ) ) ).

% distrib_imp1
tff(fact_1037_Un__empty__right,axiom,
    ! [A2: set_o] : sup_sup_set_o(A2,bot_bot_set_o) = A2 ).

% Un_empty_right
tff(fact_1038_Un__empty__right,axiom,
    ! [A2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),bot_bot_set_nat) = A2 ).

% Un_empty_right
tff(fact_1039_Un__empty__left,axiom,
    ! [B2: set_o] : sup_sup_set_o(bot_bot_set_o,B2) = B2 ).

% Un_empty_left
tff(fact_1040_Un__empty__left,axiom,
    ! [B2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,bot_bot_set_nat),B2) = B2 ).

% Un_empty_left
tff(fact_1041_bot_Onot__eq__extremum,axiom,
    ! [A: set_o] :
      ( ( A != bot_bot_set_o )
    <=> aa_set_o_o(ord_less_set_o(bot_bot_set_o),A) ) ).

% bot.not_eq_extremum
tff(fact_1042_bot_Onot__eq__extremum,axiom,
    ! [A: set_nat] :
      ( ( A != bot_bot_set_nat )
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,bot_bot_set_nat),A) ) ).

% bot.not_eq_extremum
tff(fact_1043_bot_Onot__eq__extremum,axiom,
    ! [A: assn] :
      ( ( A != bot_bot_assn )
    <=> aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,bot_bot_assn),A) ) ).

% bot.not_eq_extremum
tff(fact_1044_bot_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != bot_bot_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,bot_bot_nat),A) ) ).

% bot.not_eq_extremum
tff(fact_1045_bot_Oextremum__strict,axiom,
    ! [A: set_o] : ~ aa_set_o_o(ord_less_set_o(A),bot_bot_set_o) ).

% bot.extremum_strict
tff(fact_1046_bot_Oextremum__strict,axiom,
    ! [A: set_nat] : ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),bot_bot_set_nat) ).

% bot.extremum_strict
tff(fact_1047_bot_Oextremum__strict,axiom,
    ! [A: assn] : ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),bot_bot_assn) ).

% bot.extremum_strict
tff(fact_1048_bot_Oextremum__strict,axiom,
    ! [A: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),bot_bot_nat) ).

% bot.extremum_strict
tff(fact_1049_not__psubset__empty,axiom,
    ! [A2: set_o] : ~ aa_set_o_o(ord_less_set_o(A2),bot_bot_set_o) ).

% not_psubset_empty
tff(fact_1050_not__psubset__empty,axiom,
    ! [A2: set_nat] : ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A2),bot_bot_set_nat) ).

% not_psubset_empty
tff(fact_1051_inf_Ostrict__coboundedI2,axiom,
    ! [B: set_nat,C2: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,B),C2)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)),C2) ) ).

% inf.strict_coboundedI2
tff(fact_1052_inf_Ostrict__coboundedI2,axiom,
    ! [B: product_unit,C2: product_unit,A: product_unit] :
      ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le361264281704409273t_unit,B),C2)
     => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le361264281704409273t_unit,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,A),B)),C2) ) ).

% inf.strict_coboundedI2
tff(fact_1053_inf_Ostrict__coboundedI2,axiom,
    ! [B: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,B),C2)
     => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)),C2) ) ).

% inf.strict_coboundedI2
tff(fact_1054_inf_Ostrict__coboundedI2,axiom,
    ! [B: assn,C2: assn,A: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),C2)
     => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B)),C2) ) ).

% inf.strict_coboundedI2
tff(fact_1055_inf_Ostrict__coboundedI2,axiom,
    ! [B: nat,C2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)),C2) ) ).

% inf.strict_coboundedI2
tff(fact_1056_inf_Ostrict__coboundedI2,axiom,
    ! [B: int,C2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C2)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B)),C2) ) ).

% inf.strict_coboundedI2
tff(fact_1057_inf_Ostrict__coboundedI1,axiom,
    ! [A: set_nat,C2: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),C2)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)),C2) ) ).

% inf.strict_coboundedI1
tff(fact_1058_inf_Ostrict__coboundedI1,axiom,
    ! [A: product_unit,C2: product_unit,B: product_unit] :
      ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le361264281704409273t_unit,A),C2)
     => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le361264281704409273t_unit,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,A),B)),C2) ) ).

% inf.strict_coboundedI1
tff(fact_1059_inf_Ostrict__coboundedI1,axiom,
    ! [A: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,A),C2)
     => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)),C2) ) ).

% inf.strict_coboundedI1
tff(fact_1060_inf_Ostrict__coboundedI1,axiom,
    ! [A: assn,C2: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),C2)
     => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B)),C2) ) ).

% inf.strict_coboundedI1
tff(fact_1061_inf_Ostrict__coboundedI1,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),C2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)),C2) ) ).

% inf.strict_coboundedI1
tff(fact_1062_inf_Ostrict__coboundedI1,axiom,
    ! [A: int,C2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),C2)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B)),C2) ) ).

% inf.strict_coboundedI1
tff(fact_1063_inf_Ostrict__order__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),B)
    <=> ( ( A = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) )
        & ( A != B ) ) ) ).

% inf.strict_order_iff
tff(fact_1064_inf_Ostrict__order__iff,axiom,
    ! [A: product_unit,B: product_unit] :
      ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le361264281704409273t_unit,A),B)
    <=> ( ( A = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,A),B) )
        & ( A != B ) ) ) ).

% inf.strict_order_iff
tff(fact_1065_inf_Ostrict__order__iff,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,A),B)
    <=> ( ( A = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) )
        & ( A != B ) ) ) ).

% inf.strict_order_iff
tff(fact_1066_inf_Ostrict__order__iff,axiom,
    ! [A: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
    <=> ( ( A = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B) )
        & ( A != B ) ) ) ).

% inf.strict_order_iff
tff(fact_1067_inf_Ostrict__order__iff,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
    <=> ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B) )
        & ( A != B ) ) ) ).

% inf.strict_order_iff
tff(fact_1068_inf_Ostrict__order__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
    <=> ( ( A = aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B) )
        & ( A != B ) ) ) ).

% inf.strict_order_iff
tff(fact_1069_inf_Ostrict__boundedE,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B),C2))
     => ~ ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),B)
         => ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),C2) ) ) ).

% inf.strict_boundedE
tff(fact_1070_inf_Ostrict__boundedE,axiom,
    ! [A: product_unit,B: product_unit,C2: product_unit] :
      ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le361264281704409273t_unit,A),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,B),C2))
     => ~ ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le361264281704409273t_unit,A),B)
         => ~ aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le361264281704409273t_unit,A),C2) ) ) ).

% inf.strict_boundedE
tff(fact_1071_inf_Ostrict__boundedE,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,A),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B),C2))
     => ~ ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,A),B)
         => ~ aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,A),C2) ) ) ).

% inf.strict_boundedE
tff(fact_1072_inf_Ostrict__boundedE,axiom,
    ! [A: assn,B: assn,C2: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,B),C2))
     => ~ ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
         => ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),C2) ) ) ).

% inf.strict_boundedE
tff(fact_1073_inf_Ostrict__boundedE,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,B),C2))
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),C2) ) ) ).

% inf.strict_boundedE
tff(fact_1074_inf_Ostrict__boundedE,axiom,
    ! [A: int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(aa_int_fun_int_int(inf_inf_int,B),C2))
     => ~ ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
         => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A),C2) ) ) ).

% inf.strict_boundedE
tff(fact_1075_inf_Oabsorb4,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,B),A)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) = B ) ) ).

% inf.absorb4
tff(fact_1076_inf_Oabsorb4,axiom,
    ! [B: product_unit,A: product_unit] :
      ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le361264281704409273t_unit,B),A)
     => ( aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,A),B) = B ) ) ).

% inf.absorb4
tff(fact_1077_inf_Oabsorb4,axiom,
    ! [B: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,B),A)
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) = B ) ) ).

% inf.absorb4
tff(fact_1078_inf_Oabsorb4,axiom,
    ! [B: assn,A: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),A)
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B) = B ) ) ).

% inf.absorb4
tff(fact_1079_inf_Oabsorb4,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B) = B ) ) ).

% inf.absorb4
tff(fact_1080_inf_Oabsorb4,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ( aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B) = B ) ) ).

% inf.absorb4
tff(fact_1081_inf_Oabsorb3,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),B)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) = A ) ) ).

% inf.absorb3
tff(fact_1082_inf_Oabsorb3,axiom,
    ! [A: product_unit,B: product_unit] :
      ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le361264281704409273t_unit,A),B)
     => ( aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,A),B) = A ) ) ).

% inf.absorb3
tff(fact_1083_inf_Oabsorb3,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,A),B)
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) = A ) ) ).

% inf.absorb3
tff(fact_1084_inf_Oabsorb3,axiom,
    ! [A: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B) = A ) ) ).

% inf.absorb3
tff(fact_1085_inf_Oabsorb3,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B) = A ) ) ).

% inf.absorb3
tff(fact_1086_inf_Oabsorb3,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B) = A ) ) ).

% inf.absorb3
tff(fact_1087_less__infI2,axiom,
    ! [B: set_nat,X: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,B),X)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)),X) ) ).

% less_infI2
tff(fact_1088_less__infI2,axiom,
    ! [B: product_unit,X: product_unit,A: product_unit] :
      ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le361264281704409273t_unit,B),X)
     => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le361264281704409273t_unit,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,A),B)),X) ) ).

% less_infI2
tff(fact_1089_less__infI2,axiom,
    ! [B: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,B),X)
     => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)),X) ) ).

% less_infI2
tff(fact_1090_less__infI2,axiom,
    ! [B: assn,X: assn,A: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),X)
     => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B)),X) ) ).

% less_infI2
tff(fact_1091_less__infI2,axiom,
    ! [B: nat,X: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),X)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)),X) ) ).

% less_infI2
tff(fact_1092_less__infI2,axiom,
    ! [B: int,X: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),X)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B)),X) ) ).

% less_infI2
tff(fact_1093_less__infI1,axiom,
    ! [A: set_nat,X: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),X)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)),X) ) ).

% less_infI1
tff(fact_1094_less__infI1,axiom,
    ! [A: product_unit,X: product_unit,B: product_unit] :
      ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le361264281704409273t_unit,A),X)
     => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le361264281704409273t_unit,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,A),B)),X) ) ).

% less_infI1
tff(fact_1095_less__infI1,axiom,
    ! [A: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,A),X)
     => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)),X) ) ).

% less_infI1
tff(fact_1096_less__infI1,axiom,
    ! [A: assn,X: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),X)
     => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B)),X) ) ).

% less_infI1
tff(fact_1097_less__infI1,axiom,
    ! [A: nat,X: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),X)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)),X) ) ).

% less_infI1
tff(fact_1098_less__infI1,axiom,
    ! [A: int,X: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),X)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B)),X) ) ).

% less_infI1
tff(fact_1099_FI__p__nomatch,axiom,
    ! [M2: list_P8527749157015355191n_assn,Ps: assn,Qs: assn,Q2: assn,P3: assn,Up: assn,Uq: assn,F: assn] :
      ( aa_assn_o(fi(M2,Ps,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Qs),Q2),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P3),Up),Uq),F)
     => aa_assn_o(fi(M2,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Ps),P3),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Qs),Q2),Up,Uq),F) ) ).

% FI_p_nomatch
tff(fact_1100_lexordp__eq_OCons__eq,axiom,
    ! [X: assn,Y: assn,Xs: list_assn,Ys: list_assn] :
      ( ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),Y)
     => ( ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,Y),X)
       => ( ord_lexordp_eq_assn(Xs,Ys)
         => ord_lexordp_eq_assn(cons_assn(X,Xs),cons_assn(Y,Ys)) ) ) ) ).

% lexordp_eq.Cons_eq
tff(fact_1101_lexordp__eq_OCons__eq,axiom,
    ! [X: nat,Y: nat,Xs: list_nat,Ys: list_nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),X)
       => ( aa_list_nat_o(aa_lis557685189327716851_nat_o(ord_lexordp_eq_nat,Xs),Ys)
         => aa_list_nat_o(aa_lis557685189327716851_nat_o(ord_lexordp_eq_nat,aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X),Xs)),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Y),Ys)) ) ) ) ).

% lexordp_eq.Cons_eq
tff(fact_1102_lexordp__eq_OCons__eq,axiom,
    ! [X: int,Y: int,Xs: list_int,Ys: list_int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
     => ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Y),X)
       => ( aa_list_int_o(aa_lis7773489400162621755_int_o(ord_lexordp_eq_int,Xs),Ys)
         => aa_list_int_o(aa_lis7773489400162621755_int_o(ord_lexordp_eq_int,aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X),Xs)),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Y),Ys)) ) ) ) ).

% lexordp_eq.Cons_eq
tff(fact_1103_lexordp__eq_OCons,axiom,
    ! [X: assn,Y: assn,Xs: list_assn,Ys: list_assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),Y)
     => ord_lexordp_eq_assn(cons_assn(X,Xs),cons_assn(Y,Ys)) ) ).

% lexordp_eq.Cons
tff(fact_1104_lexordp__eq_OCons,axiom,
    ! [X: nat,Y: nat,Xs: list_nat,Ys: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
     => aa_list_nat_o(aa_lis557685189327716851_nat_o(ord_lexordp_eq_nat,aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X),Xs)),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Y),Ys)) ) ).

% lexordp_eq.Cons
tff(fact_1105_lexordp__eq_OCons,axiom,
    ! [X: int,Y: int,Xs: list_int,Ys: list_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
     => aa_list_int_o(aa_lis7773489400162621755_int_o(ord_lexordp_eq_int,aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X),Xs)),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Y),Ys)) ) ).

% lexordp_eq.Cons
tff(fact_1106_assn__aci_I11_J,axiom,
    ! [X: assn,Y: assn,A: assn,B: assn] :
      ( aa_assn_o(syntax7398250324933576852n_assn(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,X),Y)),A)
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),B) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B),A) ) ) ).

% assn_aci(11)
tff(fact_1107_less__1__mult,axiom,
    ! [M2: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),M2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),N)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),N)) ) ) ).

% less_1_mult
tff(fact_1108_less__1__mult,axiom,
    ! [M2: int,N: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),M2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),N)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,M2),N)) ) ) ).

% less_1_mult
tff(fact_1109_less__numeral__extra_I4_J,axiom,
    ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),one_one_nat) ).

% less_numeral_extra(4)
tff(fact_1110_less__numeral__extra_I4_J,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),one_one_int) ).

% less_numeral_extra(4)
tff(fact_1111_FI__q__nomatch,axiom,
    ! [M2: list_P8527749157015355191n_assn,Up: assn,Qs: assn,Q2: assn,Uq: assn,F: assn] :
      ( aa_assn_o(fi(M2,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,sln),Up),Qs,sln,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Q2),Uq)),F)
     => aa_assn_o(fi(M2,sln,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Qs),Q2),Up,Uq),F) ) ).

% FI_q_nomatch
tff(fact_1112_merge_Oelims,axiom,
    ! [X: list_nat,Xa: list_nat,Y: list_nat] :
      ( ( merge_nat(X,Xa) = Y )
     => ( ( ( X = nil_nat )
         => ( Y != Xa ) )
       => ( ! [V2: nat,Va2: list_nat] :
              ( ( X = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,V2),Va2) )
             => ( ( Xa = nil_nat )
               => ( Y != aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,V2),Va2) ) ) )
         => ~ ! [X1: nat,L1: list_nat] :
                ( ( X = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X1),L1) )
               => ! [X23: nat,L2: list_nat] :
                    ( ( Xa = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X23),L2) )
                   => ( Y != $ite(
                          aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X1),X23),
                          aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X1),merge_nat(L1,aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X23),L2))),
                          $ite(X1 = X23,aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X1),merge_nat(L1,L2)),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X23),merge_nat(aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X1),L1),L2))) ) ) ) ) ) ) ) ).

% merge.elims
tff(fact_1113_merge_Oelims,axiom,
    ! [X: list_int,Xa: list_int,Y: list_int] :
      ( ( merge_int(X,Xa) = Y )
     => ( ( ( X = nil_int )
         => ( Y != Xa ) )
       => ( ! [V2: int,Va2: list_int] :
              ( ( X = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,V2),Va2) )
             => ( ( Xa = nil_int )
               => ( Y != aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,V2),Va2) ) ) )
         => ~ ! [X1: int,L1: list_int] :
                ( ( X = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X1),L1) )
               => ! [X23: int,L2: list_int] :
                    ( ( Xa = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X23),L2) )
                   => ( Y != $ite(
                          aa_int_o(aa_int_fun_int_o(ord_less_int,X1),X23),
                          aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X1),merge_int(L1,aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X23),L2))),
                          $ite(X1 = X23,aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X1),merge_int(L1,L2)),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X23),merge_int(aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X1),L1),L2))) ) ) ) ) ) ) ) ).

% merge.elims
tff(fact_1114_list__collect__set__map__simps_I4_J,axiom,
    ! [F: fun_nat_set_nat,X: fun_nat_nat,La: list_nat,L_a: list_nat] : list_c2452340269597857392at_nat(F,aa_list_nat_list_nat(aa_fun4736978287202700568st_nat(map_nat_nat,X),aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,La),L_a))) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,list_c2452340269597857392at_nat(F,aa_list_nat_list_nat(aa_fun4736978287202700568st_nat(map_nat_nat,X),La))),list_c2452340269597857392at_nat(F,aa_list_nat_list_nat(aa_fun4736978287202700568st_nat(map_nat_nat,X),L_a))) ).

% list_collect_set_map_simps(4)
tff(fact_1115_list__collect__set__map__simps_I4_J,axiom,
    ! [F: fun_assn_set_nat,X: fun_Pr3816910165196019820n_assn,La: list_P8527749157015355191n_assn,L_a: list_P8527749157015355191n_assn] : list_c1844713377658005960sn_nat(F,aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,X),aa_lis2797772165808192636n_assn(aa_lis4738365772350288766n_assn(append282499809098378956n_assn,La),L_a))) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,list_c1844713377658005960sn_nat(F,aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,X),La))),list_c1844713377658005960sn_nat(F,aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,X),L_a))) ).

% list_collect_set_map_simps(4)
tff(fact_1116_sorted__list__of__set_Osorted__key__list__of__set__eq__Nil__iff,axiom,
    ! [A2: set_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( linord2612477271533052124et_int(A2) = nil_int )
      <=> ( A2 = bot_bot_set_int ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_eq_Nil_iff
tff(fact_1117_sorted__list__of__set_Osorted__key__list__of__set__eq__Nil__iff,axiom,
    ! [A2: set_o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ( linord3142498349692569832_set_o(A2) = nil_o )
      <=> ( A2 = bot_bot_set_o ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_eq_Nil_iff
tff(fact_1118_sorted__list__of__set_Osorted__key__list__of__set__eq__Nil__iff,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( linord2614967742042102400et_nat(A2) = nil_nat )
      <=> ( A2 = bot_bot_set_nat ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_eq_Nil_iff
tff(fact_1119_lexordp__induct,axiom,
    ! [Xs: list_nat,Ys: list_nat,Pa: fun_li1025265955262268170_nat_o] :
      ( aa_list_nat_o(aa_lis557685189327716851_nat_o(ord_lexordp_nat,Xs),Ys)
     => ( ! [Y2: nat,Ys2: list_nat] : aa_list_nat_o(aa_lis557685189327716851_nat_o(Pa,nil_nat),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Y2),Ys2))
       => ( ! [X2: nat,Xs2: list_nat,Y2: nat,Ys2: list_nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Y2)
             => aa_list_nat_o(aa_lis557685189327716851_nat_o(Pa,aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X2),Xs2)),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Y2),Ys2)) )
         => ( ! [X2: nat,Xs2: list_nat,Ys2: list_nat] :
                ( aa_list_nat_o(aa_lis557685189327716851_nat_o(ord_lexordp_nat,Xs2),Ys2)
               => ( aa_list_nat_o(aa_lis557685189327716851_nat_o(Pa,Xs2),Ys2)
                 => aa_list_nat_o(aa_lis557685189327716851_nat_o(Pa,aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X2),Xs2)),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X2),Ys2)) ) )
           => aa_list_nat_o(aa_lis557685189327716851_nat_o(Pa,Xs),Ys) ) ) ) ) ).

% lexordp_induct
tff(fact_1120_lexordp__induct,axiom,
    ! [Xs: list_int,Ys: list_int,Pa: fun_li1334793589135804498_int_o] :
      ( aa_list_int_o(aa_lis7773489400162621755_int_o(ord_lexordp_int,Xs),Ys)
     => ( ! [Y2: int,Ys2: list_int] : aa_list_int_o(aa_lis7773489400162621755_int_o(Pa,nil_int),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Y2),Ys2))
       => ( ! [X2: int,Xs2: list_int,Y2: int,Ys2: list_int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Y2)
             => aa_list_int_o(aa_lis7773489400162621755_int_o(Pa,aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X2),Xs2)),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Y2),Ys2)) )
         => ( ! [X2: int,Xs2: list_int,Ys2: list_int] :
                ( aa_list_int_o(aa_lis7773489400162621755_int_o(ord_lexordp_int,Xs2),Ys2)
               => ( aa_list_int_o(aa_lis7773489400162621755_int_o(Pa,Xs2),Ys2)
                 => aa_list_int_o(aa_lis7773489400162621755_int_o(Pa,aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X2),Xs2)),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X2),Ys2)) ) )
           => aa_list_int_o(aa_lis7773489400162621755_int_o(Pa,Xs),Ys) ) ) ) ) ).

% lexordp_induct
tff(fact_1121_lexordp__cases,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( aa_list_nat_o(aa_lis557685189327716851_nat_o(ord_lexordp_nat,Xs),Ys)
     => ( ( ( Xs = nil_nat )
         => ! [Y2: nat,Ys4: list_nat] : Ys != aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Y2),Ys4) )
       => ( ! [X2: nat] :
              ( ? [Xs4: list_nat] : Xs = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X2),Xs4)
             => ! [Y2: nat] :
                  ( ? [Ys4: list_nat] : Ys = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Y2),Ys4)
                 => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Y2) ) )
         => ~ ! [X2: nat,Xs4: list_nat] :
                ( ( Xs = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X2),Xs4) )
               => ! [Ys4: list_nat] :
                    ( ( Ys = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X2),Ys4) )
                   => ~ aa_list_nat_o(aa_lis557685189327716851_nat_o(ord_lexordp_nat,Xs4),Ys4) ) ) ) ) ) ).

% lexordp_cases
tff(fact_1122_lexordp__cases,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( aa_list_int_o(aa_lis7773489400162621755_int_o(ord_lexordp_int,Xs),Ys)
     => ( ( ( Xs = nil_int )
         => ! [Y2: int,Ys4: list_int] : Ys != aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Y2),Ys4) )
       => ( ! [X2: int] :
              ( ? [Xs4: list_int] : Xs = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X2),Xs4)
             => ! [Y2: int] :
                  ( ? [Ys4: list_int] : Ys = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Y2),Ys4)
                 => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Y2) ) )
         => ~ ! [X2: int,Xs4: list_int] :
                ( ( Xs = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X2),Xs4) )
               => ! [Ys4: list_int] :
                    ( ( Ys = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X2),Ys4) )
                   => ~ aa_list_int_o(aa_lis7773489400162621755_int_o(ord_lexordp_int,Xs4),Ys4) ) ) ) ) ) ).

% lexordp_cases
tff(fact_1123_lexordp_Osimps,axiom,
    ! [A1: list_assn,A22: list_assn] :
      ( ord_lexordp_assn(A1,A22)
    <=> ( ? [Y3: assn,Ys3: list_assn] :
            ( ( A1 = nil_assn )
            & ( A22 = cons_assn(Y3,Ys3) ) )
        | ? [X3: assn,Y3: assn,Xs3: list_assn,Ys3: list_assn] :
            ( ( A1 = cons_assn(X3,Xs3) )
            & ( A22 = cons_assn(Y3,Ys3) )
            & aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X3),Y3) )
        | ? [X3: assn,Y3: assn,Xs3: list_assn,Ys3: list_assn] :
            ( ( A1 = cons_assn(X3,Xs3) )
            & ( A22 = cons_assn(Y3,Ys3) )
            & ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X3),Y3)
            & ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,Y3),X3)
            & ord_lexordp_assn(Xs3,Ys3) ) ) ) ).

% lexordp.simps
tff(fact_1124_lexordp_Osimps,axiom,
    ! [A1: list_nat,A22: list_nat] :
      ( aa_list_nat_o(aa_lis557685189327716851_nat_o(ord_lexordp_nat,A1),A22)
    <=> ( ? [Y3: nat,Ys3: list_nat] :
            ( ( A1 = nil_nat )
            & ( A22 = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Y3),Ys3) ) )
        | ? [X3: nat,Y3: nat,Xs3: list_nat,Ys3: list_nat] :
            ( ( A1 = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X3),Xs3) )
            & ( A22 = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Y3),Ys3) )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y3) )
        | ? [X3: nat,Y3: nat,Xs3: list_nat,Ys3: list_nat] :
            ( ( A1 = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X3),Xs3) )
            & ( A22 = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Y3),Ys3) )
            & ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y3)
            & ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y3),X3)
            & aa_list_nat_o(aa_lis557685189327716851_nat_o(ord_lexordp_nat,Xs3),Ys3) ) ) ) ).

% lexordp.simps
tff(fact_1125_lexordp_Osimps,axiom,
    ! [A1: list_int,A22: list_int] :
      ( aa_list_int_o(aa_lis7773489400162621755_int_o(ord_lexordp_int,A1),A22)
    <=> ( ? [Y3: int,Ys3: list_int] :
            ( ( A1 = nil_int )
            & ( A22 = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Y3),Ys3) ) )
        | ? [X3: int,Y3: int,Xs3: list_int,Ys3: list_int] :
            ( ( A1 = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X3),Xs3) )
            & ( A22 = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Y3),Ys3) )
            & aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Y3) )
        | ? [X3: int,Y3: int,Xs3: list_int,Ys3: list_int] :
            ( ( A1 = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X3),Xs3) )
            & ( A22 = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Y3),Ys3) )
            & ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Y3)
            & ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Y3),X3)
            & aa_list_int_o(aa_lis7773489400162621755_int_o(ord_lexordp_int,Xs3),Ys3) ) ) ) ).

% lexordp.simps
tff(fact_1126_same__append__eq,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs3: list_nat] :
      ( ( aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Xs),Ys) = aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Xs),Zs3) )
    <=> ( Ys = Zs3 ) ) ).

% same_append_eq
tff(fact_1127_same__append__eq,axiom,
    ! [Xs: list_int,Ys: list_int,Zs3: list_int] :
      ( ( aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Xs),Ys) = aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Xs),Zs3) )
    <=> ( Ys = Zs3 ) ) ).

% same_append_eq
tff(fact_1128_append__same__eq,axiom,
    ! [Ys: list_nat,Xs: list_nat,Zs3: list_nat] :
      ( ( aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Ys),Xs) = aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Zs3),Xs) )
    <=> ( Ys = Zs3 ) ) ).

% append_same_eq
tff(fact_1129_append__same__eq,axiom,
    ! [Ys: list_int,Xs: list_int,Zs3: list_int] :
      ( ( aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Ys),Xs) = aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Zs3),Xs) )
    <=> ( Ys = Zs3 ) ) ).

% append_same_eq
tff(fact_1130_append__assoc,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs3: list_nat] : aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Xs),Ys)),Zs3) = aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Xs),aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Ys),Zs3)) ).

% append_assoc
tff(fact_1131_append__assoc,axiom,
    ! [Xs: list_int,Ys: list_int,Zs3: list_int] : aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Xs),Ys)),Zs3) = aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Xs),aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Ys),Zs3)) ).

% append_assoc
tff(fact_1132_append_Oassoc,axiom,
    ! [A: list_nat,B: list_nat,C2: list_nat] : aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,A),B)),C2) = aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,A),aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,B),C2)) ).

% append.assoc
tff(fact_1133_append_Oassoc,axiom,
    ! [A: list_int,B: list_int,C2: list_int] : aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,A),B)),C2) = aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,A),aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,B),C2)) ).

% append.assoc
tff(fact_1134_merge__pure__or,axiom,
    ! [A: $o,B: $o] :
      aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_o_assn(pure_assn,(A))),aa_o_assn(pure_assn,(B))) = aa_o_assn(pure_assn,
        ( (A)
        | (B) )) ).

% merge_pure_or
tff(fact_1135_append__is__Nil__conv,axiom,
    ! [Xs: list_a,Ys: list_a] :
      ( ( aa_list_a_list_a(aa_lis2442066584718795531list_a(append_a,Xs),Ys) = nil_a )
    <=> ( ( Xs = nil_a )
        & ( Ys = nil_a ) ) ) ).

% append_is_Nil_conv
tff(fact_1136_append__is__Nil__conv,axiom,
    ! [Xs: list_b,Ys: list_b] :
      ( ( aa_list_b_list_b(aa_lis927650220358621516list_b(append_b,Xs),Ys) = nil_b )
    <=> ( ( Xs = nil_b )
        & ( Ys = nil_b ) ) ) ).

% append_is_Nil_conv
tff(fact_1137_append__is__Nil__conv,axiom,
    ! [Xs: list_P8527749157015355191n_assn,Ys: list_P8527749157015355191n_assn] :
      ( ( aa_lis2797772165808192636n_assn(aa_lis4738365772350288766n_assn(append282499809098378956n_assn,Xs),Ys) = nil_Pr5671120429643327159n_assn )
    <=> ( ( Xs = nil_Pr5671120429643327159n_assn )
        & ( Ys = nil_Pr5671120429643327159n_assn ) ) ) ).

% append_is_Nil_conv
tff(fact_1138_append__is__Nil__conv,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Xs),Ys) = nil_nat )
    <=> ( ( Xs = nil_nat )
        & ( Ys = nil_nat ) ) ) ).

% append_is_Nil_conv
tff(fact_1139_append__is__Nil__conv,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Xs),Ys) = nil_int )
    <=> ( ( Xs = nil_int )
        & ( Ys = nil_int ) ) ) ).

% append_is_Nil_conv
tff(fact_1140_Nil__is__append__conv,axiom,
    ! [Xs: list_a,Ys: list_a] :
      ( ( nil_a = aa_list_a_list_a(aa_lis2442066584718795531list_a(append_a,Xs),Ys) )
    <=> ( ( Xs = nil_a )
        & ( Ys = nil_a ) ) ) ).

% Nil_is_append_conv
tff(fact_1141_Nil__is__append__conv,axiom,
    ! [Xs: list_b,Ys: list_b] :
      ( ( nil_b = aa_list_b_list_b(aa_lis927650220358621516list_b(append_b,Xs),Ys) )
    <=> ( ( Xs = nil_b )
        & ( Ys = nil_b ) ) ) ).

% Nil_is_append_conv
tff(fact_1142_Nil__is__append__conv,axiom,
    ! [Xs: list_P8527749157015355191n_assn,Ys: list_P8527749157015355191n_assn] :
      ( ( nil_Pr5671120429643327159n_assn = aa_lis2797772165808192636n_assn(aa_lis4738365772350288766n_assn(append282499809098378956n_assn,Xs),Ys) )
    <=> ( ( Xs = nil_Pr5671120429643327159n_assn )
        & ( Ys = nil_Pr5671120429643327159n_assn ) ) ) ).

% Nil_is_append_conv
tff(fact_1143_Nil__is__append__conv,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( nil_nat = aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Xs),Ys) )
    <=> ( ( Xs = nil_nat )
        & ( Ys = nil_nat ) ) ) ).

% Nil_is_append_conv
tff(fact_1144_Nil__is__append__conv,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( nil_int = aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Xs),Ys) )
    <=> ( ( Xs = nil_int )
        & ( Ys = nil_int ) ) ) ).

% Nil_is_append_conv
tff(fact_1145_self__append__conv2,axiom,
    ! [Y: list_a,Xs: list_a] :
      ( ( Y = aa_list_a_list_a(aa_lis2442066584718795531list_a(append_a,Xs),Y) )
    <=> ( Xs = nil_a ) ) ).

% self_append_conv2
tff(fact_1146_self__append__conv2,axiom,
    ! [Y: list_b,Xs: list_b] :
      ( ( Y = aa_list_b_list_b(aa_lis927650220358621516list_b(append_b,Xs),Y) )
    <=> ( Xs = nil_b ) ) ).

% self_append_conv2
tff(fact_1147_self__append__conv2,axiom,
    ! [Y: list_P8527749157015355191n_assn,Xs: list_P8527749157015355191n_assn] :
      ( ( Y = aa_lis2797772165808192636n_assn(aa_lis4738365772350288766n_assn(append282499809098378956n_assn,Xs),Y) )
    <=> ( Xs = nil_Pr5671120429643327159n_assn ) ) ).

% self_append_conv2
tff(fact_1148_self__append__conv2,axiom,
    ! [Y: list_nat,Xs: list_nat] :
      ( ( Y = aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Xs),Y) )
    <=> ( Xs = nil_nat ) ) ).

% self_append_conv2
tff(fact_1149_self__append__conv2,axiom,
    ! [Y: list_int,Xs: list_int] :
      ( ( Y = aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Xs),Y) )
    <=> ( Xs = nil_int ) ) ).

% self_append_conv2
tff(fact_1150_append__self__conv2,axiom,
    ! [Xs: list_a,Ys: list_a] :
      ( ( aa_list_a_list_a(aa_lis2442066584718795531list_a(append_a,Xs),Ys) = Ys )
    <=> ( Xs = nil_a ) ) ).

% append_self_conv2
tff(fact_1151_append__self__conv2,axiom,
    ! [Xs: list_b,Ys: list_b] :
      ( ( aa_list_b_list_b(aa_lis927650220358621516list_b(append_b,Xs),Ys) = Ys )
    <=> ( Xs = nil_b ) ) ).

% append_self_conv2
tff(fact_1152_append__self__conv2,axiom,
    ! [Xs: list_P8527749157015355191n_assn,Ys: list_P8527749157015355191n_assn] :
      ( ( aa_lis2797772165808192636n_assn(aa_lis4738365772350288766n_assn(append282499809098378956n_assn,Xs),Ys) = Ys )
    <=> ( Xs = nil_Pr5671120429643327159n_assn ) ) ).

% append_self_conv2
tff(fact_1153_append__self__conv2,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Xs),Ys) = Ys )
    <=> ( Xs = nil_nat ) ) ).

% append_self_conv2
tff(fact_1154_append__self__conv2,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Xs),Ys) = Ys )
    <=> ( Xs = nil_int ) ) ).

% append_self_conv2
tff(fact_1155_self__append__conv,axiom,
    ! [Y: list_a,Ys: list_a] :
      ( ( Y = aa_list_a_list_a(aa_lis2442066584718795531list_a(append_a,Y),Ys) )
    <=> ( Ys = nil_a ) ) ).

% self_append_conv
tff(fact_1156_self__append__conv,axiom,
    ! [Y: list_b,Ys: list_b] :
      ( ( Y = aa_list_b_list_b(aa_lis927650220358621516list_b(append_b,Y),Ys) )
    <=> ( Ys = nil_b ) ) ).

% self_append_conv
tff(fact_1157_self__append__conv,axiom,
    ! [Y: list_P8527749157015355191n_assn,Ys: list_P8527749157015355191n_assn] :
      ( ( Y = aa_lis2797772165808192636n_assn(aa_lis4738365772350288766n_assn(append282499809098378956n_assn,Y),Ys) )
    <=> ( Ys = nil_Pr5671120429643327159n_assn ) ) ).

% self_append_conv
tff(fact_1158_self__append__conv,axiom,
    ! [Y: list_nat,Ys: list_nat] :
      ( ( Y = aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Y),Ys) )
    <=> ( Ys = nil_nat ) ) ).

% self_append_conv
tff(fact_1159_self__append__conv,axiom,
    ! [Y: list_int,Ys: list_int] :
      ( ( Y = aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Y),Ys) )
    <=> ( Ys = nil_int ) ) ).

% self_append_conv
tff(fact_1160_append__self__conv,axiom,
    ! [Xs: list_a,Ys: list_a] :
      ( ( aa_list_a_list_a(aa_lis2442066584718795531list_a(append_a,Xs),Ys) = Xs )
    <=> ( Ys = nil_a ) ) ).

% append_self_conv
tff(fact_1161_append__self__conv,axiom,
    ! [Xs: list_b,Ys: list_b] :
      ( ( aa_list_b_list_b(aa_lis927650220358621516list_b(append_b,Xs),Ys) = Xs )
    <=> ( Ys = nil_b ) ) ).

% append_self_conv
tff(fact_1162_append__self__conv,axiom,
    ! [Xs: list_P8527749157015355191n_assn,Ys: list_P8527749157015355191n_assn] :
      ( ( aa_lis2797772165808192636n_assn(aa_lis4738365772350288766n_assn(append282499809098378956n_assn,Xs),Ys) = Xs )
    <=> ( Ys = nil_Pr5671120429643327159n_assn ) ) ).

% append_self_conv
tff(fact_1163_append__self__conv,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Xs),Ys) = Xs )
    <=> ( Ys = nil_nat ) ) ).

% append_self_conv
tff(fact_1164_append__self__conv,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Xs),Ys) = Xs )
    <=> ( Ys = nil_int ) ) ).

% append_self_conv
tff(fact_1165_append__Nil2,axiom,
    ! [Xs: list_a] : aa_list_a_list_a(aa_lis2442066584718795531list_a(append_a,Xs),nil_a) = Xs ).

% append_Nil2
tff(fact_1166_append__Nil2,axiom,
    ! [Xs: list_b] : aa_list_b_list_b(aa_lis927650220358621516list_b(append_b,Xs),nil_b) = Xs ).

% append_Nil2
tff(fact_1167_append__Nil2,axiom,
    ! [Xs: list_P8527749157015355191n_assn] : aa_lis2797772165808192636n_assn(aa_lis4738365772350288766n_assn(append282499809098378956n_assn,Xs),nil_Pr5671120429643327159n_assn) = Xs ).

% append_Nil2
tff(fact_1168_append__Nil2,axiom,
    ! [Xs: list_nat] : aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Xs),nil_nat) = Xs ).

% append_Nil2
tff(fact_1169_append__Nil2,axiom,
    ! [Xs: list_int] : aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Xs),nil_int) = Xs ).

% append_Nil2
tff(fact_1170_append_Oright__neutral,axiom,
    ! [A: list_a] : aa_list_a_list_a(aa_lis2442066584718795531list_a(append_a,A),nil_a) = A ).

% append.right_neutral
tff(fact_1171_append_Oright__neutral,axiom,
    ! [A: list_b] : aa_list_b_list_b(aa_lis927650220358621516list_b(append_b,A),nil_b) = A ).

% append.right_neutral
tff(fact_1172_append_Oright__neutral,axiom,
    ! [A: list_P8527749157015355191n_assn] : aa_lis2797772165808192636n_assn(aa_lis4738365772350288766n_assn(append282499809098378956n_assn,A),nil_Pr5671120429643327159n_assn) = A ).

% append.right_neutral
tff(fact_1173_append_Oright__neutral,axiom,
    ! [A: list_nat] : aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,A),nil_nat) = A ).

% append.right_neutral
tff(fact_1174_append_Oright__neutral,axiom,
    ! [A: list_int] : aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,A),nil_int) = A ).

% append.right_neutral
tff(fact_1175_empty__append__eq__id,axiom,
    ! [X4: list_a] : aa_list_a_list_a(aa_lis2442066584718795531list_a(append_a,nil_a),X4) = X4 ).

% empty_append_eq_id
tff(fact_1176_empty__append__eq__id,axiom,
    ! [X4: list_b] : aa_list_b_list_b(aa_lis927650220358621516list_b(append_b,nil_b),X4) = X4 ).

% empty_append_eq_id
tff(fact_1177_empty__append__eq__id,axiom,
    ! [X4: list_P8527749157015355191n_assn] : aa_lis2797772165808192636n_assn(aa_lis4738365772350288766n_assn(append282499809098378956n_assn,nil_Pr5671120429643327159n_assn),X4) = X4 ).

% empty_append_eq_id
tff(fact_1178_empty__append__eq__id,axiom,
    ! [X4: list_nat] : aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,nil_nat),X4) = X4 ).

% empty_append_eq_id
tff(fact_1179_empty__append__eq__id,axiom,
    ! [X4: list_int] : aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,nil_int),X4) = X4 ).

% empty_append_eq_id
tff(fact_1180_map__append,axiom,
    ! [F: fun_nat_int,Xs: list_nat,Ys: list_nat] : map_nat_int(F,aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Xs),Ys)) = aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,map_nat_int(F,Xs)),map_nat_int(F,Ys)) ).

% map_append
tff(fact_1181_map__append,axiom,
    ! [F: fun_int_nat,Xs: list_int,Ys: list_int] : map_int_nat(F,aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Xs),Ys)) = aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,map_int_nat(F,Xs)),map_int_nat(F,Ys)) ).

% map_append
tff(fact_1182_map__append,axiom,
    ! [F: fun_int_int,Xs: list_int,Ys: list_int] : map_int_int(F,aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Xs),Ys)) = aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,map_int_int(F,Xs)),map_int_int(F,Ys)) ).

% map_append
tff(fact_1183_map__append,axiom,
    ! [F: fun_nat_nat,Xs: list_nat,Ys: list_nat] : aa_list_nat_list_nat(aa_fun4736978287202700568st_nat(map_nat_nat,F),aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Xs),Ys)) = aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,aa_list_nat_list_nat(aa_fun4736978287202700568st_nat(map_nat_nat,F),Xs)),aa_list_nat_list_nat(aa_fun4736978287202700568st_nat(map_nat_nat,F),Ys)) ).

% map_append
tff(fact_1184_map__append,axiom,
    ! [F: fun_Pr3816910165196019820n_assn,Xs: list_P8527749157015355191n_assn,Ys: list_P8527749157015355191n_assn] : aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,F),aa_lis2797772165808192636n_assn(aa_lis4738365772350288766n_assn(append282499809098378956n_assn,Xs),Ys)) = append_assn(aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,F),Xs),aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,F),Ys)) ).

% map_append
tff(fact_1185_rev__is__Nil__conv,axiom,
    ! [Xs: list_a] :
      ( ( rev_a(Xs) = nil_a )
    <=> ( Xs = nil_a ) ) ).

% rev_is_Nil_conv
tff(fact_1186_rev__is__Nil__conv,axiom,
    ! [Xs: list_b] :
      ( ( rev_b(Xs) = nil_b )
    <=> ( Xs = nil_b ) ) ).

% rev_is_Nil_conv
tff(fact_1187_rev__is__Nil__conv,axiom,
    ! [Xs: list_P8527749157015355191n_assn] :
      ( ( rev_Pr4855572775806611735n_assn(Xs) = nil_Pr5671120429643327159n_assn )
    <=> ( Xs = nil_Pr5671120429643327159n_assn ) ) ).

% rev_is_Nil_conv
tff(fact_1188_rev__is__Nil__conv,axiom,
    ! [Xs: list_nat] :
      ( ( rev_nat(Xs) = nil_nat )
    <=> ( Xs = nil_nat ) ) ).

% rev_is_Nil_conv
tff(fact_1189_rev__is__Nil__conv,axiom,
    ! [Xs: list_int] :
      ( ( rev_int(Xs) = nil_int )
    <=> ( Xs = nil_int ) ) ).

% rev_is_Nil_conv
tff(fact_1190_Nil__is__rev__conv,axiom,
    ! [Xs: list_a] :
      ( ( nil_a = rev_a(Xs) )
    <=> ( Xs = nil_a ) ) ).

% Nil_is_rev_conv
tff(fact_1191_Nil__is__rev__conv,axiom,
    ! [Xs: list_b] :
      ( ( nil_b = rev_b(Xs) )
    <=> ( Xs = nil_b ) ) ).

% Nil_is_rev_conv
tff(fact_1192_Nil__is__rev__conv,axiom,
    ! [Xs: list_P8527749157015355191n_assn] :
      ( ( nil_Pr5671120429643327159n_assn = rev_Pr4855572775806611735n_assn(Xs) )
    <=> ( Xs = nil_Pr5671120429643327159n_assn ) ) ).

% Nil_is_rev_conv
tff(fact_1193_Nil__is__rev__conv,axiom,
    ! [Xs: list_nat] :
      ( ( nil_nat = rev_nat(Xs) )
    <=> ( Xs = nil_nat ) ) ).

% Nil_is_rev_conv
tff(fact_1194_Nil__is__rev__conv,axiom,
    ! [Xs: list_int] :
      ( ( nil_int = rev_int(Xs) )
    <=> ( Xs = nil_int ) ) ).

% Nil_is_rev_conv
tff(fact_1195_rev__append,axiom,
    ! [Xs: list_nat,Ys: list_nat] : rev_nat(aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Xs),Ys)) = aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,rev_nat(Ys)),rev_nat(Xs)) ).

% rev_append
tff(fact_1196_rev__append,axiom,
    ! [Xs: list_int,Ys: list_int] : rev_int(aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Xs),Ys)) = aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,rev_int(Ys)),rev_int(Xs)) ).

% rev_append
tff(fact_1197_lexordp__simps_I1_J,axiom,
    ! [Ys: list_nat] :
      ( aa_list_nat_o(aa_lis557685189327716851_nat_o(ord_lexordp_nat,nil_nat),Ys)
    <=> ( Ys != nil_nat ) ) ).

% lexordp_simps(1)
tff(fact_1198_lexordp__simps_I1_J,axiom,
    ! [Ys: list_int] :
      ( aa_list_int_o(aa_lis7773489400162621755_int_o(ord_lexordp_int,nil_int),Ys)
    <=> ( Ys != nil_int ) ) ).

% lexordp_simps(1)
tff(fact_1199_lexordp__simps_I2_J,axiom,
    ! [Xs: list_nat] : ~ aa_list_nat_o(aa_lis557685189327716851_nat_o(ord_lexordp_nat,Xs),nil_nat) ).

% lexordp_simps(2)
tff(fact_1200_lexordp__simps_I2_J,axiom,
    ! [Xs: list_int] : ~ aa_list_int_o(aa_lis7773489400162621755_int_o(ord_lexordp_int,Xs),nil_int) ).

% lexordp_simps(2)
tff(fact_1201_append1__eq__conv,axiom,
    ! [Xs: list_a,X: a,Ys: list_a,Y: a] :
      ( ( aa_list_a_list_a(aa_lis2442066584718795531list_a(append_a,Xs),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X),nil_a)) = aa_list_a_list_a(aa_lis2442066584718795531list_a(append_a,Ys),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,Y),nil_a)) )
    <=> ( ( Xs = Ys )
        & ( X = Y ) ) ) ).

% append1_eq_conv
tff(fact_1202_append1__eq__conv,axiom,
    ! [Xs: list_b,X: b,Ys: list_b,Y: b] :
      ( ( aa_list_b_list_b(aa_lis927650220358621516list_b(append_b,Xs),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X),nil_b)) = aa_list_b_list_b(aa_lis927650220358621516list_b(append_b,Ys),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,Y),nil_b)) )
    <=> ( ( Xs = Ys )
        & ( X = Y ) ) ) ).

% append1_eq_conv
tff(fact_1203_append1__eq__conv,axiom,
    ! [Xs: list_nat,X: nat,Ys: list_nat,Y: nat] :
      ( ( aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Xs),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X),nil_nat)) = aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Ys),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Y),nil_nat)) )
    <=> ( ( Xs = Ys )
        & ( X = Y ) ) ) ).

% append1_eq_conv
tff(fact_1204_append1__eq__conv,axiom,
    ! [Xs: list_int,X: int,Ys: list_int,Y: int] :
      ( ( aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Xs),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X),nil_int)) = aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Ys),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Y),nil_int)) )
    <=> ( ( Xs = Ys )
        & ( X = Y ) ) ) ).

% append1_eq_conv
tff(fact_1205_append1__eq__conv,axiom,
    ! [Xs: list_P8527749157015355191n_assn,X: produc6575502325842934193n_assn,Ys: list_P8527749157015355191n_assn,Y: produc6575502325842934193n_assn] :
      ( ( aa_lis2797772165808192636n_assn(aa_lis4738365772350288766n_assn(append282499809098378956n_assn,Xs),aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X),nil_Pr5671120429643327159n_assn)) = aa_lis2797772165808192636n_assn(aa_lis4738365772350288766n_assn(append282499809098378956n_assn,Ys),aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,Y),nil_Pr5671120429643327159n_assn)) )
    <=> ( ( Xs = Ys )
        & ( X = Y ) ) ) ).

% append1_eq_conv
tff(fact_1206_list__e__eq__lel_I2_J,axiom,
    ! [L12: list_a,E2: a,L22: list_a,E3: a] :
      ( ( aa_list_a_list_a(aa_lis2442066584718795531list_a(append_a,L12),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,E2),L22)) = aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,E3),nil_a) )
    <=> ( ( L12 = nil_a )
        & ( E2 = E3 )
        & ( L22 = nil_a ) ) ) ).

% list_e_eq_lel(2)
tff(fact_1207_list__e__eq__lel_I2_J,axiom,
    ! [L12: list_b,E2: b,L22: list_b,E3: b] :
      ( ( aa_list_b_list_b(aa_lis927650220358621516list_b(append_b,L12),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,E2),L22)) = aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,E3),nil_b) )
    <=> ( ( L12 = nil_b )
        & ( E2 = E3 )
        & ( L22 = nil_b ) ) ) ).

% list_e_eq_lel(2)
tff(fact_1208_list__e__eq__lel_I2_J,axiom,
    ! [L12: list_nat,E2: nat,L22: list_nat,E3: nat] :
      ( ( aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,L12),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,E2),L22)) = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,E3),nil_nat) )
    <=> ( ( L12 = nil_nat )
        & ( E2 = E3 )
        & ( L22 = nil_nat ) ) ) ).

% list_e_eq_lel(2)
tff(fact_1209_list__e__eq__lel_I2_J,axiom,
    ! [L12: list_int,E2: int,L22: list_int,E3: int] :
      ( ( aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,L12),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,E2),L22)) = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,E3),nil_int) )
    <=> ( ( L12 = nil_int )
        & ( E2 = E3 )
        & ( L22 = nil_int ) ) ) ).

% list_e_eq_lel(2)
tff(fact_1210_list__e__eq__lel_I2_J,axiom,
    ! [L12: list_P8527749157015355191n_assn,E2: produc6575502325842934193n_assn,L22: list_P8527749157015355191n_assn,E3: produc6575502325842934193n_assn] :
      ( ( aa_lis2797772165808192636n_assn(aa_lis4738365772350288766n_assn(append282499809098378956n_assn,L12),aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,E2),L22)) = aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,E3),nil_Pr5671120429643327159n_assn) )
    <=> ( ( L12 = nil_Pr5671120429643327159n_assn )
        & ( E2 = E3 )
        & ( L22 = nil_Pr5671120429643327159n_assn ) ) ) ).

% list_e_eq_lel(2)
tff(fact_1211_list__e__eq__lel_I1_J,axiom,
    ! [E3: a,L12: list_a,E2: a,L22: list_a] :
      ( ( aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,E3),nil_a) = aa_list_a_list_a(aa_lis2442066584718795531list_a(append_a,L12),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,E2),L22)) )
    <=> ( ( L12 = nil_a )
        & ( E2 = E3 )
        & ( L22 = nil_a ) ) ) ).

% list_e_eq_lel(1)
tff(fact_1212_list__e__eq__lel_I1_J,axiom,
    ! [E3: b,L12: list_b,E2: b,L22: list_b] :
      ( ( aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,E3),nil_b) = aa_list_b_list_b(aa_lis927650220358621516list_b(append_b,L12),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,E2),L22)) )
    <=> ( ( L12 = nil_b )
        & ( E2 = E3 )
        & ( L22 = nil_b ) ) ) ).

% list_e_eq_lel(1)
tff(fact_1213_list__e__eq__lel_I1_J,axiom,
    ! [E3: nat,L12: list_nat,E2: nat,L22: list_nat] :
      ( ( aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,E3),nil_nat) = aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,L12),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,E2),L22)) )
    <=> ( ( L12 = nil_nat )
        & ( E2 = E3 )
        & ( L22 = nil_nat ) ) ) ).

% list_e_eq_lel(1)
tff(fact_1214_list__e__eq__lel_I1_J,axiom,
    ! [E3: int,L12: list_int,E2: int,L22: list_int] :
      ( ( aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,E3),nil_int) = aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,L12),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,E2),L22)) )
    <=> ( ( L12 = nil_int )
        & ( E2 = E3 )
        & ( L22 = nil_int ) ) ) ).

% list_e_eq_lel(1)
tff(fact_1215_list__e__eq__lel_I1_J,axiom,
    ! [E3: produc6575502325842934193n_assn,L12: list_P8527749157015355191n_assn,E2: produc6575502325842934193n_assn,L22: list_P8527749157015355191n_assn] :
      ( ( aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,E3),nil_Pr5671120429643327159n_assn) = aa_lis2797772165808192636n_assn(aa_lis4738365772350288766n_assn(append282499809098378956n_assn,L12),aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,E2),L22)) )
    <=> ( ( L12 = nil_Pr5671120429643327159n_assn )
        & ( E2 = E3 )
        & ( L22 = nil_Pr5671120429643327159n_assn ) ) ) ).

% list_e_eq_lel(1)
tff(fact_1216_list__se__match_I4_J,axiom,
    ! [L22: list_a,A: a,L12: list_a] :
      ( ( L22 != nil_a )
     => ( ( aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,A),nil_a) = aa_list_a_list_a(aa_lis2442066584718795531list_a(append_a,L12),L22) )
      <=> ( ( L12 = nil_a )
          & ( L22 = aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,A),nil_a) ) ) ) ) ).

% list_se_match(4)
tff(fact_1217_list__se__match_I4_J,axiom,
    ! [L22: list_b,A: b,L12: list_b] :
      ( ( L22 != nil_b )
     => ( ( aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,A),nil_b) = aa_list_b_list_b(aa_lis927650220358621516list_b(append_b,L12),L22) )
      <=> ( ( L12 = nil_b )
          & ( L22 = aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,A),nil_b) ) ) ) ) ).

% list_se_match(4)
tff(fact_1218_list__se__match_I4_J,axiom,
    ! [L22: list_nat,A: nat,L12: list_nat] :
      ( ( L22 != nil_nat )
     => ( ( aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,A),nil_nat) = aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,L12),L22) )
      <=> ( ( L12 = nil_nat )
          & ( L22 = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,A),nil_nat) ) ) ) ) ).

% list_se_match(4)
tff(fact_1219_list__se__match_I4_J,axiom,
    ! [L22: list_int,A: int,L12: list_int] :
      ( ( L22 != nil_int )
     => ( ( aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,A),nil_int) = aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,L12),L22) )
      <=> ( ( L12 = nil_int )
          & ( L22 = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,A),nil_int) ) ) ) ) ).

% list_se_match(4)
tff(fact_1220_list__se__match_I4_J,axiom,
    ! [L22: list_P8527749157015355191n_assn,A: produc6575502325842934193n_assn,L12: list_P8527749157015355191n_assn] :
      ( ( L22 != nil_Pr5671120429643327159n_assn )
     => ( ( aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,A),nil_Pr5671120429643327159n_assn) = aa_lis2797772165808192636n_assn(aa_lis4738365772350288766n_assn(append282499809098378956n_assn,L12),L22) )
      <=> ( ( L12 = nil_Pr5671120429643327159n_assn )
          & ( L22 = aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,A),nil_Pr5671120429643327159n_assn) ) ) ) ) ).

% list_se_match(4)
tff(fact_1221_list__se__match_I3_J,axiom,
    ! [L12: list_a,A: a,L22: list_a] :
      ( ( L12 != nil_a )
     => ( ( aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,A),nil_a) = aa_list_a_list_a(aa_lis2442066584718795531list_a(append_a,L12),L22) )
      <=> ( ( L12 = aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,A),nil_a) )
          & ( L22 = nil_a ) ) ) ) ).

% list_se_match(3)
tff(fact_1222_list__se__match_I3_J,axiom,
    ! [L12: list_b,A: b,L22: list_b] :
      ( ( L12 != nil_b )
     => ( ( aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,A),nil_b) = aa_list_b_list_b(aa_lis927650220358621516list_b(append_b,L12),L22) )
      <=> ( ( L12 = aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,A),nil_b) )
          & ( L22 = nil_b ) ) ) ) ).

% list_se_match(3)
tff(fact_1223_list__se__match_I3_J,axiom,
    ! [L12: list_nat,A: nat,L22: list_nat] :
      ( ( L12 != nil_nat )
     => ( ( aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,A),nil_nat) = aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,L12),L22) )
      <=> ( ( L12 = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,A),nil_nat) )
          & ( L22 = nil_nat ) ) ) ) ).

% list_se_match(3)
tff(fact_1224_list__se__match_I3_J,axiom,
    ! [L12: list_int,A: int,L22: list_int] :
      ( ( L12 != nil_int )
     => ( ( aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,A),nil_int) = aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,L12),L22) )
      <=> ( ( L12 = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,A),nil_int) )
          & ( L22 = nil_int ) ) ) ) ).

% list_se_match(3)
tff(fact_1225_list__se__match_I3_J,axiom,
    ! [L12: list_P8527749157015355191n_assn,A: produc6575502325842934193n_assn,L22: list_P8527749157015355191n_assn] :
      ( ( L12 != nil_Pr5671120429643327159n_assn )
     => ( ( aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,A),nil_Pr5671120429643327159n_assn) = aa_lis2797772165808192636n_assn(aa_lis4738365772350288766n_assn(append282499809098378956n_assn,L12),L22) )
      <=> ( ( L12 = aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,A),nil_Pr5671120429643327159n_assn) )
          & ( L22 = nil_Pr5671120429643327159n_assn ) ) ) ) ).

% list_se_match(3)
tff(fact_1226_list__se__match_I2_J,axiom,
    ! [L22: list_a,L12: list_a,A: a] :
      ( ( L22 != nil_a )
     => ( ( aa_list_a_list_a(aa_lis2442066584718795531list_a(append_a,L12),L22) = aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,A),nil_a) )
      <=> ( ( L12 = nil_a )
          & ( L22 = aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,A),nil_a) ) ) ) ) ).

% list_se_match(2)
tff(fact_1227_list__se__match_I2_J,axiom,
    ! [L22: list_b,L12: list_b,A: b] :
      ( ( L22 != nil_b )
     => ( ( aa_list_b_list_b(aa_lis927650220358621516list_b(append_b,L12),L22) = aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,A),nil_b) )
      <=> ( ( L12 = nil_b )
          & ( L22 = aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,A),nil_b) ) ) ) ) ).

% list_se_match(2)
tff(fact_1228_list__se__match_I2_J,axiom,
    ! [L22: list_nat,L12: list_nat,A: nat] :
      ( ( L22 != nil_nat )
     => ( ( aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,L12),L22) = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,A),nil_nat) )
      <=> ( ( L12 = nil_nat )
          & ( L22 = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,A),nil_nat) ) ) ) ) ).

% list_se_match(2)
tff(fact_1229_list__se__match_I2_J,axiom,
    ! [L22: list_int,L12: list_int,A: int] :
      ( ( L22 != nil_int )
     => ( ( aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,L12),L22) = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,A),nil_int) )
      <=> ( ( L12 = nil_int )
          & ( L22 = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,A),nil_int) ) ) ) ) ).

% list_se_match(2)
tff(fact_1230_list__se__match_I2_J,axiom,
    ! [L22: list_P8527749157015355191n_assn,L12: list_P8527749157015355191n_assn,A: produc6575502325842934193n_assn] :
      ( ( L22 != nil_Pr5671120429643327159n_assn )
     => ( ( aa_lis2797772165808192636n_assn(aa_lis4738365772350288766n_assn(append282499809098378956n_assn,L12),L22) = aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,A),nil_Pr5671120429643327159n_assn) )
      <=> ( ( L12 = nil_Pr5671120429643327159n_assn )
          & ( L22 = aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,A),nil_Pr5671120429643327159n_assn) ) ) ) ) ).

% list_se_match(2)
tff(fact_1231_list__se__match_I1_J,axiom,
    ! [L12: list_a,L22: list_a,A: a] :
      ( ( L12 != nil_a )
     => ( ( aa_list_a_list_a(aa_lis2442066584718795531list_a(append_a,L12),L22) = aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,A),nil_a) )
      <=> ( ( L12 = aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,A),nil_a) )
          & ( L22 = nil_a ) ) ) ) ).

% list_se_match(1)
tff(fact_1232_list__se__match_I1_J,axiom,
    ! [L12: list_b,L22: list_b,A: b] :
      ( ( L12 != nil_b )
     => ( ( aa_list_b_list_b(aa_lis927650220358621516list_b(append_b,L12),L22) = aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,A),nil_b) )
      <=> ( ( L12 = aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,A),nil_b) )
          & ( L22 = nil_b ) ) ) ) ).

% list_se_match(1)
tff(fact_1233_list__se__match_I1_J,axiom,
    ! [L12: list_nat,L22: list_nat,A: nat] :
      ( ( L12 != nil_nat )
     => ( ( aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,L12),L22) = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,A),nil_nat) )
      <=> ( ( L12 = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,A),nil_nat) )
          & ( L22 = nil_nat ) ) ) ) ).

% list_se_match(1)
tff(fact_1234_list__se__match_I1_J,axiom,
    ! [L12: list_int,L22: list_int,A: int] :
      ( ( L12 != nil_int )
     => ( ( aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,L12),L22) = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,A),nil_int) )
      <=> ( ( L12 = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,A),nil_int) )
          & ( L22 = nil_int ) ) ) ) ).

% list_se_match(1)
tff(fact_1235_list__se__match_I1_J,axiom,
    ! [L12: list_P8527749157015355191n_assn,L22: list_P8527749157015355191n_assn,A: produc6575502325842934193n_assn] :
      ( ( L12 != nil_Pr5671120429643327159n_assn )
     => ( ( aa_lis2797772165808192636n_assn(aa_lis4738365772350288766n_assn(append282499809098378956n_assn,L12),L22) = aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,A),nil_Pr5671120429643327159n_assn) )
      <=> ( ( L12 = aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,A),nil_Pr5671120429643327159n_assn) )
          & ( L22 = nil_Pr5671120429643327159n_assn ) ) ) ) ).

% list_se_match(1)
tff(fact_1236_list__ee__eq__leel_I2_J,axiom,
    ! [L12: list_a,E1: a,E22: a,L22: list_a,E12: a,E23: a] :
      ( ( aa_list_a_list_a(aa_lis2442066584718795531list_a(append_a,L12),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,E1),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,E22),L22))) = aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,E12),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,E23),nil_a)) )
    <=> ( ( L12 = nil_a )
        & ( E12 = E1 )
        & ( E23 = E22 )
        & ( L22 = nil_a ) ) ) ).

% list_ee_eq_leel(2)
tff(fact_1237_list__ee__eq__leel_I2_J,axiom,
    ! [L12: list_b,E1: b,E22: b,L22: list_b,E12: b,E23: b] :
      ( ( aa_list_b_list_b(aa_lis927650220358621516list_b(append_b,L12),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,E1),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,E22),L22))) = aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,E12),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,E23),nil_b)) )
    <=> ( ( L12 = nil_b )
        & ( E12 = E1 )
        & ( E23 = E22 )
        & ( L22 = nil_b ) ) ) ).

% list_ee_eq_leel(2)
tff(fact_1238_list__ee__eq__leel_I2_J,axiom,
    ! [L12: list_nat,E1: nat,E22: nat,L22: list_nat,E12: nat,E23: nat] :
      ( ( aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,L12),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,E1),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,E22),L22))) = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,E12),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,E23),nil_nat)) )
    <=> ( ( L12 = nil_nat )
        & ( E12 = E1 )
        & ( E23 = E22 )
        & ( L22 = nil_nat ) ) ) ).

% list_ee_eq_leel(2)
tff(fact_1239_list__ee__eq__leel_I2_J,axiom,
    ! [L12: list_int,E1: int,E22: int,L22: list_int,E12: int,E23: int] :
      ( ( aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,L12),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,E1),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,E22),L22))) = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,E12),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,E23),nil_int)) )
    <=> ( ( L12 = nil_int )
        & ( E12 = E1 )
        & ( E23 = E22 )
        & ( L22 = nil_int ) ) ) ).

% list_ee_eq_leel(2)
tff(fact_1240_list__ee__eq__leel_I2_J,axiom,
    ! [L12: list_P8527749157015355191n_assn,E1: produc6575502325842934193n_assn,E22: produc6575502325842934193n_assn,L22: list_P8527749157015355191n_assn,E12: produc6575502325842934193n_assn,E23: produc6575502325842934193n_assn] :
      ( ( aa_lis2797772165808192636n_assn(aa_lis4738365772350288766n_assn(append282499809098378956n_assn,L12),aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,E1),aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,E22),L22))) = aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,E12),aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,E23),nil_Pr5671120429643327159n_assn)) )
    <=> ( ( L12 = nil_Pr5671120429643327159n_assn )
        & ( E12 = E1 )
        & ( E23 = E22 )
        & ( L22 = nil_Pr5671120429643327159n_assn ) ) ) ).

% list_ee_eq_leel(2)
tff(fact_1241_list__ee__eq__leel_I1_J,axiom,
    ! [E12: a,E23: a,L12: list_a,E1: a,E22: a,L22: list_a] :
      ( ( aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,E12),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,E23),nil_a)) = aa_list_a_list_a(aa_lis2442066584718795531list_a(append_a,L12),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,E1),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,E22),L22))) )
    <=> ( ( L12 = nil_a )
        & ( E12 = E1 )
        & ( E23 = E22 )
        & ( L22 = nil_a ) ) ) ).

% list_ee_eq_leel(1)
tff(fact_1242_list__ee__eq__leel_I1_J,axiom,
    ! [E12: b,E23: b,L12: list_b,E1: b,E22: b,L22: list_b] :
      ( ( aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,E12),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,E23),nil_b)) = aa_list_b_list_b(aa_lis927650220358621516list_b(append_b,L12),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,E1),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,E22),L22))) )
    <=> ( ( L12 = nil_b )
        & ( E12 = E1 )
        & ( E23 = E22 )
        & ( L22 = nil_b ) ) ) ).

% list_ee_eq_leel(1)
tff(fact_1243_list__ee__eq__leel_I1_J,axiom,
    ! [E12: nat,E23: nat,L12: list_nat,E1: nat,E22: nat,L22: list_nat] :
      ( ( aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,E12),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,E23),nil_nat)) = aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,L12),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,E1),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,E22),L22))) )
    <=> ( ( L12 = nil_nat )
        & ( E12 = E1 )
        & ( E23 = E22 )
        & ( L22 = nil_nat ) ) ) ).

% list_ee_eq_leel(1)
tff(fact_1244_list__ee__eq__leel_I1_J,axiom,
    ! [E12: int,E23: int,L12: list_int,E1: int,E22: int,L22: list_int] :
      ( ( aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,E12),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,E23),nil_int)) = aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,L12),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,E1),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,E22),L22))) )
    <=> ( ( L12 = nil_int )
        & ( E12 = E1 )
        & ( E23 = E22 )
        & ( L22 = nil_int ) ) ) ).

% list_ee_eq_leel(1)
tff(fact_1245_list__ee__eq__leel_I1_J,axiom,
    ! [E12: produc6575502325842934193n_assn,E23: produc6575502325842934193n_assn,L12: list_P8527749157015355191n_assn,E1: produc6575502325842934193n_assn,E22: produc6575502325842934193n_assn,L22: list_P8527749157015355191n_assn] :
      ( ( aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,E12),aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,E23),nil_Pr5671120429643327159n_assn)) = aa_lis2797772165808192636n_assn(aa_lis4738365772350288766n_assn(append282499809098378956n_assn,L12),aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,E1),aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,E22),L22))) )
    <=> ( ( L12 = nil_Pr5671120429643327159n_assn )
        & ( E12 = E1 )
        & ( E23 = E22 )
        & ( L22 = nil_Pr5671120429643327159n_assn ) ) ) ).

% list_ee_eq_leel(1)
tff(fact_1246_rev__singleton__conv,axiom,
    ! [Xs: list_a,X: a] :
      ( ( rev_a(Xs) = aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X),nil_a) )
    <=> ( Xs = aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X),nil_a) ) ) ).

% rev_singleton_conv
tff(fact_1247_rev__singleton__conv,axiom,
    ! [Xs: list_b,X: b] :
      ( ( rev_b(Xs) = aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X),nil_b) )
    <=> ( Xs = aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X),nil_b) ) ) ).

% rev_singleton_conv
tff(fact_1248_rev__singleton__conv,axiom,
    ! [Xs: list_nat,X: nat] :
      ( ( rev_nat(Xs) = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X),nil_nat) )
    <=> ( Xs = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X),nil_nat) ) ) ).

% rev_singleton_conv
tff(fact_1249_rev__singleton__conv,axiom,
    ! [Xs: list_int,X: int] :
      ( ( rev_int(Xs) = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X),nil_int) )
    <=> ( Xs = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X),nil_int) ) ) ).

% rev_singleton_conv
tff(fact_1250_rev__singleton__conv,axiom,
    ! [Xs: list_P8527749157015355191n_assn,X: produc6575502325842934193n_assn] :
      ( ( rev_Pr4855572775806611735n_assn(Xs) = aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X),nil_Pr5671120429643327159n_assn) )
    <=> ( Xs = aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X),nil_Pr5671120429643327159n_assn) ) ) ).

% rev_singleton_conv
tff(fact_1251_singleton__rev__conv,axiom,
    ! [X: a,Xs: list_a] :
      ( ( aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X),nil_a) = rev_a(Xs) )
    <=> ( aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X),nil_a) = Xs ) ) ).

% singleton_rev_conv
tff(fact_1252_singleton__rev__conv,axiom,
    ! [X: b,Xs: list_b] :
      ( ( aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X),nil_b) = rev_b(Xs) )
    <=> ( aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X),nil_b) = Xs ) ) ).

% singleton_rev_conv
tff(fact_1253_singleton__rev__conv,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X),nil_nat) = rev_nat(Xs) )
    <=> ( aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X),nil_nat) = Xs ) ) ).

% singleton_rev_conv
tff(fact_1254_singleton__rev__conv,axiom,
    ! [X: int,Xs: list_int] :
      ( ( aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X),nil_int) = rev_int(Xs) )
    <=> ( aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X),nil_int) = Xs ) ) ).

% singleton_rev_conv
tff(fact_1255_singleton__rev__conv,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] :
      ( ( aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X),nil_Pr5671120429643327159n_assn) = rev_Pr4855572775806611735n_assn(Xs) )
    <=> ( aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X),nil_Pr5671120429643327159n_assn) = Xs ) ) ).

% singleton_rev_conv
tff(fact_1256_lexordp__simps_I3_J,axiom,
    ! [X: assn,Xs: list_assn,Y: assn,Ys: list_assn] :
      ( ord_lexordp_assn(cons_assn(X,Xs),cons_assn(Y,Ys))
    <=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),Y)
        | ( ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,Y),X)
          & ord_lexordp_assn(Xs,Ys) ) ) ) ).

% lexordp_simps(3)
tff(fact_1257_lexordp__simps_I3_J,axiom,
    ! [X: nat,Xs: list_nat,Y: nat,Ys: list_nat] :
      ( aa_list_nat_o(aa_lis557685189327716851_nat_o(ord_lexordp_nat,aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X),Xs)),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Y),Ys))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),X)
          & aa_list_nat_o(aa_lis557685189327716851_nat_o(ord_lexordp_nat,Xs),Ys) ) ) ) ).

% lexordp_simps(3)
tff(fact_1258_lexordp__simps_I3_J,axiom,
    ! [X: int,Xs: list_int,Y: int,Ys: list_int] :
      ( aa_list_int_o(aa_lis7773489400162621755_int_o(ord_lexordp_int,aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X),Xs)),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Y),Ys))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
        | ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Y),X)
          & aa_list_int_o(aa_lis7773489400162621755_int_o(ord_lexordp_int,Xs),Ys) ) ) ) ).

% lexordp_simps(3)
tff(fact_1259_sorted__list__of__set_Ofold__insort__key_Oinfinite,axiom,
    ! [A2: set_int] :
      ( ~ aa_set_int_o(finite_finite_int,A2)
     => ( linord2612477271533052124et_int(A2) = nil_int ) ) ).

% sorted_list_of_set.fold_insort_key.infinite
tff(fact_1260_sorted__list__of__set_Ofold__insort__key_Oinfinite,axiom,
    ! [A2: set_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,A2)
     => ( linord2614967742042102400et_nat(A2) = nil_nat ) ) ).

% sorted_list_of_set.fold_insort_key.infinite
tff(fact_1261_list__collect__set__simps_I4_J,axiom,
    ! [F: fun_nat_set_nat,La: list_nat,L_a: list_nat] : list_c2452340269597857392at_nat(F,aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,La),L_a)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,list_c2452340269597857392at_nat(F,La)),list_c2452340269597857392at_nat(F,L_a)) ).

% list_collect_set_simps(4)
tff(fact_1262_list__collect__set__simps_I4_J,axiom,
    ! [F: fun_int_set_nat,La: list_int,L_a: list_int] : list_c3451693981498911948nt_nat(F,aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,La),L_a)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,list_c3451693981498911948nt_nat(F,La)),list_c3451693981498911948nt_nat(F,L_a)) ).

% list_collect_set_simps(4)
tff(fact_1263_bind__simps_I2_J,axiom,
    ! [X: a,Xs: list_a,F: fun_a_list_b] : bind_a_b(aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X),Xs),F) = aa_list_b_list_b(aa_lis927650220358621516list_b(append_b,aa_a_list_b(F,X)),bind_a_b(Xs,F)) ).

% bind_simps(2)
tff(fact_1264_bind__simps_I2_J,axiom,
    ! [X: a,Xs: list_a,F: fun_a_list_a] : bind_a_a(aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X),Xs),F) = aa_list_a_list_a(aa_lis2442066584718795531list_a(append_a,aa_a_list_a(F,X)),bind_a_a(Xs,F)) ).

% bind_simps(2)
tff(fact_1265_bind__simps_I2_J,axiom,
    ! [X: a,Xs: list_a,F: fun_a_list_nat] : bind_a_nat(aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X),Xs),F) = aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,aa_a_list_nat(F,X)),bind_a_nat(Xs,F)) ).

% bind_simps(2)
tff(fact_1266_bind__simps_I2_J,axiom,
    ! [X: a,Xs: list_a,F: fun_a_list_int] : bind_a_int(aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X),Xs),F) = aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,aa_a_list_int(F,X)),bind_a_int(Xs,F)) ).

% bind_simps(2)
tff(fact_1267_bind__simps_I2_J,axiom,
    ! [X: nat,Xs: list_nat,F: fun_nat_list_nat] : bind_nat_nat(aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X),Xs),F) = aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,aa_nat_list_nat(F,X)),bind_nat_nat(Xs,F)) ).

% bind_simps(2)
tff(fact_1268_bind__simps_I2_J,axiom,
    ! [X: nat,Xs: list_nat,F: fun_nat_list_int] : bind_nat_int(aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X),Xs),F) = aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,aa_nat_list_int(F,X)),bind_nat_int(Xs,F)) ).

% bind_simps(2)
tff(fact_1269_bind__simps_I2_J,axiom,
    ! [X: int,Xs: list_int,F: fun_int_list_nat] : bind_int_nat(aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X),Xs),F) = aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,aa_int_list_nat(F,X)),bind_int_nat(Xs,F)) ).

% bind_simps(2)
tff(fact_1270_bind__simps_I2_J,axiom,
    ! [X: int,Xs: list_int,F: fun_int_list_int] : bind_int_int(aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X),Xs),F) = aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,aa_int_list_int(F,X)),bind_int_int(Xs,F)) ).

% bind_simps(2)
tff(fact_1271_bind__simps_I2_J,axiom,
    ! [X: a,Xs: list_a,F: fun_a_681019998106348156n_assn] : bind_a3542047475819770682n_assn(aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X),Xs),F) = aa_lis2797772165808192636n_assn(aa_lis4738365772350288766n_assn(append282499809098378956n_assn,aa_a_l8800941881919933557n_assn(F,X)),bind_a3542047475819770682n_assn(Xs,F)) ).

% bind_simps(2)
tff(fact_1272_bind__simps_I2_J,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn,F: fun_Pr7454088522235451300st_nat] : bind_P8084169516273685562sn_nat(aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X),Xs),F) = aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,aa_Pro579495954165322453st_nat(F,X)),bind_P8084169516273685562sn_nat(Xs,F)) ).

% bind_simps(2)
tff(fact_1273_revg__fun,axiom,
    ! [A: list_nat,B: list_nat] : revg_nat(A,B) = aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,rev_nat(A)),B) ).

% revg_fun
tff(fact_1274_revg__fun,axiom,
    ! [A: list_int,B: list_int] : revg_int(A,B) = aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,rev_int(A)),B) ).

% revg_fun
tff(fact_1275_rev__eq__Cons__iff,axiom,
    ! [Xs: list_a,Y: a,Ys: list_a] :
      ( ( rev_a(Xs) = aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,Y),Ys) )
    <=> ( Xs = aa_list_a_list_a(aa_lis2442066584718795531list_a(append_a,rev_a(Ys)),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,Y),nil_a)) ) ) ).

% rev_eq_Cons_iff
tff(fact_1276_rev__eq__Cons__iff,axiom,
    ! [Xs: list_b,Y: b,Ys: list_b] :
      ( ( rev_b(Xs) = aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,Y),Ys) )
    <=> ( Xs = aa_list_b_list_b(aa_lis927650220358621516list_b(append_b,rev_b(Ys)),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,Y),nil_b)) ) ) ).

% rev_eq_Cons_iff
tff(fact_1277_rev__eq__Cons__iff,axiom,
    ! [Xs: list_nat,Y: nat,Ys: list_nat] :
      ( ( rev_nat(Xs) = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Y),Ys) )
    <=> ( Xs = aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,rev_nat(Ys)),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Y),nil_nat)) ) ) ).

% rev_eq_Cons_iff
tff(fact_1278_rev__eq__Cons__iff,axiom,
    ! [Xs: list_int,Y: int,Ys: list_int] :
      ( ( rev_int(Xs) = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Y),Ys) )
    <=> ( Xs = aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,rev_int(Ys)),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Y),nil_int)) ) ) ).

% rev_eq_Cons_iff
tff(fact_1279_rev__eq__Cons__iff,axiom,
    ! [Xs: list_P8527749157015355191n_assn,Y: produc6575502325842934193n_assn,Ys: list_P8527749157015355191n_assn] :
      ( ( rev_Pr4855572775806611735n_assn(Xs) = aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,Y),Ys) )
    <=> ( Xs = aa_lis2797772165808192636n_assn(aa_lis4738365772350288766n_assn(append282499809098378956n_assn,rev_Pr4855572775806611735n_assn(Ys)),aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,Y),nil_Pr5671120429643327159n_assn)) ) ) ).

% rev_eq_Cons_iff
tff(fact_1280_psubsetD,axiom,
    ! [A2: set_fu3555266933746557007_nat_o,B2: set_fu3555266933746557007_nat_o,C2: fun_Pr6460944699146269935_nat_o] :
      ( aa_set5091322549519198405at_o_o(ord_le1274849205035319547_nat_o(A2),B2)
     => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C2),A2)
       => aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C2),B2) ) ) ).

% psubsetD
tff(fact_1281_append__eq__append__conv2,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs3: list_nat,Ts: list_nat] :
      ( ( aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Xs),Ys) = aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Zs3),Ts) )
    <=> ? [Us: list_nat] :
          ( ( ( Xs = aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Zs3),Us) )
            & ( aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Us),Ys) = Ts ) )
          | ( ( aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Xs),Us) = Zs3 )
            & ( Ys = aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Us),Ts) ) ) ) ) ).

% append_eq_append_conv2
tff(fact_1282_append__eq__append__conv2,axiom,
    ! [Xs: list_int,Ys: list_int,Zs3: list_int,Ts: list_int] :
      ( ( aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Xs),Ys) = aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Zs3),Ts) )
    <=> ? [Us: list_int] :
          ( ( ( Xs = aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Zs3),Us) )
            & ( aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Us),Ys) = Ts ) )
          | ( ( aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Xs),Us) = Zs3 )
            & ( Ys = aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Us),Ts) ) ) ) ) ).

% append_eq_append_conv2
tff(fact_1283_append__eq__appendI,axiom,
    ! [Xs: list_nat,Xs1: list_nat,Zs3: list_nat,Ys: list_nat,Us2: list_nat] :
      ( ( aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Xs),Xs1) = Zs3 )
     => ( ( Ys = aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Xs1),Us2) )
       => ( aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Xs),Ys) = aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Zs3),Us2) ) ) ) ).

% append_eq_appendI
tff(fact_1284_append__eq__appendI,axiom,
    ! [Xs: list_int,Xs1: list_int,Zs3: list_int,Ys: list_int,Us2: list_int] :
      ( ( aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Xs),Xs1) = Zs3 )
     => ( ( Ys = aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Xs1),Us2) )
       => ( aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Xs),Ys) = aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Zs3),Us2) ) ) ) ).

% append_eq_appendI
tff(fact_1285_lexordp__append__leftI,axiom,
    ! [Us2: list_nat,Vs: list_nat,Xs: list_nat] :
      ( aa_list_nat_o(aa_lis557685189327716851_nat_o(ord_lexordp_nat,Us2),Vs)
     => aa_list_nat_o(aa_lis557685189327716851_nat_o(ord_lexordp_nat,aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Xs),Us2)),aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Xs),Vs)) ) ).

% lexordp_append_leftI
tff(fact_1286_lexordp__append__leftI,axiom,
    ! [Us2: list_int,Vs: list_int,Xs: list_int] :
      ( aa_list_int_o(aa_lis7773489400162621755_int_o(ord_lexordp_int,Us2),Vs)
     => aa_list_int_o(aa_lis7773489400162621755_int_o(ord_lexordp_int,aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Xs),Us2)),aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Xs),Vs)) ) ).

% lexordp_append_leftI
tff(fact_1287_lexordp__append__leftD,axiom,
    ! [Xs: list_assn,Us2: list_assn,Vs: list_assn] :
      ( ord_lexordp_assn(append_assn(Xs,Us2),append_assn(Xs,Vs))
     => ( ! [A3: assn] : ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A3),A3)
       => ord_lexordp_assn(Us2,Vs) ) ) ).

% lexordp_append_leftD
tff(fact_1288_lexordp__append__leftD,axiom,
    ! [Xs: list_nat,Us2: list_nat,Vs: list_nat] :
      ( aa_list_nat_o(aa_lis557685189327716851_nat_o(ord_lexordp_nat,aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Xs),Us2)),aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Xs),Vs))
     => ( ! [A3: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A3),A3)
       => aa_list_nat_o(aa_lis557685189327716851_nat_o(ord_lexordp_nat,Us2),Vs) ) ) ).

% lexordp_append_leftD
tff(fact_1289_lexordp__append__leftD,axiom,
    ! [Xs: list_int,Us2: list_int,Vs: list_int] :
      ( aa_list_int_o(aa_lis7773489400162621755_int_o(ord_lexordp_int,aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Xs),Us2)),aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Xs),Vs))
     => ( ! [A3: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A3),A3)
       => aa_list_int_o(aa_lis7773489400162621755_int_o(ord_lexordp_int,Us2),Vs) ) ) ).

% lexordp_append_leftD
tff(fact_1290_lexordp__append__rightI,axiom,
    ! [Ys: list_nat,Xs: list_nat] :
      ( ( Ys != nil_nat )
     => aa_list_nat_o(aa_lis557685189327716851_nat_o(ord_lexordp_nat,Xs),aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Xs),Ys)) ) ).

% lexordp_append_rightI
tff(fact_1291_lexordp__append__rightI,axiom,
    ! [Ys: list_int,Xs: list_int] :
      ( ( Ys != nil_int )
     => aa_list_int_o(aa_lis7773489400162621755_int_o(ord_lexordp_int,Xs),aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Xs),Ys)) ) ).

% lexordp_append_rightI
tff(fact_1292_rev_Osimps_I2_J,axiom,
    ! [X: a,Xs: list_a] : rev_a(aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X),Xs)) = aa_list_a_list_a(aa_lis2442066584718795531list_a(append_a,rev_a(Xs)),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X),nil_a)) ).

% rev.simps(2)
tff(fact_1293_rev_Osimps_I2_J,axiom,
    ! [X: b,Xs: list_b] : rev_b(aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X),Xs)) = aa_list_b_list_b(aa_lis927650220358621516list_b(append_b,rev_b(Xs)),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X),nil_b)) ).

% rev.simps(2)
tff(fact_1294_rev_Osimps_I2_J,axiom,
    ! [X: nat,Xs: list_nat] : rev_nat(aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X),Xs)) = aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,rev_nat(Xs)),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X),nil_nat)) ).

% rev.simps(2)
tff(fact_1295_rev_Osimps_I2_J,axiom,
    ! [X: int,Xs: list_int] : rev_int(aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X),Xs)) = aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,rev_int(Xs)),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X),nil_int)) ).

% rev.simps(2)
tff(fact_1296_rev_Osimps_I2_J,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] : rev_Pr4855572775806611735n_assn(aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X),Xs)) = aa_lis2797772165808192636n_assn(aa_lis4738365772350288766n_assn(append282499809098378956n_assn,rev_Pr4855572775806611735n_assn(Xs)),aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X),nil_Pr5671120429643327159n_assn)) ).

% rev.simps(2)
tff(fact_1297_lexordp__iff,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( aa_list_nat_o(aa_lis557685189327716851_nat_o(ord_lexordp_nat,Xs),Ys)
    <=> ( ? [X3: nat,Vs2: list_nat] : Ys = aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Xs),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X3),Vs2))
        | ? [Us: list_nat,A4: nat,B4: nat,Vs2: list_nat,Ws: list_nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A4),B4)
            & ( Xs = aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Us),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,A4),Vs2)) )
            & ( Ys = aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Us),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,B4),Ws)) ) ) ) ) ).

% lexordp_iff
tff(fact_1298_lexordp__iff,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( aa_list_int_o(aa_lis7773489400162621755_int_o(ord_lexordp_int,Xs),Ys)
    <=> ( ? [X3: int,Vs2: list_int] : Ys = aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Xs),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X3),Vs2))
        | ? [Us: list_int,A4: int,B4: int,Vs2: list_int,Ws: list_int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,A4),B4)
            & ( Xs = aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Us),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,A4),Vs2)) )
            & ( Ys = aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Us),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,B4),Ws)) ) ) ) ) ).

% lexordp_iff
tff(fact_1299_lexordp__append__left__rightI,axiom,
    ! [X: assn,Y: assn,Us2: list_assn,Xs: list_assn,Ys: list_assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),Y)
     => ord_lexordp_assn(append_assn(Us2,cons_assn(X,Xs)),append_assn(Us2,cons_assn(Y,Ys))) ) ).

% lexordp_append_left_rightI
tff(fact_1300_lexordp__append__left__rightI,axiom,
    ! [X: nat,Y: nat,Us2: list_nat,Xs: list_nat,Ys: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
     => aa_list_nat_o(aa_lis557685189327716851_nat_o(ord_lexordp_nat,aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Us2),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X),Xs))),aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Us2),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Y),Ys))) ) ).

% lexordp_append_left_rightI
tff(fact_1301_lexordp__append__left__rightI,axiom,
    ! [X: int,Y: int,Us2: list_int,Xs: list_int,Ys: list_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
     => aa_list_int_o(aa_lis7773489400162621755_int_o(ord_lexordp_int,aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Us2),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X),Xs))),aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Us2),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Y),Ys))) ) ).

% lexordp_append_left_rightI
tff(fact_1302_rev_Osimps_I1_J,axiom,
    rev_a(nil_a) = nil_a ).

% rev.simps(1)
tff(fact_1303_rev_Osimps_I1_J,axiom,
    rev_b(nil_b) = nil_b ).

% rev.simps(1)
tff(fact_1304_rev_Osimps_I1_J,axiom,
    rev_Pr4855572775806611735n_assn(nil_Pr5671120429643327159n_assn) = nil_Pr5671120429643327159n_assn ).

% rev.simps(1)
tff(fact_1305_rev_Osimps_I1_J,axiom,
    rev_nat(nil_nat) = nil_nat ).

% rev.simps(1)
tff(fact_1306_rev_Osimps_I1_J,axiom,
    rev_int(nil_int) = nil_int ).

% rev.simps(1)
tff(fact_1307_rev__map,axiom,
    ! [F: fun_nat_nat,Xs: list_nat] : rev_nat(aa_list_nat_list_nat(aa_fun4736978287202700568st_nat(map_nat_nat,F),Xs)) = aa_list_nat_list_nat(aa_fun4736978287202700568st_nat(map_nat_nat,F),rev_nat(Xs)) ).

% rev_map
tff(fact_1308_rev__map,axiom,
    ! [F: fun_Pr3816910165196019820n_assn,Xs: list_P8527749157015355191n_assn] : rev_assn(aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,F),Xs)) = aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,F),rev_Pr4855572775806611735n_assn(Xs)) ).

% rev_map
tff(fact_1309_append__Cons,axiom,
    ! [X: nat,Xs: list_nat,Ys: list_nat] : aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X),Xs)),Ys) = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X),aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Xs),Ys)) ).

% append_Cons
tff(fact_1310_append__Cons,axiom,
    ! [X: int,Xs: list_int,Ys: list_int] : aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X),Xs)),Ys) = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X),aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Xs),Ys)) ).

% append_Cons
tff(fact_1311_append__Cons,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn,Ys: list_P8527749157015355191n_assn] : aa_lis2797772165808192636n_assn(aa_lis4738365772350288766n_assn(append282499809098378956n_assn,aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X),Xs)),Ys) = aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X),aa_lis2797772165808192636n_assn(aa_lis4738365772350288766n_assn(append282499809098378956n_assn,Xs),Ys)) ).

% append_Cons
tff(fact_1312_Cons__eq__appendI,axiom,
    ! [X: nat,Xs1: list_nat,Ys: list_nat,Xs: list_nat,Zs3: list_nat] :
      ( ( aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X),Xs1) = Ys )
     => ( ( Xs = aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Xs1),Zs3) )
       => ( aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X),Xs) = aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Ys),Zs3) ) ) ) ).

% Cons_eq_appendI
tff(fact_1313_Cons__eq__appendI,axiom,
    ! [X: int,Xs1: list_int,Ys: list_int,Xs: list_int,Zs3: list_int] :
      ( ( aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X),Xs1) = Ys )
     => ( ( Xs = aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Xs1),Zs3) )
       => ( aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X),Xs) = aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Ys),Zs3) ) ) ) ).

% Cons_eq_appendI
tff(fact_1314_Cons__eq__appendI,axiom,
    ! [X: produc6575502325842934193n_assn,Xs1: list_P8527749157015355191n_assn,Ys: list_P8527749157015355191n_assn,Xs: list_P8527749157015355191n_assn,Zs3: list_P8527749157015355191n_assn] :
      ( ( aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X),Xs1) = Ys )
     => ( ( Xs = aa_lis2797772165808192636n_assn(aa_lis4738365772350288766n_assn(append282499809098378956n_assn,Xs1),Zs3) )
       => ( aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X),Xs) = aa_lis2797772165808192636n_assn(aa_lis4738365772350288766n_assn(append282499809098378956n_assn,Ys),Zs3) ) ) ) ).

% Cons_eq_appendI
tff(fact_1315_list__match__lel__lel,axiom,
    ! [C1: list_nat,Qs: nat,C22: list_nat,C12: list_nat,Qs2: nat,C23: list_nat] :
      ( ( aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,C1),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Qs),C22)) = aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,C12),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Qs2),C23)) )
     => ( ! [C21: list_nat] :
            ( ( C1 = aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,C12),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Qs2),C21)) )
           => ( C23 != aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,C21),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Qs),C22)) ) )
       => ( ( ( C12 = C1 )
           => ( ( Qs2 = Qs )
             => ( C23 != C22 ) ) )
         => ~ ! [C212: list_nat] :
                ( ( C12 = aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,C1),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Qs),C212)) )
               => ( C22 != aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,C212),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Qs2),C23)) ) ) ) ) ) ).

% list_match_lel_lel
tff(fact_1316_list__match__lel__lel,axiom,
    ! [C1: list_int,Qs: int,C22: list_int,C12: list_int,Qs2: int,C23: list_int] :
      ( ( aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,C1),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Qs),C22)) = aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,C12),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Qs2),C23)) )
     => ( ! [C21: list_int] :
            ( ( C1 = aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,C12),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Qs2),C21)) )
           => ( C23 != aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,C21),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Qs),C22)) ) )
       => ( ( ( C12 = C1 )
           => ( ( Qs2 = Qs )
             => ( C23 != C22 ) ) )
         => ~ ! [C212: list_int] :
                ( ( C12 = aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,C1),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Qs),C212)) )
               => ( C22 != aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,C212),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Qs2),C23)) ) ) ) ) ) ).

% list_match_lel_lel
tff(fact_1317_list__match__lel__lel,axiom,
    ! [C1: list_P8527749157015355191n_assn,Qs: produc6575502325842934193n_assn,C22: list_P8527749157015355191n_assn,C12: list_P8527749157015355191n_assn,Qs2: produc6575502325842934193n_assn,C23: list_P8527749157015355191n_assn] :
      ( ( aa_lis2797772165808192636n_assn(aa_lis4738365772350288766n_assn(append282499809098378956n_assn,C1),aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,Qs),C22)) = aa_lis2797772165808192636n_assn(aa_lis4738365772350288766n_assn(append282499809098378956n_assn,C12),aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,Qs2),C23)) )
     => ( ! [C21: list_P8527749157015355191n_assn] :
            ( ( C1 = aa_lis2797772165808192636n_assn(aa_lis4738365772350288766n_assn(append282499809098378956n_assn,C12),aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,Qs2),C21)) )
           => ( C23 != aa_lis2797772165808192636n_assn(aa_lis4738365772350288766n_assn(append282499809098378956n_assn,C21),aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,Qs),C22)) ) )
       => ( ( ( C12 = C1 )
           => ( ( Qs2 = Qs )
             => ( C23 != C22 ) ) )
         => ~ ! [C212: list_P8527749157015355191n_assn] :
                ( ( C12 = aa_lis2797772165808192636n_assn(aa_lis4738365772350288766n_assn(append282499809098378956n_assn,C1),aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,Qs),C212)) )
               => ( C22 != aa_lis2797772165808192636n_assn(aa_lis4738365772350288766n_assn(append282499809098378956n_assn,C212),aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,Qs2),C23)) ) ) ) ) ) ).

% list_match_lel_lel
tff(fact_1318_eq__Nil__appendI,axiom,
    ! [Xs: list_a,Ys: list_a] :
      ( ( Xs = Ys )
     => ( Xs = aa_list_a_list_a(aa_lis2442066584718795531list_a(append_a,nil_a),Ys) ) ) ).

% eq_Nil_appendI
tff(fact_1319_eq__Nil__appendI,axiom,
    ! [Xs: list_b,Ys: list_b] :
      ( ( Xs = Ys )
     => ( Xs = aa_list_b_list_b(aa_lis927650220358621516list_b(append_b,nil_b),Ys) ) ) ).

% eq_Nil_appendI
tff(fact_1320_eq__Nil__appendI,axiom,
    ! [Xs: list_P8527749157015355191n_assn,Ys: list_P8527749157015355191n_assn] :
      ( ( Xs = Ys )
     => ( Xs = aa_lis2797772165808192636n_assn(aa_lis4738365772350288766n_assn(append282499809098378956n_assn,nil_Pr5671120429643327159n_assn),Ys) ) ) ).

% eq_Nil_appendI
tff(fact_1321_eq__Nil__appendI,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( Xs = Ys )
     => ( Xs = aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,nil_nat),Ys) ) ) ).

% eq_Nil_appendI
tff(fact_1322_eq__Nil__appendI,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( Xs = Ys )
     => ( Xs = aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,nil_int),Ys) ) ) ).

% eq_Nil_appendI
tff(fact_1323_append_Oleft__neutral,axiom,
    ! [A: list_a] : aa_list_a_list_a(aa_lis2442066584718795531list_a(append_a,nil_a),A) = A ).

% append.left_neutral
tff(fact_1324_append_Oleft__neutral,axiom,
    ! [A: list_b] : aa_list_b_list_b(aa_lis927650220358621516list_b(append_b,nil_b),A) = A ).

% append.left_neutral
tff(fact_1325_append_Oleft__neutral,axiom,
    ! [A: list_P8527749157015355191n_assn] : aa_lis2797772165808192636n_assn(aa_lis4738365772350288766n_assn(append282499809098378956n_assn,nil_Pr5671120429643327159n_assn),A) = A ).

% append.left_neutral
tff(fact_1326_append_Oleft__neutral,axiom,
    ! [A: list_nat] : aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,nil_nat),A) = A ).

% append.left_neutral
tff(fact_1327_append_Oleft__neutral,axiom,
    ! [A: list_int] : aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,nil_int),A) = A ).

% append.left_neutral
tff(fact_1328_append__Nil,axiom,
    ! [Ys: list_a] : aa_list_a_list_a(aa_lis2442066584718795531list_a(append_a,nil_a),Ys) = Ys ).

% append_Nil
tff(fact_1329_append__Nil,axiom,
    ! [Ys: list_b] : aa_list_b_list_b(aa_lis927650220358621516list_b(append_b,nil_b),Ys) = Ys ).

% append_Nil
tff(fact_1330_append__Nil,axiom,
    ! [Ys: list_P8527749157015355191n_assn] : aa_lis2797772165808192636n_assn(aa_lis4738365772350288766n_assn(append282499809098378956n_assn,nil_Pr5671120429643327159n_assn),Ys) = Ys ).

% append_Nil
tff(fact_1331_append__Nil,axiom,
    ! [Ys: list_nat] : aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,nil_nat),Ys) = Ys ).

% append_Nil
tff(fact_1332_append__Nil,axiom,
    ! [Ys: list_int] : aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,nil_int),Ys) = Ys ).

% append_Nil
tff(fact_1333_map__tailrec__rev,axiom,
    ! [F: fun_nat_nat,As2: list_nat,Bs: list_nat] : map_ta7164188454487880599at_nat(F,As2,Bs) = aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,rev_nat(aa_list_nat_list_nat(aa_fun4736978287202700568st_nat(map_nat_nat,F),As2))),Bs) ).

% map_tailrec_rev
tff(fact_1334_map__tailrec__rev,axiom,
    ! [F: fun_Pr3816910165196019820n_assn,As2: list_P8527749157015355191n_assn,Bs: list_assn] : map_ta5611102776663852196n_assn(F,As2,Bs) = append_assn(rev_assn(aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,F),As2)),Bs) ).

% map_tailrec_rev
tff(fact_1335_Misc_Omap__eq__append__conv,axiom,
    ! [F: fun_int_nat,Ls: list_int,Fl: list_nat,Fl2: list_nat] :
      ( ( map_int_nat(F,Ls) = aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Fl),Fl2) )
    <=> ? [L3: list_int,L4: list_int] :
          ( ( Ls = aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,L3),L4) )
          & ( map_int_nat(F,L3) = Fl )
          & ( map_int_nat(F,L4) = Fl2 ) ) ) ).

% Misc.map_eq_append_conv
tff(fact_1336_Misc_Omap__eq__append__conv,axiom,
    ! [F: fun_nat_int,Ls: list_nat,Fl: list_int,Fl2: list_int] :
      ( ( map_nat_int(F,Ls) = aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Fl),Fl2) )
    <=> ? [L3: list_nat,L4: list_nat] :
          ( ( Ls = aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,L3),L4) )
          & ( map_nat_int(F,L3) = Fl )
          & ( map_nat_int(F,L4) = Fl2 ) ) ) ).

% Misc.map_eq_append_conv
tff(fact_1337_Misc_Omap__eq__append__conv,axiom,
    ! [F: fun_int_int,Ls: list_int,Fl: list_int,Fl2: list_int] :
      ( ( map_int_int(F,Ls) = aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Fl),Fl2) )
    <=> ? [L3: list_int,L4: list_int] :
          ( ( Ls = aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,L3),L4) )
          & ( map_int_int(F,L3) = Fl )
          & ( map_int_int(F,L4) = Fl2 ) ) ) ).

% Misc.map_eq_append_conv
tff(fact_1338_Misc_Omap__eq__append__conv,axiom,
    ! [F: fun_nat_nat,Ls: list_nat,Fl: list_nat,Fl2: list_nat] :
      ( ( aa_list_nat_list_nat(aa_fun4736978287202700568st_nat(map_nat_nat,F),Ls) = aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Fl),Fl2) )
    <=> ? [L3: list_nat,L4: list_nat] :
          ( ( Ls = aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,L3),L4) )
          & ( aa_list_nat_list_nat(aa_fun4736978287202700568st_nat(map_nat_nat,F),L3) = Fl )
          & ( aa_list_nat_list_nat(aa_fun4736978287202700568st_nat(map_nat_nat,F),L4) = Fl2 ) ) ) ).

% Misc.map_eq_append_conv
tff(fact_1339_Misc_Omap__eq__append__conv,axiom,
    ! [F: fun_Pr3816910165196019820n_assn,Ls: list_P8527749157015355191n_assn,Fl: list_assn,Fl2: list_assn] :
      ( ( aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,F),Ls) = append_assn(Fl,Fl2) )
    <=> ? [L3: list_P8527749157015355191n_assn,L4: list_P8527749157015355191n_assn] :
          ( ( Ls = aa_lis2797772165808192636n_assn(aa_lis4738365772350288766n_assn(append282499809098378956n_assn,L3),L4) )
          & ( aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,F),L3) = Fl )
          & ( aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,F),L4) = Fl2 ) ) ) ).

% Misc.map_eq_append_conv
tff(fact_1340_Misc_Oappend__eq__map__conv,axiom,
    ! [Fl: list_nat,Fl2: list_nat,F: fun_int_nat,Ls: list_int] :
      ( ( aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Fl),Fl2) = map_int_nat(F,Ls) )
    <=> ? [L3: list_int,L4: list_int] :
          ( ( Ls = aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,L3),L4) )
          & ( map_int_nat(F,L3) = Fl )
          & ( map_int_nat(F,L4) = Fl2 ) ) ) ).

% Misc.append_eq_map_conv
tff(fact_1341_Misc_Oappend__eq__map__conv,axiom,
    ! [Fl: list_int,Fl2: list_int,F: fun_nat_int,Ls: list_nat] :
      ( ( aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Fl),Fl2) = map_nat_int(F,Ls) )
    <=> ? [L3: list_nat,L4: list_nat] :
          ( ( Ls = aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,L3),L4) )
          & ( map_nat_int(F,L3) = Fl )
          & ( map_nat_int(F,L4) = Fl2 ) ) ) ).

% Misc.append_eq_map_conv
tff(fact_1342_Misc_Oappend__eq__map__conv,axiom,
    ! [Fl: list_int,Fl2: list_int,F: fun_int_int,Ls: list_int] :
      ( ( aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Fl),Fl2) = map_int_int(F,Ls) )
    <=> ? [L3: list_int,L4: list_int] :
          ( ( Ls = aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,L3),L4) )
          & ( map_int_int(F,L3) = Fl )
          & ( map_int_int(F,L4) = Fl2 ) ) ) ).

% Misc.append_eq_map_conv
tff(fact_1343_Misc_Oappend__eq__map__conv,axiom,
    ! [Fl: list_nat,Fl2: list_nat,F: fun_nat_nat,Ls: list_nat] :
      ( ( aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Fl),Fl2) = aa_list_nat_list_nat(aa_fun4736978287202700568st_nat(map_nat_nat,F),Ls) )
    <=> ? [L3: list_nat,L4: list_nat] :
          ( ( Ls = aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,L3),L4) )
          & ( aa_list_nat_list_nat(aa_fun4736978287202700568st_nat(map_nat_nat,F),L3) = Fl )
          & ( aa_list_nat_list_nat(aa_fun4736978287202700568st_nat(map_nat_nat,F),L4) = Fl2 ) ) ) ).

% Misc.append_eq_map_conv
tff(fact_1344_Misc_Oappend__eq__map__conv,axiom,
    ! [Fl: list_assn,Fl2: list_assn,F: fun_Pr3816910165196019820n_assn,Ls: list_P8527749157015355191n_assn] :
      ( ( append_assn(Fl,Fl2) = aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,F),Ls) )
    <=> ? [L3: list_P8527749157015355191n_assn,L4: list_P8527749157015355191n_assn] :
          ( ( Ls = aa_lis2797772165808192636n_assn(aa_lis4738365772350288766n_assn(append282499809098378956n_assn,L3),L4) )
          & ( aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,F),L3) = Fl )
          & ( aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,F),L4) = Fl2 ) ) ) ).

% Misc.append_eq_map_conv
tff(fact_1345_map__eq__appendE,axiom,
    ! [F: fun_int_nat,Ls: list_int,Fl: list_nat,Fl2: list_nat] :
      ( ( map_int_nat(F,Ls) = aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Fl),Fl2) )
     => ~ ! [L: list_int,L5: list_int] :
            ( ( Ls = aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,L),L5) )
           => ( ( map_int_nat(F,L) = Fl )
             => ( map_int_nat(F,L5) != Fl2 ) ) ) ) ).

% map_eq_appendE
tff(fact_1346_map__eq__appendE,axiom,
    ! [F: fun_nat_int,Ls: list_nat,Fl: list_int,Fl2: list_int] :
      ( ( map_nat_int(F,Ls) = aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Fl),Fl2) )
     => ~ ! [L: list_nat,L5: list_nat] :
            ( ( Ls = aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,L),L5) )
           => ( ( map_nat_int(F,L) = Fl )
             => ( map_nat_int(F,L5) != Fl2 ) ) ) ) ).

% map_eq_appendE
tff(fact_1347_map__eq__appendE,axiom,
    ! [F: fun_int_int,Ls: list_int,Fl: list_int,Fl2: list_int] :
      ( ( map_int_int(F,Ls) = aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Fl),Fl2) )
     => ~ ! [L: list_int,L5: list_int] :
            ( ( Ls = aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,L),L5) )
           => ( ( map_int_int(F,L) = Fl )
             => ( map_int_int(F,L5) != Fl2 ) ) ) ) ).

% map_eq_appendE
tff(fact_1348_map__eq__appendE,axiom,
    ! [F: fun_nat_nat,Ls: list_nat,Fl: list_nat,Fl2: list_nat] :
      ( ( aa_list_nat_list_nat(aa_fun4736978287202700568st_nat(map_nat_nat,F),Ls) = aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Fl),Fl2) )
     => ~ ! [L: list_nat,L5: list_nat] :
            ( ( Ls = aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,L),L5) )
           => ( ( aa_list_nat_list_nat(aa_fun4736978287202700568st_nat(map_nat_nat,F),L) = Fl )
             => ( aa_list_nat_list_nat(aa_fun4736978287202700568st_nat(map_nat_nat,F),L5) != Fl2 ) ) ) ) ).

% map_eq_appendE
tff(fact_1349_map__eq__appendE,axiom,
    ! [F: fun_Pr3816910165196019820n_assn,Ls: list_P8527749157015355191n_assn,Fl: list_assn,Fl2: list_assn] :
      ( ( aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,F),Ls) = append_assn(Fl,Fl2) )
     => ~ ! [L: list_P8527749157015355191n_assn,L5: list_P8527749157015355191n_assn] :
            ( ( Ls = aa_lis2797772165808192636n_assn(aa_lis4738365772350288766n_assn(append282499809098378956n_assn,L),L5) )
           => ( ( aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,F),L) = Fl )
             => ( aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,F),L5) != Fl2 ) ) ) ) ).

% map_eq_appendE
tff(fact_1350_append__eq__mapE,axiom,
    ! [Fl: list_nat,Fl2: list_nat,F: fun_int_nat,Ls: list_int] :
      ( ( aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Fl),Fl2) = map_int_nat(F,Ls) )
     => ~ ! [L: list_int,L5: list_int] :
            ( ( Ls = aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,L),L5) )
           => ( ( map_int_nat(F,L) = Fl )
             => ( map_int_nat(F,L5) != Fl2 ) ) ) ) ).

% append_eq_mapE
tff(fact_1351_append__eq__mapE,axiom,
    ! [Fl: list_int,Fl2: list_int,F: fun_nat_int,Ls: list_nat] :
      ( ( aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Fl),Fl2) = map_nat_int(F,Ls) )
     => ~ ! [L: list_nat,L5: list_nat] :
            ( ( Ls = aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,L),L5) )
           => ( ( map_nat_int(F,L) = Fl )
             => ( map_nat_int(F,L5) != Fl2 ) ) ) ) ).

% append_eq_mapE
tff(fact_1352_append__eq__mapE,axiom,
    ! [Fl: list_int,Fl2: list_int,F: fun_int_int,Ls: list_int] :
      ( ( aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Fl),Fl2) = map_int_int(F,Ls) )
     => ~ ! [L: list_int,L5: list_int] :
            ( ( Ls = aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,L),L5) )
           => ( ( map_int_int(F,L) = Fl )
             => ( map_int_int(F,L5) != Fl2 ) ) ) ) ).

% append_eq_mapE
tff(fact_1353_append__eq__mapE,axiom,
    ! [Fl: list_nat,Fl2: list_nat,F: fun_nat_nat,Ls: list_nat] :
      ( ( aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Fl),Fl2) = aa_list_nat_list_nat(aa_fun4736978287202700568st_nat(map_nat_nat,F),Ls) )
     => ~ ! [L: list_nat,L5: list_nat] :
            ( ( Ls = aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,L),L5) )
           => ( ( aa_list_nat_list_nat(aa_fun4736978287202700568st_nat(map_nat_nat,F),L) = Fl )
             => ( aa_list_nat_list_nat(aa_fun4736978287202700568st_nat(map_nat_nat,F),L5) != Fl2 ) ) ) ) ).

% append_eq_mapE
tff(fact_1354_append__eq__mapE,axiom,
    ! [Fl: list_assn,Fl2: list_assn,F: fun_Pr3816910165196019820n_assn,Ls: list_P8527749157015355191n_assn] :
      ( ( append_assn(Fl,Fl2) = aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,F),Ls) )
     => ~ ! [L: list_P8527749157015355191n_assn,L5: list_P8527749157015355191n_assn] :
            ( ( Ls = aa_lis2797772165808192636n_assn(aa_lis4738365772350288766n_assn(append282499809098378956n_assn,L),L5) )
           => ( ( aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,F),L) = Fl )
             => ( aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,F),L5) != Fl2 ) ) ) ) ).

% append_eq_mapE
tff(fact_1355_List_Omap__eq__append__conv,axiom,
    ! [F: fun_int_nat,Xs: list_int,Ys: list_nat,Zs3: list_nat] :
      ( ( map_int_nat(F,Xs) = aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Ys),Zs3) )
    <=> ? [Us: list_int,Vs2: list_int] :
          ( ( Xs = aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Us),Vs2) )
          & ( Ys = map_int_nat(F,Us) )
          & ( Zs3 = map_int_nat(F,Vs2) ) ) ) ).

% List.map_eq_append_conv
tff(fact_1356_List_Omap__eq__append__conv,axiom,
    ! [F: fun_nat_int,Xs: list_nat,Ys: list_int,Zs3: list_int] :
      ( ( map_nat_int(F,Xs) = aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Ys),Zs3) )
    <=> ? [Us: list_nat,Vs2: list_nat] :
          ( ( Xs = aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Us),Vs2) )
          & ( Ys = map_nat_int(F,Us) )
          & ( Zs3 = map_nat_int(F,Vs2) ) ) ) ).

% List.map_eq_append_conv
tff(fact_1357_List_Omap__eq__append__conv,axiom,
    ! [F: fun_int_int,Xs: list_int,Ys: list_int,Zs3: list_int] :
      ( ( map_int_int(F,Xs) = aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Ys),Zs3) )
    <=> ? [Us: list_int,Vs2: list_int] :
          ( ( Xs = aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Us),Vs2) )
          & ( Ys = map_int_int(F,Us) )
          & ( Zs3 = map_int_int(F,Vs2) ) ) ) ).

% List.map_eq_append_conv
tff(fact_1358_List_Omap__eq__append__conv,axiom,
    ! [F: fun_nat_nat,Xs: list_nat,Ys: list_nat,Zs3: list_nat] :
      ( ( aa_list_nat_list_nat(aa_fun4736978287202700568st_nat(map_nat_nat,F),Xs) = aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Ys),Zs3) )
    <=> ? [Us: list_nat,Vs2: list_nat] :
          ( ( Xs = aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Us),Vs2) )
          & ( Ys = aa_list_nat_list_nat(aa_fun4736978287202700568st_nat(map_nat_nat,F),Us) )
          & ( Zs3 = aa_list_nat_list_nat(aa_fun4736978287202700568st_nat(map_nat_nat,F),Vs2) ) ) ) ).

% List.map_eq_append_conv
tff(fact_1359_List_Omap__eq__append__conv,axiom,
    ! [F: fun_Pr3816910165196019820n_assn,Xs: list_P8527749157015355191n_assn,Ys: list_assn,Zs3: list_assn] :
      ( ( aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,F),Xs) = append_assn(Ys,Zs3) )
    <=> ? [Us: list_P8527749157015355191n_assn,Vs2: list_P8527749157015355191n_assn] :
          ( ( Xs = aa_lis2797772165808192636n_assn(aa_lis4738365772350288766n_assn(append282499809098378956n_assn,Us),Vs2) )
          & ( Ys = aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,F),Us) )
          & ( Zs3 = aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,F),Vs2) ) ) ) ).

% List.map_eq_append_conv
tff(fact_1360_List_Oappend__eq__map__conv,axiom,
    ! [Ys: list_nat,Zs3: list_nat,F: fun_int_nat,Xs: list_int] :
      ( ( aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Ys),Zs3) = map_int_nat(F,Xs) )
    <=> ? [Us: list_int,Vs2: list_int] :
          ( ( Xs = aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Us),Vs2) )
          & ( Ys = map_int_nat(F,Us) )
          & ( Zs3 = map_int_nat(F,Vs2) ) ) ) ).

% List.append_eq_map_conv
tff(fact_1361_List_Oappend__eq__map__conv,axiom,
    ! [Ys: list_int,Zs3: list_int,F: fun_nat_int,Xs: list_nat] :
      ( ( aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Ys),Zs3) = map_nat_int(F,Xs) )
    <=> ? [Us: list_nat,Vs2: list_nat] :
          ( ( Xs = aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Us),Vs2) )
          & ( Ys = map_nat_int(F,Us) )
          & ( Zs3 = map_nat_int(F,Vs2) ) ) ) ).

% List.append_eq_map_conv
tff(fact_1362_List_Oappend__eq__map__conv,axiom,
    ! [Ys: list_int,Zs3: list_int,F: fun_int_int,Xs: list_int] :
      ( ( aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Ys),Zs3) = map_int_int(F,Xs) )
    <=> ? [Us: list_int,Vs2: list_int] :
          ( ( Xs = aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Us),Vs2) )
          & ( Ys = map_int_int(F,Us) )
          & ( Zs3 = map_int_int(F,Vs2) ) ) ) ).

% List.append_eq_map_conv
tff(fact_1363_List_Oappend__eq__map__conv,axiom,
    ! [Ys: list_nat,Zs3: list_nat,F: fun_nat_nat,Xs: list_nat] :
      ( ( aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Ys),Zs3) = aa_list_nat_list_nat(aa_fun4736978287202700568st_nat(map_nat_nat,F),Xs) )
    <=> ? [Us: list_nat,Vs2: list_nat] :
          ( ( Xs = aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Us),Vs2) )
          & ( Ys = aa_list_nat_list_nat(aa_fun4736978287202700568st_nat(map_nat_nat,F),Us) )
          & ( Zs3 = aa_list_nat_list_nat(aa_fun4736978287202700568st_nat(map_nat_nat,F),Vs2) ) ) ) ).

% List.append_eq_map_conv
tff(fact_1364_List_Oappend__eq__map__conv,axiom,
    ! [Ys: list_assn,Zs3: list_assn,F: fun_Pr3816910165196019820n_assn,Xs: list_P8527749157015355191n_assn] :
      ( ( append_assn(Ys,Zs3) = aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,F),Xs) )
    <=> ? [Us: list_P8527749157015355191n_assn,Vs2: list_P8527749157015355191n_assn] :
          ( ( Xs = aa_lis2797772165808192636n_assn(aa_lis4738365772350288766n_assn(append282499809098378956n_assn,Us),Vs2) )
          & ( Ys = aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,F),Us) )
          & ( Zs3 = aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,F),Vs2) ) ) ) ).

% List.append_eq_map_conv
tff(fact_1365_star__or__dist2,axiom,
    ! [C3: assn,A2: assn,B2: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,C3),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A2),B2)) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,C3),A2)),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,C3),B2)) ).

% star_or_dist2
tff(fact_1366_star__or__dist1,axiom,
    ! [A2: assn,B2: assn,C3: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A2),B2)),C3) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A2),C3)),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B2),C3)) ).

% star_or_dist1
tff(fact_1367_ord_Olexordp__eq__pref,axiom,
    ! [Less: fun_nat_fun_nat_o,U: list_nat,V: list_nat] : aa_list_nat_o(aa_lis557685189327716851_nat_o(lexordp_eq_nat(Less),U),aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,U),V)) ).

% ord.lexordp_eq_pref
tff(fact_1368_ord_Olexordp__eq__pref,axiom,
    ! [Less: fun_int_fun_int_o,U: list_int,V: list_int] : aa_list_int_o(aa_lis7773489400162621755_int_o(lexordp_eq_int(Less),U),aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,U),V)) ).

% ord.lexordp_eq_pref
tff(fact_1369_lexordp__irreflexive,axiom,
    ! [Xs: list_assn] :
      ( ! [X2: assn] : ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X2),X2)
     => ~ ord_lexordp_assn(Xs,Xs) ) ).

% lexordp_irreflexive
tff(fact_1370_lexordp__irreflexive,axiom,
    ! [Xs: list_nat] :
      ( ! [X2: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),X2)
     => ~ aa_list_nat_o(aa_lis557685189327716851_nat_o(ord_lexordp_nat,Xs),Xs) ) ).

% lexordp_irreflexive
tff(fact_1371_lexordp__irreflexive,axiom,
    ! [Xs: list_int] :
      ( ! [X2: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X2),X2)
     => ~ aa_list_int_o(aa_lis7773489400162621755_int_o(ord_lexordp_int,Xs),Xs) ) ).

% lexordp_irreflexive
tff(fact_1372_norm__assertion__simps_I6_J,axiom,
    ! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),bot_bot_assn) = X ).

% norm_assertion_simps(6)
tff(fact_1373_norm__assertion__simps_I5_J,axiom,
    ! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,bot_bot_assn),X) = X ).

% norm_assertion_simps(5)
tff(fact_1374_lexordp__eq__pref,axiom,
    ! [U: list_nat,V: list_nat] : aa_list_nat_o(aa_lis557685189327716851_nat_o(ord_lexordp_eq_nat,U),aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,U),V)) ).

% lexordp_eq_pref
tff(fact_1375_lexordp__eq__pref,axiom,
    ! [U: list_int,V: list_int] : aa_list_int_o(aa_lis7773489400162621755_int_o(ord_lexordp_eq_int,U),aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,U),V)) ).

% lexordp_eq_pref
tff(fact_1376_sorted__list__of__set_Osorted__key__list__of__set__inject,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( linord2614967742042102400et_nat(A2) = linord2614967742042102400et_nat(B2) )
     => ( aa_set_nat_o(finite_finite_nat,A2)
       => ( aa_set_nat_o(finite_finite_nat,B2)
         => ( A2 = B2 ) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_inject
tff(fact_1377_merge_Osimps_I1_J,axiom,
    ! [L22: list_nat] : merge_nat(nil_nat,L22) = L22 ).

% merge.simps(1)
tff(fact_1378_merge_Osimps_I1_J,axiom,
    ! [L22: list_int] : merge_int(nil_int,L22) = L22 ).

% merge.simps(1)
tff(fact_1379_rev__induct,axiom,
    ! [Pa: fun_list_a_o,Xs: list_a] :
      ( aa_list_a_o(Pa,nil_a)
     => ( ! [X2: a,Xs2: list_a] :
            ( aa_list_a_o(Pa,Xs2)
           => aa_list_a_o(Pa,aa_list_a_list_a(aa_lis2442066584718795531list_a(append_a,Xs2),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X2),nil_a))) )
       => aa_list_a_o(Pa,Xs) ) ) ).

% rev_induct
tff(fact_1380_rev__induct,axiom,
    ! [Pa: fun_list_b_o,Xs: list_b] :
      ( aa_list_b_o(Pa,nil_b)
     => ( ! [X2: b,Xs2: list_b] :
            ( aa_list_b_o(Pa,Xs2)
           => aa_list_b_o(Pa,aa_list_b_list_b(aa_lis927650220358621516list_b(append_b,Xs2),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X2),nil_b))) )
       => aa_list_b_o(Pa,Xs) ) ) ).

% rev_induct
tff(fact_1381_rev__induct,axiom,
    ! [Pa: fun_list_nat_o,Xs: list_nat] :
      ( aa_list_nat_o(Pa,nil_nat)
     => ( ! [X2: nat,Xs2: list_nat] :
            ( aa_list_nat_o(Pa,Xs2)
           => aa_list_nat_o(Pa,aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Xs2),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X2),nil_nat))) )
       => aa_list_nat_o(Pa,Xs) ) ) ).

% rev_induct
tff(fact_1382_rev__induct,axiom,
    ! [Pa: fun_list_int_o,Xs: list_int] :
      ( aa_list_int_o(Pa,nil_int)
     => ( ! [X2: int,Xs2: list_int] :
            ( aa_list_int_o(Pa,Xs2)
           => aa_list_int_o(Pa,aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Xs2),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X2),nil_int))) )
       => aa_list_int_o(Pa,Xs) ) ) ).

% rev_induct
tff(fact_1383_rev__induct,axiom,
    ! [Pa: fun_li2812663585457640420assn_o,Xs: list_P8527749157015355191n_assn] :
      ( aa_lis1404944545740551261assn_o(Pa,nil_Pr5671120429643327159n_assn)
     => ( ! [X2: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn] :
            ( aa_lis1404944545740551261assn_o(Pa,Xs2)
           => aa_lis1404944545740551261assn_o(Pa,aa_lis2797772165808192636n_assn(aa_lis4738365772350288766n_assn(append282499809098378956n_assn,Xs2),aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X2),nil_Pr5671120429643327159n_assn))) )
       => aa_lis1404944545740551261assn_o(Pa,Xs) ) ) ).

% rev_induct
tff(fact_1384_rev__exhaust,axiom,
    ! [Xs: list_a] :
      ( ( Xs != nil_a )
     => ~ ! [Ys2: list_a,Y2: a] : Xs != aa_list_a_list_a(aa_lis2442066584718795531list_a(append_a,Ys2),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,Y2),nil_a)) ) ).

% rev_exhaust
tff(fact_1385_rev__exhaust,axiom,
    ! [Xs: list_b] :
      ( ( Xs != nil_b )
     => ~ ! [Ys2: list_b,Y2: b] : Xs != aa_list_b_list_b(aa_lis927650220358621516list_b(append_b,Ys2),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,Y2),nil_b)) ) ).

% rev_exhaust
tff(fact_1386_rev__exhaust,axiom,
    ! [Xs: list_nat] :
      ( ( Xs != nil_nat )
     => ~ ! [Ys2: list_nat,Y2: nat] : Xs != aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Ys2),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Y2),nil_nat)) ) ).

% rev_exhaust
tff(fact_1387_rev__exhaust,axiom,
    ! [Xs: list_int] :
      ( ( Xs != nil_int )
     => ~ ! [Ys2: list_int,Y2: int] : Xs != aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Ys2),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Y2),nil_int)) ) ).

% rev_exhaust
tff(fact_1388_rev__exhaust,axiom,
    ! [Xs: list_P8527749157015355191n_assn] :
      ( ( Xs != nil_Pr5671120429643327159n_assn )
     => ~ ! [Ys2: list_P8527749157015355191n_assn,Y2: produc6575502325842934193n_assn] : Xs != aa_lis2797772165808192636n_assn(aa_lis4738365772350288766n_assn(append282499809098378956n_assn,Ys2),aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,Y2),nil_Pr5671120429643327159n_assn)) ) ).

% rev_exhaust
tff(fact_1389_Cons__eq__append__conv,axiom,
    ! [X: a,Xs: list_a,Ys: list_a,Zs3: list_a] :
      ( ( aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X),Xs) = aa_list_a_list_a(aa_lis2442066584718795531list_a(append_a,Ys),Zs3) )
    <=> ( ( ( Ys = nil_a )
          & ( aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X),Xs) = Zs3 ) )
        | ? [Ys5: list_a] :
            ( ( aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X),Ys5) = Ys )
            & ( Xs = aa_list_a_list_a(aa_lis2442066584718795531list_a(append_a,Ys5),Zs3) ) ) ) ) ).

% Cons_eq_append_conv
tff(fact_1390_Cons__eq__append__conv,axiom,
    ! [X: b,Xs: list_b,Ys: list_b,Zs3: list_b] :
      ( ( aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X),Xs) = aa_list_b_list_b(aa_lis927650220358621516list_b(append_b,Ys),Zs3) )
    <=> ( ( ( Ys = nil_b )
          & ( aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X),Xs) = Zs3 ) )
        | ? [Ys5: list_b] :
            ( ( aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X),Ys5) = Ys )
            & ( Xs = aa_list_b_list_b(aa_lis927650220358621516list_b(append_b,Ys5),Zs3) ) ) ) ) ).

% Cons_eq_append_conv
tff(fact_1391_Cons__eq__append__conv,axiom,
    ! [X: nat,Xs: list_nat,Ys: list_nat,Zs3: list_nat] :
      ( ( aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X),Xs) = aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Ys),Zs3) )
    <=> ( ( ( Ys = nil_nat )
          & ( aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X),Xs) = Zs3 ) )
        | ? [Ys5: list_nat] :
            ( ( aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X),Ys5) = Ys )
            & ( Xs = aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Ys5),Zs3) ) ) ) ) ).

% Cons_eq_append_conv
tff(fact_1392_Cons__eq__append__conv,axiom,
    ! [X: int,Xs: list_int,Ys: list_int,Zs3: list_int] :
      ( ( aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X),Xs) = aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Ys),Zs3) )
    <=> ( ( ( Ys = nil_int )
          & ( aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X),Xs) = Zs3 ) )
        | ? [Ys5: list_int] :
            ( ( aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X),Ys5) = Ys )
            & ( Xs = aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Ys5),Zs3) ) ) ) ) ).

% Cons_eq_append_conv
tff(fact_1393_Cons__eq__append__conv,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn,Ys: list_P8527749157015355191n_assn,Zs3: list_P8527749157015355191n_assn] :
      ( ( aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X),Xs) = aa_lis2797772165808192636n_assn(aa_lis4738365772350288766n_assn(append282499809098378956n_assn,Ys),Zs3) )
    <=> ( ( ( Ys = nil_Pr5671120429643327159n_assn )
          & ( aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X),Xs) = Zs3 ) )
        | ? [Ys5: list_P8527749157015355191n_assn] :
            ( ( aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X),Ys5) = Ys )
            & ( Xs = aa_lis2797772165808192636n_assn(aa_lis4738365772350288766n_assn(append282499809098378956n_assn,Ys5),Zs3) ) ) ) ) ).

% Cons_eq_append_conv
tff(fact_1394_append__eq__Cons__conv,axiom,
    ! [Ys: list_a,Zs3: list_a,X: a,Xs: list_a] :
      ( ( aa_list_a_list_a(aa_lis2442066584718795531list_a(append_a,Ys),Zs3) = aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X),Xs) )
    <=> ( ( ( Ys = nil_a )
          & ( Zs3 = aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X),Xs) ) )
        | ? [Ys5: list_a] :
            ( ( Ys = aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X),Ys5) )
            & ( aa_list_a_list_a(aa_lis2442066584718795531list_a(append_a,Ys5),Zs3) = Xs ) ) ) ) ).

% append_eq_Cons_conv
tff(fact_1395_append__eq__Cons__conv,axiom,
    ! [Ys: list_b,Zs3: list_b,X: b,Xs: list_b] :
      ( ( aa_list_b_list_b(aa_lis927650220358621516list_b(append_b,Ys),Zs3) = aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X),Xs) )
    <=> ( ( ( Ys = nil_b )
          & ( Zs3 = aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X),Xs) ) )
        | ? [Ys5: list_b] :
            ( ( Ys = aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X),Ys5) )
            & ( aa_list_b_list_b(aa_lis927650220358621516list_b(append_b,Ys5),Zs3) = Xs ) ) ) ) ).

% append_eq_Cons_conv
tff(fact_1396_append__eq__Cons__conv,axiom,
    ! [Ys: list_nat,Zs3: list_nat,X: nat,Xs: list_nat] :
      ( ( aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Ys),Zs3) = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X),Xs) )
    <=> ( ( ( Ys = nil_nat )
          & ( Zs3 = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X),Xs) ) )
        | ? [Ys5: list_nat] :
            ( ( Ys = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X),Ys5) )
            & ( aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Ys5),Zs3) = Xs ) ) ) ) ).

% append_eq_Cons_conv
tff(fact_1397_append__eq__Cons__conv,axiom,
    ! [Ys: list_int,Zs3: list_int,X: int,Xs: list_int] :
      ( ( aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Ys),Zs3) = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X),Xs) )
    <=> ( ( ( Ys = nil_int )
          & ( Zs3 = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X),Xs) ) )
        | ? [Ys5: list_int] :
            ( ( Ys = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X),Ys5) )
            & ( aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Ys5),Zs3) = Xs ) ) ) ) ).

% append_eq_Cons_conv
tff(fact_1398_append__eq__Cons__conv,axiom,
    ! [Ys: list_P8527749157015355191n_assn,Zs3: list_P8527749157015355191n_assn,X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] :
      ( ( aa_lis2797772165808192636n_assn(aa_lis4738365772350288766n_assn(append282499809098378956n_assn,Ys),Zs3) = aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X),Xs) )
    <=> ( ( ( Ys = nil_Pr5671120429643327159n_assn )
          & ( Zs3 = aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X),Xs) ) )
        | ? [Ys5: list_P8527749157015355191n_assn] :
            ( ( Ys = aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X),Ys5) )
            & ( aa_lis2797772165808192636n_assn(aa_lis4738365772350288766n_assn(append282499809098378956n_assn,Ys5),Zs3) = Xs ) ) ) ) ).

% append_eq_Cons_conv
tff(fact_1399_rev__nonempty__induct,axiom,
    ! [Xs: list_a,Pa: fun_list_a_o] :
      ( ( Xs != nil_a )
     => ( ! [X2: a] : aa_list_a_o(Pa,aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X2),nil_a))
       => ( ! [X2: a,Xs2: list_a] :
              ( ( Xs2 != nil_a )
             => ( aa_list_a_o(Pa,Xs2)
               => aa_list_a_o(Pa,aa_list_a_list_a(aa_lis2442066584718795531list_a(append_a,Xs2),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X2),nil_a))) ) )
         => aa_list_a_o(Pa,Xs) ) ) ) ).

% rev_nonempty_induct
tff(fact_1400_rev__nonempty__induct,axiom,
    ! [Xs: list_b,Pa: fun_list_b_o] :
      ( ( Xs != nil_b )
     => ( ! [X2: b] : aa_list_b_o(Pa,aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X2),nil_b))
       => ( ! [X2: b,Xs2: list_b] :
              ( ( Xs2 != nil_b )
             => ( aa_list_b_o(Pa,Xs2)
               => aa_list_b_o(Pa,aa_list_b_list_b(aa_lis927650220358621516list_b(append_b,Xs2),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X2),nil_b))) ) )
         => aa_list_b_o(Pa,Xs) ) ) ) ).

% rev_nonempty_induct
tff(fact_1401_rev__nonempty__induct,axiom,
    ! [Xs: list_nat,Pa: fun_list_nat_o] :
      ( ( Xs != nil_nat )
     => ( ! [X2: nat] : aa_list_nat_o(Pa,aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X2),nil_nat))
       => ( ! [X2: nat,Xs2: list_nat] :
              ( ( Xs2 != nil_nat )
             => ( aa_list_nat_o(Pa,Xs2)
               => aa_list_nat_o(Pa,aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Xs2),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X2),nil_nat))) ) )
         => aa_list_nat_o(Pa,Xs) ) ) ) ).

% rev_nonempty_induct
tff(fact_1402_rev__nonempty__induct,axiom,
    ! [Xs: list_int,Pa: fun_list_int_o] :
      ( ( Xs != nil_int )
     => ( ! [X2: int] : aa_list_int_o(Pa,aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X2),nil_int))
       => ( ! [X2: int,Xs2: list_int] :
              ( ( Xs2 != nil_int )
             => ( aa_list_int_o(Pa,Xs2)
               => aa_list_int_o(Pa,aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Xs2),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X2),nil_int))) ) )
         => aa_list_int_o(Pa,Xs) ) ) ) ).

% rev_nonempty_induct
tff(fact_1403_rev__nonempty__induct,axiom,
    ! [Xs: list_P8527749157015355191n_assn,Pa: fun_li2812663585457640420assn_o] :
      ( ( Xs != nil_Pr5671120429643327159n_assn )
     => ( ! [X2: produc6575502325842934193n_assn] : aa_lis1404944545740551261assn_o(Pa,aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X2),nil_Pr5671120429643327159n_assn))
       => ( ! [X2: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn] :
              ( ( Xs2 != nil_Pr5671120429643327159n_assn )
             => ( aa_lis1404944545740551261assn_o(Pa,Xs2)
               => aa_lis1404944545740551261assn_o(Pa,aa_lis2797772165808192636n_assn(aa_lis4738365772350288766n_assn(append282499809098378956n_assn,Xs2),aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X2),nil_Pr5671120429643327159n_assn))) ) )
         => aa_lis1404944545740551261assn_o(Pa,Xs) ) ) ) ).

% rev_nonempty_induct
tff(fact_1404_neq__Nil__revE,axiom,
    ! [La: list_a] :
      ( ( La != nil_a )
     => ~ ! [Ll: list_a,E: a] : La != aa_list_a_list_a(aa_lis2442066584718795531list_a(append_a,Ll),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,E),nil_a)) ) ).

% neq_Nil_revE
tff(fact_1405_neq__Nil__revE,axiom,
    ! [La: list_b] :
      ( ( La != nil_b )
     => ~ ! [Ll: list_b,E: b] : La != aa_list_b_list_b(aa_lis927650220358621516list_b(append_b,Ll),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,E),nil_b)) ) ).

% neq_Nil_revE
tff(fact_1406_neq__Nil__revE,axiom,
    ! [La: list_nat] :
      ( ( La != nil_nat )
     => ~ ! [Ll: list_nat,E: nat] : La != aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Ll),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,E),nil_nat)) ) ).

% neq_Nil_revE
tff(fact_1407_neq__Nil__revE,axiom,
    ! [La: list_int] :
      ( ( La != nil_int )
     => ~ ! [Ll: list_int,E: int] : La != aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Ll),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,E),nil_int)) ) ).

% neq_Nil_revE
tff(fact_1408_neq__Nil__revE,axiom,
    ! [La: list_P8527749157015355191n_assn] :
      ( ( La != nil_Pr5671120429643327159n_assn )
     => ~ ! [Ll: list_P8527749157015355191n_assn,E: produc6575502325842934193n_assn] : La != aa_lis2797772165808192636n_assn(aa_lis4738365772350288766n_assn(append282499809098378956n_assn,Ll),aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,E),nil_Pr5671120429643327159n_assn)) ) ).

% neq_Nil_revE
tff(fact_1409_rev__induct2_H,axiom,
    ! [Pa: fun_li7061788742611047022st_a_o,Xs: list_a,Ys: list_a] :
      ( aa_list_a_o(aa_lis4963054610627903775st_a_o(Pa,nil_a),nil_a)
     => ( ! [X2: a,Xs2: list_a] : aa_list_a_o(aa_lis4963054610627903775st_a_o(Pa,aa_list_a_list_a(aa_lis2442066584718795531list_a(append_a,Xs2),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X2),nil_a))),nil_a)
       => ( ! [Y2: a,Ys2: list_a] : aa_list_a_o(aa_lis4963054610627903775st_a_o(Pa,nil_a),aa_list_a_list_a(aa_lis2442066584718795531list_a(append_a,Ys2),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,Y2),nil_a)))
         => ( ! [X2: a,Xs2: list_a,Y2: a,Ys2: list_a] :
                ( aa_list_a_o(aa_lis4963054610627903775st_a_o(Pa,Xs2),Ys2)
               => aa_list_a_o(aa_lis4963054610627903775st_a_o(Pa,aa_list_a_list_a(aa_lis2442066584718795531list_a(append_a,Xs2),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X2),nil_a))),aa_list_a_list_a(aa_lis2442066584718795531list_a(append_a,Ys2),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,Y2),nil_a))) )
           => aa_list_a_o(aa_lis4963054610627903775st_a_o(Pa,Xs),Ys) ) ) ) ) ).

% rev_induct2'
tff(fact_1410_rev__induct2_H,axiom,
    ! [Pa: fun_li6426139742006060333st_b_o,Xs: list_a,Ys: list_b] :
      ( aa_list_b_o(aa_lis2175599028655227742st_b_o(Pa,nil_a),nil_b)
     => ( ! [X2: a,Xs2: list_a] : aa_list_b_o(aa_lis2175599028655227742st_b_o(Pa,aa_list_a_list_a(aa_lis2442066584718795531list_a(append_a,Xs2),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X2),nil_a))),nil_b)
       => ( ! [Y2: b,Ys2: list_b] : aa_list_b_o(aa_lis2175599028655227742st_b_o(Pa,nil_a),aa_list_b_list_b(aa_lis927650220358621516list_b(append_b,Ys2),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,Y2),nil_b)))
         => ( ! [X2: a,Xs2: list_a,Y2: b,Ys2: list_b] :
                ( aa_list_b_o(aa_lis2175599028655227742st_b_o(Pa,Xs2),Ys2)
               => aa_list_b_o(aa_lis2175599028655227742st_b_o(Pa,aa_list_a_list_a(aa_lis2442066584718795531list_a(append_a,Xs2),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X2),nil_a))),aa_list_b_list_b(aa_lis927650220358621516list_b(append_b,Ys2),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,Y2),nil_b))) )
           => aa_list_b_o(aa_lis2175599028655227742st_b_o(Pa,Xs),Ys) ) ) ) ) ).

% rev_induct2'
tff(fact_1411_rev__induct2_H,axiom,
    ! [Pa: fun_li100135987217049455st_a_o,Xs: list_b,Ys: list_a] :
      ( aa_list_a_o(aa_lis7224773892088682016st_a_o(Pa,nil_b),nil_a)
     => ( ! [X2: b,Xs2: list_b] : aa_list_a_o(aa_lis7224773892088682016st_a_o(Pa,aa_list_b_list_b(aa_lis927650220358621516list_b(append_b,Xs2),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X2),nil_b))),nil_a)
       => ( ! [Y2: a,Ys2: list_a] : aa_list_a_o(aa_lis7224773892088682016st_a_o(Pa,nil_b),aa_list_a_list_a(aa_lis2442066584718795531list_a(append_a,Ys2),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,Y2),nil_a)))
         => ( ! [X2: b,Xs2: list_b,Y2: a,Ys2: list_a] :
                ( aa_list_a_o(aa_lis7224773892088682016st_a_o(Pa,Xs2),Ys2)
               => aa_list_a_o(aa_lis7224773892088682016st_a_o(Pa,aa_list_b_list_b(aa_lis927650220358621516list_b(append_b,Xs2),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X2),nil_b))),aa_list_a_list_a(aa_lis2442066584718795531list_a(append_a,Ys2),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,Y2),nil_a))) )
           => aa_list_a_o(aa_lis7224773892088682016st_a_o(Pa,Xs),Ys) ) ) ) ) ).

% rev_induct2'
tff(fact_1412_rev__induct2_H,axiom,
    ! [Pa: fun_li8687859023466838574st_b_o,Xs: list_b,Ys: list_b] :
      ( aa_list_b_o(aa_lis4437318310116005983st_b_o(Pa,nil_b),nil_b)
     => ( ! [X2: b,Xs2: list_b] : aa_list_b_o(aa_lis4437318310116005983st_b_o(Pa,aa_list_b_list_b(aa_lis927650220358621516list_b(append_b,Xs2),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X2),nil_b))),nil_b)
       => ( ! [Y2: b,Ys2: list_b] : aa_list_b_o(aa_lis4437318310116005983st_b_o(Pa,nil_b),aa_list_b_list_b(aa_lis927650220358621516list_b(append_b,Ys2),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,Y2),nil_b)))
         => ( ! [X2: b,Xs2: list_b,Y2: b,Ys2: list_b] :
                ( aa_list_b_o(aa_lis4437318310116005983st_b_o(Pa,Xs2),Ys2)
               => aa_list_b_o(aa_lis4437318310116005983st_b_o(Pa,aa_list_b_list_b(aa_lis927650220358621516list_b(append_b,Xs2),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X2),nil_b))),aa_list_b_list_b(aa_lis927650220358621516list_b(append_b,Ys2),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,Y2),nil_b))) )
           => aa_list_b_o(aa_lis4437318310116005983st_b_o(Pa,Xs),Ys) ) ) ) ) ).

% rev_induct2'
tff(fact_1413_rev__induct2_H,axiom,
    ! [Pa: fun_li2975960090505800144_nat_o,Xs: list_a,Ys: list_nat] :
      ( aa_list_nat_o(aa_lis6662471382306767049_nat_o(Pa,nil_a),nil_nat)
     => ( ! [X2: a,Xs2: list_a] : aa_list_nat_o(aa_lis6662471382306767049_nat_o(Pa,aa_list_a_list_a(aa_lis2442066584718795531list_a(append_a,Xs2),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X2),nil_a))),nil_nat)
       => ( ! [Y2: nat,Ys2: list_nat] : aa_list_nat_o(aa_lis6662471382306767049_nat_o(Pa,nil_a),aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Ys2),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Y2),nil_nat)))
         => ( ! [X2: a,Xs2: list_a,Y2: nat,Ys2: list_nat] :
                ( aa_list_nat_o(aa_lis6662471382306767049_nat_o(Pa,Xs2),Ys2)
               => aa_list_nat_o(aa_lis6662471382306767049_nat_o(Pa,aa_list_a_list_a(aa_lis2442066584718795531list_a(append_a,Xs2),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X2),nil_a))),aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Ys2),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Y2),nil_nat))) )
           => aa_list_nat_o(aa_lis6662471382306767049_nat_o(Pa,Xs),Ys) ) ) ) ) ).

% rev_induct2'
tff(fact_1414_rev__induct2_H,axiom,
    ! [Pa: fun_li9190550672693380751_nat_o,Xs: list_b,Ys: list_nat] :
      ( aa_list_nat_o(aa_lis3653689927639571848_nat_o(Pa,nil_b),nil_nat)
     => ( ! [X2: b,Xs2: list_b] : aa_list_nat_o(aa_lis3653689927639571848_nat_o(Pa,aa_list_b_list_b(aa_lis927650220358621516list_b(append_b,Xs2),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X2),nil_b))),nil_nat)
       => ( ! [Y2: nat,Ys2: list_nat] : aa_list_nat_o(aa_lis3653689927639571848_nat_o(Pa,nil_b),aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Ys2),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Y2),nil_nat)))
         => ( ! [X2: b,Xs2: list_b,Y2: nat,Ys2: list_nat] :
                ( aa_list_nat_o(aa_lis3653689927639571848_nat_o(Pa,Xs2),Ys2)
               => aa_list_nat_o(aa_lis3653689927639571848_nat_o(Pa,aa_list_b_list_b(aa_lis927650220358621516list_b(append_b,Xs2),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X2),nil_b))),aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Ys2),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Y2),nil_nat))) )
           => aa_list_nat_o(aa_lis3653689927639571848_nat_o(Pa,Xs),Ys) ) ) ) ) ).

% rev_induct2'
tff(fact_1415_rev__induct2_H,axiom,
    ! [Pa: fun_li3047910582728761460_int_o,Xs: list_a,Ys: list_int] :
      ( aa_list_int_o(aa_lis4417326414636321133_int_o(Pa,nil_a),nil_int)
     => ( ! [X2: a,Xs2: list_a] : aa_list_int_o(aa_lis4417326414636321133_int_o(Pa,aa_list_a_list_a(aa_lis2442066584718795531list_a(append_a,Xs2),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X2),nil_a))),nil_int)
       => ( ! [Y2: int,Ys2: list_int] : aa_list_int_o(aa_lis4417326414636321133_int_o(Pa,nil_a),aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Ys2),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Y2),nil_int)))
         => ( ! [X2: a,Xs2: list_a,Y2: int,Ys2: list_int] :
                ( aa_list_int_o(aa_lis4417326414636321133_int_o(Pa,Xs2),Ys2)
               => aa_list_int_o(aa_lis4417326414636321133_int_o(Pa,aa_list_a_list_a(aa_lis2442066584718795531list_a(append_a,Xs2),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X2),nil_a))),aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Ys2),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Y2),nil_int))) )
           => aa_list_int_o(aa_lis4417326414636321133_int_o(Pa,Xs),Ys) ) ) ) ) ).

% rev_induct2'
tff(fact_1416_rev__induct2_H,axiom,
    ! [Pa: fun_li39129128061566259_int_o,Xs: list_b,Ys: list_int] :
      ( aa_list_int_o(aa_lis1408544959969125932_int_o(Pa,nil_b),nil_int)
     => ( ! [X2: b,Xs2: list_b] : aa_list_int_o(aa_lis1408544959969125932_int_o(Pa,aa_list_b_list_b(aa_lis927650220358621516list_b(append_b,Xs2),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X2),nil_b))),nil_int)
       => ( ! [Y2: int,Ys2: list_int] : aa_list_int_o(aa_lis1408544959969125932_int_o(Pa,nil_b),aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Ys2),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Y2),nil_int)))
         => ( ! [X2: b,Xs2: list_b,Y2: int,Ys2: list_int] :
                ( aa_list_int_o(aa_lis1408544959969125932_int_o(Pa,Xs2),Ys2)
               => aa_list_int_o(aa_lis1408544959969125932_int_o(Pa,aa_list_b_list_b(aa_lis927650220358621516list_b(append_b,Xs2),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X2),nil_b))),aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Ys2),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Y2),nil_int))) )
           => aa_list_int_o(aa_lis1408544959969125932_int_o(Pa,Xs),Ys) ) ) ) ) ).

% rev_induct2'
tff(fact_1417_rev__induct2_H,axiom,
    ! [Pa: fun_li5607945090506445748st_a_o,Xs: list_nat,Ys: list_a] :
      ( aa_list_a_o(aa_lis6242202807773918837st_a_o(Pa,nil_nat),nil_a)
     => ( ! [X2: nat,Xs2: list_nat] : aa_list_a_o(aa_lis6242202807773918837st_a_o(Pa,aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Xs2),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X2),nil_nat))),nil_a)
       => ( ! [Y2: a,Ys2: list_a] : aa_list_a_o(aa_lis6242202807773918837st_a_o(Pa,nil_nat),aa_list_a_list_a(aa_lis2442066584718795531list_a(append_a,Ys2),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,Y2),nil_a)))
         => ( ! [X2: nat,Xs2: list_nat,Y2: a,Ys2: list_a] :
                ( aa_list_a_o(aa_lis6242202807773918837st_a_o(Pa,Xs2),Ys2)
               => aa_list_a_o(aa_lis6242202807773918837st_a_o(Pa,aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Xs2),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X2),nil_nat))),aa_list_a_list_a(aa_lis2442066584718795531list_a(append_a,Ys2),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,Y2),nil_a))) )
           => aa_list_a_o(aa_lis6242202807773918837st_a_o(Pa,Xs),Ys) ) ) ) ) ).

% rev_induct2'
tff(fact_1418_rev__induct2_H,axiom,
    ! [Pa: fun_li4972296089901459059st_b_o,Xs: list_nat,Ys: list_b] :
      ( aa_list_b_o(aa_lis3454747225801242804st_b_o(Pa,nil_nat),nil_b)
     => ( ! [X2: nat,Xs2: list_nat] : aa_list_b_o(aa_lis3454747225801242804st_b_o(Pa,aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Xs2),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X2),nil_nat))),nil_b)
       => ( ! [Y2: b,Ys2: list_b] : aa_list_b_o(aa_lis3454747225801242804st_b_o(Pa,nil_nat),aa_list_b_list_b(aa_lis927650220358621516list_b(append_b,Ys2),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,Y2),nil_b)))
         => ( ! [X2: nat,Xs2: list_nat,Y2: b,Ys2: list_b] :
                ( aa_list_b_o(aa_lis3454747225801242804st_b_o(Pa,Xs2),Ys2)
               => aa_list_b_o(aa_lis3454747225801242804st_b_o(Pa,aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Xs2),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X2),nil_nat))),aa_list_b_list_b(aa_lis927650220358621516list_b(append_b,Ys2),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,Y2),nil_b))) )
           => aa_list_b_o(aa_lis3454747225801242804st_b_o(Pa,Xs),Ys) ) ) ) ) ).

% rev_induct2'
tff(fact_1419_neq__Nil__rev__conv,axiom,
    ! [La: list_a] :
      ( ( La != nil_a )
    <=> ? [Xs3: list_a,X3: a] : La = aa_list_a_list_a(aa_lis2442066584718795531list_a(append_a,Xs3),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X3),nil_a)) ) ).

% neq_Nil_rev_conv
tff(fact_1420_neq__Nil__rev__conv,axiom,
    ! [La: list_b] :
      ( ( La != nil_b )
    <=> ? [Xs3: list_b,X3: b] : La = aa_list_b_list_b(aa_lis927650220358621516list_b(append_b,Xs3),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X3),nil_b)) ) ).

% neq_Nil_rev_conv
tff(fact_1421_neq__Nil__rev__conv,axiom,
    ! [La: list_nat] :
      ( ( La != nil_nat )
    <=> ? [Xs3: list_nat,X3: nat] : La = aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Xs3),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X3),nil_nat)) ) ).

% neq_Nil_rev_conv
tff(fact_1422_neq__Nil__rev__conv,axiom,
    ! [La: list_int] :
      ( ( La != nil_int )
    <=> ? [Xs3: list_int,X3: int] : La = aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Xs3),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X3),nil_int)) ) ).

% neq_Nil_rev_conv
tff(fact_1423_neq__Nil__rev__conv,axiom,
    ! [La: list_P8527749157015355191n_assn] :
      ( ( La != nil_Pr5671120429643327159n_assn )
    <=> ? [Xs3: list_P8527749157015355191n_assn,X3: produc6575502325842934193n_assn] : La = aa_lis2797772165808192636n_assn(aa_lis4738365772350288766n_assn(append282499809098378956n_assn,Xs3),aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X3),nil_Pr5671120429643327159n_assn)) ) ).

% neq_Nil_rev_conv
tff(fact_1424_rev__nonempty__induct2_H,axiom,
    ! [Xs: list_a,Ys: list_a,Pa: fun_li7061788742611047022st_a_o] :
      ( ( Xs != nil_a )
     => ( ( Ys != nil_a )
       => ( ! [X2: a,Y2: a] : aa_list_a_o(aa_lis4963054610627903775st_a_o(Pa,aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X2),nil_a)),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,Y2),nil_a))
         => ( ! [X2: a,Xs2: list_a,Y2: a] :
                ( ( Xs2 != nil_a )
               => aa_list_a_o(aa_lis4963054610627903775st_a_o(Pa,aa_list_a_list_a(aa_lis2442066584718795531list_a(append_a,Xs2),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X2),nil_a))),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,Y2),nil_a)) )
           => ( ! [X2: a,Y2: a,Ys2: list_a] :
                  ( ( Ys2 != nil_a )
                 => aa_list_a_o(aa_lis4963054610627903775st_a_o(Pa,aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X2),nil_a)),aa_list_a_list_a(aa_lis2442066584718795531list_a(append_a,Ys2),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,Y2),nil_a))) )
             => ( ! [X2: a,Xs2: list_a,Y2: a,Ys2: list_a] :
                    ( aa_list_a_o(aa_lis4963054610627903775st_a_o(Pa,Xs2),Ys2)
                   => ( ( Xs2 != nil_a )
                     => ( ( Ys2 != nil_a )
                       => aa_list_a_o(aa_lis4963054610627903775st_a_o(Pa,aa_list_a_list_a(aa_lis2442066584718795531list_a(append_a,Xs2),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X2),nil_a))),aa_list_a_list_a(aa_lis2442066584718795531list_a(append_a,Ys2),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,Y2),nil_a))) ) ) )
               => aa_list_a_o(aa_lis4963054610627903775st_a_o(Pa,Xs),Ys) ) ) ) ) ) ) ).

% rev_nonempty_induct2'
tff(fact_1425_rev__nonempty__induct2_H,axiom,
    ! [Xs: list_a,Ys: list_b,Pa: fun_li6426139742006060333st_b_o] :
      ( ( Xs != nil_a )
     => ( ( Ys != nil_b )
       => ( ! [X2: a,Y2: b] : aa_list_b_o(aa_lis2175599028655227742st_b_o(Pa,aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X2),nil_a)),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,Y2),nil_b))
         => ( ! [X2: a,Xs2: list_a,Y2: b] :
                ( ( Xs2 != nil_a )
               => aa_list_b_o(aa_lis2175599028655227742st_b_o(Pa,aa_list_a_list_a(aa_lis2442066584718795531list_a(append_a,Xs2),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X2),nil_a))),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,Y2),nil_b)) )
           => ( ! [X2: a,Y2: b,Ys2: list_b] :
                  ( ( Ys2 != nil_b )
                 => aa_list_b_o(aa_lis2175599028655227742st_b_o(Pa,aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X2),nil_a)),aa_list_b_list_b(aa_lis927650220358621516list_b(append_b,Ys2),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,Y2),nil_b))) )
             => ( ! [X2: a,Xs2: list_a,Y2: b,Ys2: list_b] :
                    ( aa_list_b_o(aa_lis2175599028655227742st_b_o(Pa,Xs2),Ys2)
                   => ( ( Xs2 != nil_a )
                     => ( ( Ys2 != nil_b )
                       => aa_list_b_o(aa_lis2175599028655227742st_b_o(Pa,aa_list_a_list_a(aa_lis2442066584718795531list_a(append_a,Xs2),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X2),nil_a))),aa_list_b_list_b(aa_lis927650220358621516list_b(append_b,Ys2),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,Y2),nil_b))) ) ) )
               => aa_list_b_o(aa_lis2175599028655227742st_b_o(Pa,Xs),Ys) ) ) ) ) ) ) ).

% rev_nonempty_induct2'
tff(fact_1426_rev__nonempty__induct2_H,axiom,
    ! [Xs: list_b,Ys: list_a,Pa: fun_li100135987217049455st_a_o] :
      ( ( Xs != nil_b )
     => ( ( Ys != nil_a )
       => ( ! [X2: b,Y2: a] : aa_list_a_o(aa_lis7224773892088682016st_a_o(Pa,aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X2),nil_b)),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,Y2),nil_a))
         => ( ! [X2: b,Xs2: list_b,Y2: a] :
                ( ( Xs2 != nil_b )
               => aa_list_a_o(aa_lis7224773892088682016st_a_o(Pa,aa_list_b_list_b(aa_lis927650220358621516list_b(append_b,Xs2),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X2),nil_b))),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,Y2),nil_a)) )
           => ( ! [X2: b,Y2: a,Ys2: list_a] :
                  ( ( Ys2 != nil_a )
                 => aa_list_a_o(aa_lis7224773892088682016st_a_o(Pa,aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X2),nil_b)),aa_list_a_list_a(aa_lis2442066584718795531list_a(append_a,Ys2),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,Y2),nil_a))) )
             => ( ! [X2: b,Xs2: list_b,Y2: a,Ys2: list_a] :
                    ( aa_list_a_o(aa_lis7224773892088682016st_a_o(Pa,Xs2),Ys2)
                   => ( ( Xs2 != nil_b )
                     => ( ( Ys2 != nil_a )
                       => aa_list_a_o(aa_lis7224773892088682016st_a_o(Pa,aa_list_b_list_b(aa_lis927650220358621516list_b(append_b,Xs2),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X2),nil_b))),aa_list_a_list_a(aa_lis2442066584718795531list_a(append_a,Ys2),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,Y2),nil_a))) ) ) )
               => aa_list_a_o(aa_lis7224773892088682016st_a_o(Pa,Xs),Ys) ) ) ) ) ) ) ).

% rev_nonempty_induct2'
tff(fact_1427_rev__nonempty__induct2_H,axiom,
    ! [Xs: list_b,Ys: list_b,Pa: fun_li8687859023466838574st_b_o] :
      ( ( Xs != nil_b )
     => ( ( Ys != nil_b )
       => ( ! [X2: b,Y2: b] : aa_list_b_o(aa_lis4437318310116005983st_b_o(Pa,aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X2),nil_b)),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,Y2),nil_b))
         => ( ! [X2: b,Xs2: list_b,Y2: b] :
                ( ( Xs2 != nil_b )
               => aa_list_b_o(aa_lis4437318310116005983st_b_o(Pa,aa_list_b_list_b(aa_lis927650220358621516list_b(append_b,Xs2),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X2),nil_b))),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,Y2),nil_b)) )
           => ( ! [X2: b,Y2: b,Ys2: list_b] :
                  ( ( Ys2 != nil_b )
                 => aa_list_b_o(aa_lis4437318310116005983st_b_o(Pa,aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X2),nil_b)),aa_list_b_list_b(aa_lis927650220358621516list_b(append_b,Ys2),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,Y2),nil_b))) )
             => ( ! [X2: b,Xs2: list_b,Y2: b,Ys2: list_b] :
                    ( aa_list_b_o(aa_lis4437318310116005983st_b_o(Pa,Xs2),Ys2)
                   => ( ( Xs2 != nil_b )
                     => ( ( Ys2 != nil_b )
                       => aa_list_b_o(aa_lis4437318310116005983st_b_o(Pa,aa_list_b_list_b(aa_lis927650220358621516list_b(append_b,Xs2),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X2),nil_b))),aa_list_b_list_b(aa_lis927650220358621516list_b(append_b,Ys2),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,Y2),nil_b))) ) ) )
               => aa_list_b_o(aa_lis4437318310116005983st_b_o(Pa,Xs),Ys) ) ) ) ) ) ) ).

% rev_nonempty_induct2'
tff(fact_1428_rev__nonempty__induct2_H,axiom,
    ! [Xs: list_a,Ys: list_nat,Pa: fun_li2975960090505800144_nat_o] :
      ( ( Xs != nil_a )
     => ( ( Ys != nil_nat )
       => ( ! [X2: a,Y2: nat] : aa_list_nat_o(aa_lis6662471382306767049_nat_o(Pa,aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X2),nil_a)),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Y2),nil_nat))
         => ( ! [X2: a,Xs2: list_a,Y2: nat] :
                ( ( Xs2 != nil_a )
               => aa_list_nat_o(aa_lis6662471382306767049_nat_o(Pa,aa_list_a_list_a(aa_lis2442066584718795531list_a(append_a,Xs2),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X2),nil_a))),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Y2),nil_nat)) )
           => ( ! [X2: a,Y2: nat,Ys2: list_nat] :
                  ( ( Ys2 != nil_nat )
                 => aa_list_nat_o(aa_lis6662471382306767049_nat_o(Pa,aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X2),nil_a)),aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Ys2),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Y2),nil_nat))) )
             => ( ! [X2: a,Xs2: list_a,Y2: nat,Ys2: list_nat] :
                    ( aa_list_nat_o(aa_lis6662471382306767049_nat_o(Pa,Xs2),Ys2)
                   => ( ( Xs2 != nil_a )
                     => ( ( Ys2 != nil_nat )
                       => aa_list_nat_o(aa_lis6662471382306767049_nat_o(Pa,aa_list_a_list_a(aa_lis2442066584718795531list_a(append_a,Xs2),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X2),nil_a))),aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Ys2),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Y2),nil_nat))) ) ) )
               => aa_list_nat_o(aa_lis6662471382306767049_nat_o(Pa,Xs),Ys) ) ) ) ) ) ) ).

% rev_nonempty_induct2'
tff(fact_1429_rev__nonempty__induct2_H,axiom,
    ! [Xs: list_b,Ys: list_nat,Pa: fun_li9190550672693380751_nat_o] :
      ( ( Xs != nil_b )
     => ( ( Ys != nil_nat )
       => ( ! [X2: b,Y2: nat] : aa_list_nat_o(aa_lis3653689927639571848_nat_o(Pa,aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X2),nil_b)),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Y2),nil_nat))
         => ( ! [X2: b,Xs2: list_b,Y2: nat] :
                ( ( Xs2 != nil_b )
               => aa_list_nat_o(aa_lis3653689927639571848_nat_o(Pa,aa_list_b_list_b(aa_lis927650220358621516list_b(append_b,Xs2),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X2),nil_b))),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Y2),nil_nat)) )
           => ( ! [X2: b,Y2: nat,Ys2: list_nat] :
                  ( ( Ys2 != nil_nat )
                 => aa_list_nat_o(aa_lis3653689927639571848_nat_o(Pa,aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X2),nil_b)),aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Ys2),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Y2),nil_nat))) )
             => ( ! [X2: b,Xs2: list_b,Y2: nat,Ys2: list_nat] :
                    ( aa_list_nat_o(aa_lis3653689927639571848_nat_o(Pa,Xs2),Ys2)
                   => ( ( Xs2 != nil_b )
                     => ( ( Ys2 != nil_nat )
                       => aa_list_nat_o(aa_lis3653689927639571848_nat_o(Pa,aa_list_b_list_b(aa_lis927650220358621516list_b(append_b,Xs2),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X2),nil_b))),aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Ys2),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Y2),nil_nat))) ) ) )
               => aa_list_nat_o(aa_lis3653689927639571848_nat_o(Pa,Xs),Ys) ) ) ) ) ) ) ).

% rev_nonempty_induct2'
tff(fact_1430_rev__nonempty__induct2_H,axiom,
    ! [Xs: list_a,Ys: list_int,Pa: fun_li3047910582728761460_int_o] :
      ( ( Xs != nil_a )
     => ( ( Ys != nil_int )
       => ( ! [X2: a,Y2: int] : aa_list_int_o(aa_lis4417326414636321133_int_o(Pa,aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X2),nil_a)),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Y2),nil_int))
         => ( ! [X2: a,Xs2: list_a,Y2: int] :
                ( ( Xs2 != nil_a )
               => aa_list_int_o(aa_lis4417326414636321133_int_o(Pa,aa_list_a_list_a(aa_lis2442066584718795531list_a(append_a,Xs2),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X2),nil_a))),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Y2),nil_int)) )
           => ( ! [X2: a,Y2: int,Ys2: list_int] :
                  ( ( Ys2 != nil_int )
                 => aa_list_int_o(aa_lis4417326414636321133_int_o(Pa,aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X2),nil_a)),aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Ys2),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Y2),nil_int))) )
             => ( ! [X2: a,Xs2: list_a,Y2: int,Ys2: list_int] :
                    ( aa_list_int_o(aa_lis4417326414636321133_int_o(Pa,Xs2),Ys2)
                   => ( ( Xs2 != nil_a )
                     => ( ( Ys2 != nil_int )
                       => aa_list_int_o(aa_lis4417326414636321133_int_o(Pa,aa_list_a_list_a(aa_lis2442066584718795531list_a(append_a,Xs2),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X2),nil_a))),aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Ys2),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Y2),nil_int))) ) ) )
               => aa_list_int_o(aa_lis4417326414636321133_int_o(Pa,Xs),Ys) ) ) ) ) ) ) ).

% rev_nonempty_induct2'
tff(fact_1431_rev__nonempty__induct2_H,axiom,
    ! [Xs: list_b,Ys: list_int,Pa: fun_li39129128061566259_int_o] :
      ( ( Xs != nil_b )
     => ( ( Ys != nil_int )
       => ( ! [X2: b,Y2: int] : aa_list_int_o(aa_lis1408544959969125932_int_o(Pa,aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X2),nil_b)),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Y2),nil_int))
         => ( ! [X2: b,Xs2: list_b,Y2: int] :
                ( ( Xs2 != nil_b )
               => aa_list_int_o(aa_lis1408544959969125932_int_o(Pa,aa_list_b_list_b(aa_lis927650220358621516list_b(append_b,Xs2),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X2),nil_b))),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Y2),nil_int)) )
           => ( ! [X2: b,Y2: int,Ys2: list_int] :
                  ( ( Ys2 != nil_int )
                 => aa_list_int_o(aa_lis1408544959969125932_int_o(Pa,aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X2),nil_b)),aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Ys2),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Y2),nil_int))) )
             => ( ! [X2: b,Xs2: list_b,Y2: int,Ys2: list_int] :
                    ( aa_list_int_o(aa_lis1408544959969125932_int_o(Pa,Xs2),Ys2)
                   => ( ( Xs2 != nil_b )
                     => ( ( Ys2 != nil_int )
                       => aa_list_int_o(aa_lis1408544959969125932_int_o(Pa,aa_list_b_list_b(aa_lis927650220358621516list_b(append_b,Xs2),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X2),nil_b))),aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Ys2),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Y2),nil_int))) ) ) )
               => aa_list_int_o(aa_lis1408544959969125932_int_o(Pa,Xs),Ys) ) ) ) ) ) ) ).

% rev_nonempty_induct2'
tff(fact_1432_rev__nonempty__induct2_H,axiom,
    ! [Xs: list_nat,Ys: list_a,Pa: fun_li5607945090506445748st_a_o] :
      ( ( Xs != nil_nat )
     => ( ( Ys != nil_a )
       => ( ! [X2: nat,Y2: a] : aa_list_a_o(aa_lis6242202807773918837st_a_o(Pa,aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X2),nil_nat)),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,Y2),nil_a))
         => ( ! [X2: nat,Xs2: list_nat,Y2: a] :
                ( ( Xs2 != nil_nat )
               => aa_list_a_o(aa_lis6242202807773918837st_a_o(Pa,aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Xs2),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X2),nil_nat))),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,Y2),nil_a)) )
           => ( ! [X2: nat,Y2: a,Ys2: list_a] :
                  ( ( Ys2 != nil_a )
                 => aa_list_a_o(aa_lis6242202807773918837st_a_o(Pa,aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X2),nil_nat)),aa_list_a_list_a(aa_lis2442066584718795531list_a(append_a,Ys2),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,Y2),nil_a))) )
             => ( ! [X2: nat,Xs2: list_nat,Y2: a,Ys2: list_a] :
                    ( aa_list_a_o(aa_lis6242202807773918837st_a_o(Pa,Xs2),Ys2)
                   => ( ( Xs2 != nil_nat )
                     => ( ( Ys2 != nil_a )
                       => aa_list_a_o(aa_lis6242202807773918837st_a_o(Pa,aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Xs2),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X2),nil_nat))),aa_list_a_list_a(aa_lis2442066584718795531list_a(append_a,Ys2),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,Y2),nil_a))) ) ) )
               => aa_list_a_o(aa_lis6242202807773918837st_a_o(Pa,Xs),Ys) ) ) ) ) ) ) ).

% rev_nonempty_induct2'
tff(fact_1433_rev__nonempty__induct2_H,axiom,
    ! [Xs: list_nat,Ys: list_b,Pa: fun_li4972296089901459059st_b_o] :
      ( ( Xs != nil_nat )
     => ( ( Ys != nil_b )
       => ( ! [X2: nat,Y2: b] : aa_list_b_o(aa_lis3454747225801242804st_b_o(Pa,aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X2),nil_nat)),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,Y2),nil_b))
         => ( ! [X2: nat,Xs2: list_nat,Y2: b] :
                ( ( Xs2 != nil_nat )
               => aa_list_b_o(aa_lis3454747225801242804st_b_o(Pa,aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Xs2),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X2),nil_nat))),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,Y2),nil_b)) )
           => ( ! [X2: nat,Y2: b,Ys2: list_b] :
                  ( ( Ys2 != nil_b )
                 => aa_list_b_o(aa_lis3454747225801242804st_b_o(Pa,aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X2),nil_nat)),aa_list_b_list_b(aa_lis927650220358621516list_b(append_b,Ys2),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,Y2),nil_b))) )
             => ( ! [X2: nat,Xs2: list_nat,Y2: b,Ys2: list_b] :
                    ( aa_list_b_o(aa_lis3454747225801242804st_b_o(Pa,Xs2),Ys2)
                   => ( ( Xs2 != nil_nat )
                     => ( ( Ys2 != nil_b )
                       => aa_list_b_o(aa_lis3454747225801242804st_b_o(Pa,aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Xs2),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X2),nil_nat))),aa_list_b_list_b(aa_lis927650220358621516list_b(append_b,Ys2),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,Y2),nil_b))) ) ) )
               => aa_list_b_o(aa_lis3454747225801242804st_b_o(Pa,Xs),Ys) ) ) ) ) ) ) ).

% rev_nonempty_induct2'
tff(fact_1434_list__Cons__eq__append__cases,axiom,
    ! [X: a,Xs: list_a,Ys: list_a,Zs3: list_a] :
      ( ( aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X),Xs) = aa_list_a_list_a(aa_lis2442066584718795531list_a(append_a,Ys),Zs3) )
     => ( ( ( Ys = nil_a )
         => ( Zs3 != aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X),Xs) ) )
       => ~ ! [Ys4: list_a] :
              ( ( Ys = aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X),Ys4) )
             => ( aa_list_a_list_a(aa_lis2442066584718795531list_a(append_a,Ys4),Zs3) != Xs ) ) ) ) ).

% list_Cons_eq_append_cases
tff(fact_1435_list__Cons__eq__append__cases,axiom,
    ! [X: b,Xs: list_b,Ys: list_b,Zs3: list_b] :
      ( ( aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X),Xs) = aa_list_b_list_b(aa_lis927650220358621516list_b(append_b,Ys),Zs3) )
     => ( ( ( Ys = nil_b )
         => ( Zs3 != aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X),Xs) ) )
       => ~ ! [Ys4: list_b] :
              ( ( Ys = aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X),Ys4) )
             => ( aa_list_b_list_b(aa_lis927650220358621516list_b(append_b,Ys4),Zs3) != Xs ) ) ) ) ).

% list_Cons_eq_append_cases
tff(fact_1436_list__Cons__eq__append__cases,axiom,
    ! [X: nat,Xs: list_nat,Ys: list_nat,Zs3: list_nat] :
      ( ( aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X),Xs) = aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Ys),Zs3) )
     => ( ( ( Ys = nil_nat )
         => ( Zs3 != aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X),Xs) ) )
       => ~ ! [Ys4: list_nat] :
              ( ( Ys = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X),Ys4) )
             => ( aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Ys4),Zs3) != Xs ) ) ) ) ).

% list_Cons_eq_append_cases
tff(fact_1437_list__Cons__eq__append__cases,axiom,
    ! [X: int,Xs: list_int,Ys: list_int,Zs3: list_int] :
      ( ( aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X),Xs) = aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Ys),Zs3) )
     => ( ( ( Ys = nil_int )
         => ( Zs3 != aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X),Xs) ) )
       => ~ ! [Ys4: list_int] :
              ( ( Ys = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X),Ys4) )
             => ( aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Ys4),Zs3) != Xs ) ) ) ) ).

% list_Cons_eq_append_cases
tff(fact_1438_list__Cons__eq__append__cases,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn,Ys: list_P8527749157015355191n_assn,Zs3: list_P8527749157015355191n_assn] :
      ( ( aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X),Xs) = aa_lis2797772165808192636n_assn(aa_lis4738365772350288766n_assn(append282499809098378956n_assn,Ys),Zs3) )
     => ( ( ( Ys = nil_Pr5671120429643327159n_assn )
         => ( Zs3 != aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X),Xs) ) )
       => ~ ! [Ys4: list_P8527749157015355191n_assn] :
              ( ( Ys = aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X),Ys4) )
             => ( aa_lis2797772165808192636n_assn(aa_lis4738365772350288766n_assn(append282499809098378956n_assn,Ys4),Zs3) != Xs ) ) ) ) ).

% list_Cons_eq_append_cases
tff(fact_1439_list__append__eq__Cons__cases,axiom,
    ! [Ys: list_a,Zs3: list_a,X: a,Xs: list_a] :
      ( ( aa_list_a_list_a(aa_lis2442066584718795531list_a(append_a,Ys),Zs3) = aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X),Xs) )
     => ( ( ( Ys = nil_a )
         => ( Zs3 != aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X),Xs) ) )
       => ~ ! [Ys4: list_a] :
              ( ( Ys = aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X),Ys4) )
             => ( aa_list_a_list_a(aa_lis2442066584718795531list_a(append_a,Ys4),Zs3) != Xs ) ) ) ) ).

% list_append_eq_Cons_cases
tff(fact_1440_list__append__eq__Cons__cases,axiom,
    ! [Ys: list_b,Zs3: list_b,X: b,Xs: list_b] :
      ( ( aa_list_b_list_b(aa_lis927650220358621516list_b(append_b,Ys),Zs3) = aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X),Xs) )
     => ( ( ( Ys = nil_b )
         => ( Zs3 != aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X),Xs) ) )
       => ~ ! [Ys4: list_b] :
              ( ( Ys = aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X),Ys4) )
             => ( aa_list_b_list_b(aa_lis927650220358621516list_b(append_b,Ys4),Zs3) != Xs ) ) ) ) ).

% list_append_eq_Cons_cases
tff(fact_1441_list__append__eq__Cons__cases,axiom,
    ! [Ys: list_nat,Zs3: list_nat,X: nat,Xs: list_nat] :
      ( ( aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Ys),Zs3) = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X),Xs) )
     => ( ( ( Ys = nil_nat )
         => ( Zs3 != aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X),Xs) ) )
       => ~ ! [Ys4: list_nat] :
              ( ( Ys = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X),Ys4) )
             => ( aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Ys4),Zs3) != Xs ) ) ) ) ).

% list_append_eq_Cons_cases
tff(fact_1442_list__append__eq__Cons__cases,axiom,
    ! [Ys: list_int,Zs3: list_int,X: int,Xs: list_int] :
      ( ( aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Ys),Zs3) = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X),Xs) )
     => ( ( ( Ys = nil_int )
         => ( Zs3 != aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X),Xs) ) )
       => ~ ! [Ys4: list_int] :
              ( ( Ys = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X),Ys4) )
             => ( aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Ys4),Zs3) != Xs ) ) ) ) ).

% list_append_eq_Cons_cases
tff(fact_1443_list__append__eq__Cons__cases,axiom,
    ! [Ys: list_P8527749157015355191n_assn,Zs3: list_P8527749157015355191n_assn,X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] :
      ( ( aa_lis2797772165808192636n_assn(aa_lis4738365772350288766n_assn(append282499809098378956n_assn,Ys),Zs3) = aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X),Xs) )
     => ( ( ( Ys = nil_Pr5671120429643327159n_assn )
         => ( Zs3 != aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X),Xs) ) )
       => ~ ! [Ys4: list_P8527749157015355191n_assn] :
              ( ( Ys = aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X),Ys4) )
             => ( aa_lis2797772165808192636n_assn(aa_lis4738365772350288766n_assn(append282499809098378956n_assn,Ys4),Zs3) != Xs ) ) ) ) ).

% list_append_eq_Cons_cases
tff(fact_1444_map__consI_I2_J,axiom,
    ! [W: list_assn,La: list_assn,F: fun_Pr3816910165196019820n_assn,Ww: list_P8527749157015355191n_assn,A: produc6575502325842934193n_assn] :
      ( ( append_assn(W,La) = append_assn(aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,F),Ww),La) )
     => ( cons_assn(aa_Pro2979818503705714077n_assn(F,A),append_assn(W,La)) = append_assn(aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,F),aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,A),Ww)),La) ) ) ).

% map_consI(2)
tff(fact_1445_map__consI_I2_J,axiom,
    ! [W: list_nat,La: list_nat,F: fun_nat_nat,Ww: list_nat,A: nat] :
      ( ( aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,W),La) = aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,aa_list_nat_list_nat(aa_fun4736978287202700568st_nat(map_nat_nat,F),Ww)),La) )
     => ( aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,aa_nat_nat(F,A)),aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,W),La)) = aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,aa_list_nat_list_nat(aa_fun4736978287202700568st_nat(map_nat_nat,F),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,A),Ww))),La) ) ) ).

% map_consI(2)
tff(fact_1446_map__consI_I2_J,axiom,
    ! [W: list_nat,La: list_nat,F: fun_int_nat,Ww: list_int,A: int] :
      ( ( aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,W),La) = aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,map_int_nat(F,Ww)),La) )
     => ( aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,aa_int_nat(F,A)),aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,W),La)) = aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,map_int_nat(F,aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,A),Ww))),La) ) ) ).

% map_consI(2)
tff(fact_1447_map__consI_I2_J,axiom,
    ! [W: list_nat,La: list_nat,F: fun_Pr7403558292872094996sn_nat,Ww: list_P8527749157015355191n_assn,A: produc6575502325842934193n_assn] :
      ( ( aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,W),La) = aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,map_Pr7570552894071451325sn_nat(F,Ww)),La) )
     => ( aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,aa_Pro8055486230010607173sn_nat(F,A)),aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,W),La)) = aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,map_Pr7570552894071451325sn_nat(F,aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,A),Ww))),La) ) ) ).

% map_consI(2)
tff(fact_1448_map__consI_I2_J,axiom,
    ! [W: list_int,La: list_int,F: fun_nat_int,Ww: list_nat,A: nat] :
      ( ( aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,W),La) = aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,map_nat_int(F,Ww)),La) )
     => ( aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,aa_nat_int(F,A)),aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,W),La)) = aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,map_nat_int(F,aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,A),Ww))),La) ) ) ).

% map_consI(2)
tff(fact_1449_map__consI_I2_J,axiom,
    ! [W: list_int,La: list_int,F: fun_int_int,Ww: list_int,A: int] :
      ( ( aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,W),La) = aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,map_int_int(F,Ww)),La) )
     => ( aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,aa_int_int(F,A)),aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,W),La)) = aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,map_int_int(F,aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,A),Ww))),La) ) ) ).

% map_consI(2)
tff(fact_1450_map__consI_I2_J,axiom,
    ! [W: list_int,La: list_int,F: fun_Pr3225707273362898288sn_int,Ww: list_P8527749157015355191n_assn,A: produc6575502325842934193n_assn] :
      ( ( aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,W),La) = aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,map_Pr7568062423562401049sn_int(F,Ww)),La) )
     => ( aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,aa_Pro8052995759501556897sn_int(F,A)),aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,W),La)) = aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,map_Pr7568062423562401049sn_int(F,aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,A),Ww))),La) ) ) ).

% map_consI(2)
tff(fact_1451_map__consI_I2_J,axiom,
    ! [W: list_P8527749157015355191n_assn,La: list_P8527749157015355191n_assn,F: fun_na5138202758309126970n_assn,Ww: list_nat,A: nat] :
      ( ( aa_lis2797772165808192636n_assn(aa_lis4738365772350288766n_assn(append282499809098378956n_assn,W),La) = aa_lis2797772165808192636n_assn(aa_lis4738365772350288766n_assn(append282499809098378956n_assn,map_na2667955367175718043n_assn(F,Ww)),La) )
     => ( aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,aa_nat3152888703114873891n_assn(F,A)),aa_lis2797772165808192636n_assn(aa_lis4738365772350288766n_assn(append282499809098378956n_assn,W),La)) = aa_lis2797772165808192636n_assn(aa_lis4738365772350288766n_assn(append282499809098378956n_assn,map_na2667955367175718043n_assn(F,aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,A),Ww))),La) ) ) ).

% map_consI(2)
tff(fact_1452_map__consI_I2_J,axiom,
    ! [W: list_P8527749157015355191n_assn,La: list_P8527749157015355191n_assn,F: fun_in6898239422062238174n_assn,Ww: list_int,A: int] :
      ( ( aa_lis2797772165808192636n_assn(aa_lis4738365772350288766n_assn(append282499809098378956n_assn,W),La) = aa_lis2797772165808192636n_assn(aa_lis4738365772350288766n_assn(append282499809098378956n_assn,map_in4427992030928829247n_assn(F,Ww)),La) )
     => ( aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,aa_int4912925366867985095n_assn(F,A)),aa_lis2797772165808192636n_assn(aa_lis4738365772350288766n_assn(append282499809098378956n_assn,W),La)) = aa_lis2797772165808192636n_assn(aa_lis4738365772350288766n_assn(append282499809098378956n_assn,map_in4427992030928829247n_assn(F,aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,A),Ww))),La) ) ) ).

% map_consI(2)
tff(fact_1453_map__consI_I2_J,axiom,
    ! [W: list_P8527749157015355191n_assn,La: list_P8527749157015355191n_assn,F: fun_Pr5530265736160528387n_assn,Ww: list_P8527749157015355191n_assn,A: produc6575502325842934193n_assn] :
      ( ( aa_lis2797772165808192636n_assn(aa_lis4738365772350288766n_assn(append282499809098378956n_assn,W),La) = aa_lis2797772165808192636n_assn(aa_lis4738365772350288766n_assn(append282499809098378956n_assn,map_Pr7925354932063753860n_assn(F,Ww)),La) )
     => ( aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,aa_Pro2851486235043456252n_assn(F,A)),aa_lis2797772165808192636n_assn(aa_lis4738365772350288766n_assn(append282499809098378956n_assn,W),La)) = aa_lis2797772165808192636n_assn(aa_lis4738365772350288766n_assn(append282499809098378956n_assn,map_Pr7925354932063753860n_assn(F,aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,A),Ww))),La) ) ) ).

% map_consI(2)
tff(fact_1454_SLN__left,axiom,
    ! [Pa: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,sln),Pa) = Pa ).

% SLN_left
tff(fact_1455_SLN__right,axiom,
    ! [Pa: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Pa),sln) = Pa ).

% SLN_right
tff(fact_1456_lexordp_OCons,axiom,
    ! [X: assn,Y: assn,Xs: list_assn,Ys: list_assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),Y)
     => ord_lexordp_assn(cons_assn(X,Xs),cons_assn(Y,Ys)) ) ).

% lexordp.Cons
tff(fact_1457_lexordp_OCons,axiom,
    ! [X: nat,Y: nat,Xs: list_nat,Ys: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
     => aa_list_nat_o(aa_lis557685189327716851_nat_o(ord_lexordp_nat,aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X),Xs)),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Y),Ys)) ) ).

% lexordp.Cons
tff(fact_1458_lexordp_OCons,axiom,
    ! [X: int,Y: int,Xs: list_int,Ys: list_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
     => aa_list_int_o(aa_lis7773489400162621755_int_o(ord_lexordp_int,aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X),Xs)),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Y),Ys)) ) ).

% lexordp.Cons
tff(fact_1459_lexordp_OCons__eq,axiom,
    ! [X: assn,Y: assn,Xs: list_assn,Ys: list_assn] :
      ( ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),Y)
     => ( ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,Y),X)
       => ( ord_lexordp_assn(Xs,Ys)
         => ord_lexordp_assn(cons_assn(X,Xs),cons_assn(Y,Ys)) ) ) ) ).

% lexordp.Cons_eq
tff(fact_1460_lexordp_OCons__eq,axiom,
    ! [X: nat,Y: nat,Xs: list_nat,Ys: list_nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),X)
       => ( aa_list_nat_o(aa_lis557685189327716851_nat_o(ord_lexordp_nat,Xs),Ys)
         => aa_list_nat_o(aa_lis557685189327716851_nat_o(ord_lexordp_nat,aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X),Xs)),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Y),Ys)) ) ) ) ).

% lexordp.Cons_eq
tff(fact_1461_lexordp_OCons__eq,axiom,
    ! [X: int,Y: int,Xs: list_int,Ys: list_int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
     => ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Y),X)
       => ( aa_list_int_o(aa_lis7773489400162621755_int_o(ord_lexordp_int,Xs),Ys)
         => aa_list_int_o(aa_lis7773489400162621755_int_o(ord_lexordp_int,aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X),Xs)),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Y),Ys)) ) ) ) ).

% lexordp.Cons_eq
tff(fact_1462_lexordp_ONil,axiom,
    ! [Y: nat,Ys: list_nat] : aa_list_nat_o(aa_lis557685189327716851_nat_o(ord_lexordp_nat,nil_nat),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Y),Ys)) ).

% lexordp.Nil
tff(fact_1463_lexordp_ONil,axiom,
    ! [Y: int,Ys: list_int] : aa_list_int_o(aa_lis7773489400162621755_int_o(ord_lexordp_int,nil_int),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Y),Ys)) ).

% lexordp.Nil
tff(fact_1464_SLN__def,axiom,
    sln = one_one_assn ).

% SLN_def
tff(fact_1465_merge_Osimps_I3_J,axiom,
    ! [X12: nat,L12: list_nat,X24: nat,L22: list_nat] :
      merge_nat(aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X12),L12),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X24),L22)) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X12),X24),
        aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X12),merge_nat(L12,aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X24),L22))),
        $ite(X12 = X24,aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X12),merge_nat(L12,L22)),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X24),merge_nat(aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X12),L12),L22))) ) ).

% merge.simps(3)
tff(fact_1466_merge_Osimps_I3_J,axiom,
    ! [X12: int,L12: list_int,X24: int,L22: list_int] :
      merge_int(aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X12),L12),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X24),L22)) = $ite(
        aa_int_o(aa_int_fun_int_o(ord_less_int,X12),X24),
        aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X12),merge_int(L12,aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X24),L22))),
        $ite(X12 = X24,aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X12),merge_int(L12,L22)),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X24),merge_int(aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X12),L12),L22))) ) ).

% merge.simps(3)
tff(fact_1467_merge_Osimps_I2_J,axiom,
    ! [V: nat,Va: list_nat] : merge_nat(aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,V),Va),nil_nat) = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,V),Va) ).

% merge.simps(2)
tff(fact_1468_merge_Osimps_I2_J,axiom,
    ! [V: int,Va: list_int] : merge_int(aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,V),Va),nil_int) = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,V),Va) ).

% merge.simps(2)
tff(fact_1469_lexordp_Ocases,axiom,
    ! [A1: list_assn,A22: list_assn] :
      ( ord_lexordp_assn(A1,A22)
     => ( ( ( A1 = nil_assn )
         => ! [Y2: assn,Ys2: list_assn] : A22 != cons_assn(Y2,Ys2) )
       => ( ! [X2: assn] :
              ( ? [Xs2: list_assn] : A1 = cons_assn(X2,Xs2)
             => ! [Y2: assn] :
                  ( ? [Ys2: list_assn] : A22 = cons_assn(Y2,Ys2)
                 => ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X2),Y2) ) )
         => ~ ! [X2: assn,Y2: assn,Xs2: list_assn] :
                ( ( A1 = cons_assn(X2,Xs2) )
               => ! [Ys2: list_assn] :
                    ( ( A22 = cons_assn(Y2,Ys2) )
                   => ( ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X2),Y2)
                     => ( ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,Y2),X2)
                       => ~ ord_lexordp_assn(Xs2,Ys2) ) ) ) ) ) ) ) ).

% lexordp.cases
tff(fact_1470_lexordp_Ocases,axiom,
    ! [A1: list_nat,A22: list_nat] :
      ( aa_list_nat_o(aa_lis557685189327716851_nat_o(ord_lexordp_nat,A1),A22)
     => ( ( ( A1 = nil_nat )
         => ! [Y2: nat,Ys2: list_nat] : A22 != aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Y2),Ys2) )
       => ( ! [X2: nat] :
              ( ? [Xs2: list_nat] : A1 = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X2),Xs2)
             => ! [Y2: nat] :
                  ( ? [Ys2: list_nat] : A22 = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Y2),Ys2)
                 => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Y2) ) )
         => ~ ! [X2: nat,Y2: nat,Xs2: list_nat] :
                ( ( A1 = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X2),Xs2) )
               => ! [Ys2: list_nat] :
                    ( ( A22 = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Y2),Ys2) )
                   => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Y2)
                     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y2),X2)
                       => ~ aa_list_nat_o(aa_lis557685189327716851_nat_o(ord_lexordp_nat,Xs2),Ys2) ) ) ) ) ) ) ) ).

% lexordp.cases
tff(fact_1471_lexordp_Ocases,axiom,
    ! [A1: list_int,A22: list_int] :
      ( aa_list_int_o(aa_lis7773489400162621755_int_o(ord_lexordp_int,A1),A22)
     => ( ( ( A1 = nil_int )
         => ! [Y2: int,Ys2: list_int] : A22 != aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Y2),Ys2) )
       => ( ! [X2: int] :
              ( ? [Xs2: list_int] : A1 = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X2),Xs2)
             => ! [Y2: int] :
                  ( ? [Ys2: list_int] : A22 = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Y2),Ys2)
                 => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Y2) ) )
         => ~ ! [X2: int,Y2: int,Xs2: list_int] :
                ( ( A1 = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X2),Xs2) )
               => ! [Ys2: list_int] :
                    ( ( A22 = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Y2),Ys2) )
                   => ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Y2)
                     => ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Y2),X2)
                       => ~ aa_list_int_o(aa_lis7773489400162621755_int_o(ord_lexordp_int,Xs2),Ys2) ) ) ) ) ) ) ) ).

% lexordp.cases
tff(fact_1472_finite__Int,axiom,
    ! [F2: set_nat,G: set_nat] :
      ( ( aa_set_nat_o(finite_finite_nat,F2)
        | aa_set_nat_o(finite_finite_nat,G) )
     => aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,F2),G)) ) ).

% finite_Int
tff(fact_1473_finite__Int,axiom,
    ! [F2: set_Pr1261947904930325089at_nat,G: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat(F2)
        | finite6177210948735845034at_nat(G) )
     => finite6177210948735845034at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,F2),G)) ) ).

% finite_Int
tff(fact_1474_ex__min__if__finite,axiom,
    ! [S: set_o] :
      ( aa_set_o_o(finite_finite_o,S)
     => ( ( S != bot_bot_set_o )
       => ? [X2: $o] :
            ( aa_set_o_o(member_o((X2)),S)
            & ~ ? [Xa3: $o] :
                  ( aa_set_o_o(member_o((Xa3)),S)
                  & aa_o_o(aa_o_fun_o_o(ord_less_o,(Xa3)),(X2)) ) ) ) ) ).

% ex_min_if_finite
tff(fact_1475_ex__min__if__finite,axiom,
    ! [S: set_assn] :
      ( finite_finite_assn(S)
     => ( ( S != bot_bot_set_assn )
       => ? [X2: assn] :
            ( member_assn(X2,S)
            & ~ ? [Xa3: assn] :
                  ( member_assn(Xa3,S)
                  & aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,Xa3),X2) ) ) ) ) ).

% ex_min_if_finite
tff(fact_1476_ex__min__if__finite,axiom,
    ! [S: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( ( S != bot_bot_set_nat )
       => ? [X2: nat] :
            ( aa_set_nat_o(member_nat2(X2),S)
            & ~ ? [Xa3: nat] :
                  ( aa_set_nat_o(member_nat2(Xa3),S)
                  & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa3),X2) ) ) ) ) ).

% ex_min_if_finite
tff(fact_1477_ex__min__if__finite,axiom,
    ! [S: set_int] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( ( S != bot_bot_set_int )
       => ? [X2: int] :
            ( aa_set_int_o(member_int2(X2),S)
            & ~ ? [Xa3: int] :
                  ( aa_set_int_o(member_int2(Xa3),S)
                  & aa_int_o(aa_int_fun_int_o(ord_less_int,Xa3),X2) ) ) ) ) ).

% ex_min_if_finite
tff(fact_1478_infinite__growing,axiom,
    ! [X5: set_o] :
      ( ( X5 != bot_bot_set_o )
     => ( ! [X2: $o] :
            ( aa_set_o_o(member_o((X2)),X5)
           => ? [Xa3: $o] :
                ( aa_set_o_o(member_o((Xa3)),X5)
                & aa_o_o(aa_o_fun_o_o(ord_less_o,(X2)),(Xa3)) ) )
       => ~ aa_set_o_o(finite_finite_o,X5) ) ) ).

% infinite_growing
tff(fact_1479_infinite__growing,axiom,
    ! [X5: set_nat] :
      ( ( X5 != bot_bot_set_nat )
     => ( ! [X2: nat] :
            ( aa_set_nat_o(member_nat2(X2),X5)
           => ? [Xa3: nat] :
                ( aa_set_nat_o(member_nat2(Xa3),X5)
                & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Xa3) ) )
       => ~ aa_set_nat_o(finite_finite_nat,X5) ) ) ).

% infinite_growing
tff(fact_1480_infinite__growing,axiom,
    ! [X5: set_int] :
      ( ( X5 != bot_bot_set_int )
     => ( ! [X2: int] :
            ( aa_set_int_o(member_int2(X2),X5)
           => ? [Xa3: int] :
                ( aa_set_int_o(member_int2(Xa3),X5)
                & aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Xa3) ) )
       => ~ aa_set_int_o(finite_finite_int,X5) ) ) ).

% infinite_growing
tff(fact_1481_FI__init,axiom,
    ! [Pa: assn,Q: assn,F2: assn] :
      ( aa_assn_o(fi(nil_Pr5671120429643327159n_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,sln),Pa),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,sln),Q),sln,sln),F2)
     => aa_assn_o(fI_QUERY(Pa,Q),F2) ) ).

% FI_init
tff(fact_1482_finite_OemptyI,axiom,
    aa_set_o_o(finite_finite_o,bot_bot_set_o) ).

% finite.emptyI
tff(fact_1483_finite_OemptyI,axiom,
    aa_set_nat_o(finite_finite_nat,bot_bot_set_nat) ).

% finite.emptyI
tff(fact_1484_infinite__imp__nonempty,axiom,
    ! [S: set_o] :
      ( ~ aa_set_o_o(finite_finite_o,S)
     => ( S != bot_bot_set_o ) ) ).

% infinite_imp_nonempty
tff(fact_1485_infinite__imp__nonempty,axiom,
    ! [S: set_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,S)
     => ( S != bot_bot_set_nat ) ) ).

% infinite_imp_nonempty
tff(fact_1486_finite__transitivity__chain,axiom,
    ! [A2: set_fu3555266933746557007_nat_o,R: fun_fu4164567611050918574at_o_o] :
      ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,A2)
     => ( ! [X2: fun_Pr6460944699146269935_nat_o] : ~ aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(R,X2),X2)
       => ( ! [X2: fun_Pr6460944699146269935_nat_o,Y2: fun_Pr6460944699146269935_nat_o,Z3: fun_Pr6460944699146269935_nat_o] :
              ( aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(R,X2),Y2)
             => ( aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(R,Y2),Z3)
               => aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(R,X2),Z3) ) )
         => ( ! [X2: fun_Pr6460944699146269935_nat_o] :
                ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X2),A2)
               => ? [Y4: fun_Pr6460944699146269935_nat_o] :
                    ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Y4),A2)
                    & aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(R,X2),Y4) ) )
           => ( A2 = bot_bo8632255151191677115_nat_o ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_1487_finite__transitivity__chain,axiom,
    ! [A2: set_o,R: fun_o_fun_o_o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ! [X2: $o] : ~ aa_o_o(aa_o_fun_o_o(R,(X2)),(X2))
       => ( ! [X2: $o,Y2: $o,Z3: $o] :
              ( aa_o_o(aa_o_fun_o_o(R,(X2)),(Y2))
             => ( aa_o_o(aa_o_fun_o_o(R,(Y2)),(Z3))
               => aa_o_o(aa_o_fun_o_o(R,(X2)),(Z3)) ) )
         => ( ! [X2: $o] :
                ( aa_set_o_o(member_o((X2)),A2)
               => ? [Y4: $o] :
                    ( aa_set_o_o(member_o((Y4)),A2)
                    & aa_o_o(aa_o_fun_o_o(R,(X2)),(Y4)) ) )
           => ( A2 = bot_bot_set_o ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_1488_finite__transitivity__chain,axiom,
    ! [A2: set_nat,R: fun_nat_fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ! [X2: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(R,X2),X2)
       => ( ! [X2: nat,Y2: nat,Z3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(R,X2),Y2)
             => ( aa_nat_o(aa_nat_fun_nat_o(R,Y2),Z3)
               => aa_nat_o(aa_nat_fun_nat_o(R,X2),Z3) ) )
         => ( ! [X2: nat] :
                ( aa_set_nat_o(member_nat2(X2),A2)
               => ? [Y4: nat] :
                    ( aa_set_nat_o(member_nat2(Y4),A2)
                    & aa_nat_o(aa_nat_fun_nat_o(R,X2),Y4) ) )
           => ( A2 = bot_bot_set_nat ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_1489_merge__list_Oelims,axiom,
    ! [X: list_list_nat,Xa: list_list_nat,Y: list_nat] :
      ( ( merge_list_nat(X,Xa) = Y )
     => ( ( ( X = nil_list_nat )
         => ( ( Xa = nil_list_nat )
           => ( Y != nil_nat ) ) )
       => ( ( ( X = nil_list_nat )
           => ! [L: list_nat] :
                ( ( Xa = cons_list_nat(L,nil_list_nat) )
               => ( Y != L ) ) )
         => ( ! [La2: list_nat,Acc2: list_list_nat] :
                ( ( X = cons_list_nat(La2,Acc2) )
               => ( ( Xa = nil_list_nat )
                 => ( Y != merge_list_nat(nil_list_nat,cons_list_nat(La2,Acc2)) ) ) )
           => ( ! [La2: list_nat,Acc2: list_list_nat] :
                  ( ( X = cons_list_nat(La2,Acc2) )
                 => ! [L: list_nat] :
                      ( ( Xa = cons_list_nat(L,nil_list_nat) )
                     => ( Y != merge_list_nat(nil_list_nat,cons_list_nat(L,cons_list_nat(La2,Acc2))) ) ) )
             => ~ ! [L1: list_nat,L2: list_nat,Ls2: list_list_nat] :
                    ( ( Xa = cons_list_nat(L1,cons_list_nat(L2,Ls2)) )
                   => ( Y != merge_list_nat(cons_list_nat(merge_nat(L1,L2),X),Ls2) ) ) ) ) ) ) ) ).

% merge_list.elims
tff(fact_1490_merge__list_Oelims,axiom,
    ! [X: list_list_int,Xa: list_list_int,Y: list_int] :
      ( ( merge_list_int(X,Xa) = Y )
     => ( ( ( X = nil_list_int )
         => ( ( Xa = nil_list_int )
           => ( Y != nil_int ) ) )
       => ( ( ( X = nil_list_int )
           => ! [L: list_int] :
                ( ( Xa = cons_list_int(L,nil_list_int) )
               => ( Y != L ) ) )
         => ( ! [La2: list_int,Acc2: list_list_int] :
                ( ( X = cons_list_int(La2,Acc2) )
               => ( ( Xa = nil_list_int )
                 => ( Y != merge_list_int(nil_list_int,cons_list_int(La2,Acc2)) ) ) )
           => ( ! [La2: list_int,Acc2: list_list_int] :
                  ( ( X = cons_list_int(La2,Acc2) )
                 => ! [L: list_int] :
                      ( ( Xa = cons_list_int(L,nil_list_int) )
                     => ( Y != merge_list_int(nil_list_int,cons_list_int(L,cons_list_int(La2,Acc2))) ) ) )
             => ~ ! [L1: list_int,L2: list_int,Ls2: list_list_int] :
                    ( ( Xa = cons_list_int(L1,cons_list_int(L2,Ls2)) )
                   => ( Y != merge_list_int(cons_list_int(merge_int(L1,L2),X),Ls2) ) ) ) ) ) ) ) ).

% merge_list.elims
tff(fact_1491_maps__simps_I1_J,axiom,
    ! [F: fun_nat_list_nat,X: nat,Xs: list_nat] : maps_nat_nat(F,aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X),Xs)) = aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,aa_nat_list_nat(F,X)),maps_nat_nat(F,Xs)) ).

% maps_simps(1)
tff(fact_1492_maps__simps_I1_J,axiom,
    ! [F: fun_nat_list_int,X: nat,Xs: list_nat] : maps_nat_int(F,aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X),Xs)) = aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,aa_nat_list_int(F,X)),maps_nat_int(F,Xs)) ).

% maps_simps(1)
tff(fact_1493_maps__simps_I1_J,axiom,
    ! [F: fun_int_list_nat,X: int,Xs: list_int] : maps_int_nat(F,aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X),Xs)) = aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,aa_int_list_nat(F,X)),maps_int_nat(F,Xs)) ).

% maps_simps(1)
tff(fact_1494_maps__simps_I1_J,axiom,
    ! [F: fun_int_list_int,X: int,Xs: list_int] : maps_int_int(F,aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X),Xs)) = aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,aa_int_list_int(F,X)),maps_int_int(F,Xs)) ).

% maps_simps(1)
tff(fact_1495_maps__simps_I1_J,axiom,
    ! [F: fun_Pr7454088522235451300st_nat,X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] : maps_P5986043149438648064sn_nat(F,aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X),Xs)) = aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,aa_Pro579495954165322453st_nat(F,X)),maps_P5986043149438648064sn_nat(F,Xs)) ).

% maps_simps(1)
tff(fact_1496_maps__simps_I1_J,axiom,
    ! [F: fun_Pr4964005377303724544st_int,X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] : maps_P5983552678929597788sn_int(F,aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X),Xs)) = aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,aa_Pro5625016971510901553st_int(F,X)),maps_P5983552678929597788sn_int(F,Xs)) ).

% maps_simps(1)
tff(fact_1497_rotate1_Osimps_I2_J,axiom,
    ! [X: a,Xs: list_a] : rotate1_a(aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X),Xs)) = aa_list_a_list_a(aa_lis2442066584718795531list_a(append_a,Xs),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X),nil_a)) ).

% rotate1.simps(2)
tff(fact_1498_rotate1_Osimps_I2_J,axiom,
    ! [X: b,Xs: list_b] : rotate1_b(aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X),Xs)) = aa_list_b_list_b(aa_lis927650220358621516list_b(append_b,Xs),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X),nil_b)) ).

% rotate1.simps(2)
tff(fact_1499_rotate1_Osimps_I2_J,axiom,
    ! [X: nat,Xs: list_nat] : rotate1_nat(aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X),Xs)) = aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Xs),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X),nil_nat)) ).

% rotate1.simps(2)
tff(fact_1500_rotate1_Osimps_I2_J,axiom,
    ! [X: int,Xs: list_int] : rotate1_int(aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X),Xs)) = aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Xs),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X),nil_int)) ).

% rotate1.simps(2)
tff(fact_1501_rotate1_Osimps_I2_J,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] : rotate328796349445179396n_assn(aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X),Xs)) = aa_lis2797772165808192636n_assn(aa_lis4738365772350288766n_assn(append282499809098378956n_assn,Xs),aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X),nil_Pr5671120429643327159n_assn)) ).

% rotate1.simps(2)
tff(fact_1502_concat__eq__append__conv,axiom,
    ! [Xss2: list_list_a,Ys: list_a,Zs3: list_a] :
      ( ( concat_a(Xss2) = aa_list_a_list_a(aa_lis2442066584718795531list_a(append_a,Ys),Zs3) )
    <=> $ite(
          Xss2 = nil_list_a,
          ( ( Ys = nil_a )
          & ( Zs3 = nil_a ) ),
          ? [Xss1: list_list_a,Xs3: list_a,Xs5: list_a,Xss22: list_list_a] :
            ( ( Xss2 = append_list_a(Xss1,cons_list_a(aa_list_a_list_a(aa_lis2442066584718795531list_a(append_a,Xs3),Xs5),Xss22)) )
            & ( Ys = aa_list_a_list_a(aa_lis2442066584718795531list_a(append_a,concat_a(Xss1)),Xs3) )
            & ( Zs3 = aa_list_a_list_a(aa_lis2442066584718795531list_a(append_a,Xs5),concat_a(Xss22)) ) ) ) ) ).

% concat_eq_append_conv
tff(fact_1503_concat__eq__append__conv,axiom,
    ! [Xss2: list_list_b,Ys: list_b,Zs3: list_b] :
      ( ( concat_b(Xss2) = aa_list_b_list_b(aa_lis927650220358621516list_b(append_b,Ys),Zs3) )
    <=> $ite(
          Xss2 = nil_list_b,
          ( ( Ys = nil_b )
          & ( Zs3 = nil_b ) ),
          ? [Xss1: list_list_b,Xs3: list_b,Xs5: list_b,Xss22: list_list_b] :
            ( ( Xss2 = append_list_b(Xss1,cons_list_b(aa_list_b_list_b(aa_lis927650220358621516list_b(append_b,Xs3),Xs5),Xss22)) )
            & ( Ys = aa_list_b_list_b(aa_lis927650220358621516list_b(append_b,concat_b(Xss1)),Xs3) )
            & ( Zs3 = aa_list_b_list_b(aa_lis927650220358621516list_b(append_b,Xs5),concat_b(Xss22)) ) ) ) ) ).

% concat_eq_append_conv
tff(fact_1504_concat__eq__append__conv,axiom,
    ! [Xss2: list_l6351802567095793725n_assn,Ys: list_P8527749157015355191n_assn,Zs3: list_P8527749157015355191n_assn] :
      ( ( concat6144485081897559666n_assn(Xss2) = aa_lis2797772165808192636n_assn(aa_lis4738365772350288766n_assn(append282499809098378956n_assn,Ys),Zs3) )
    <=> $ite(
          Xss2 = nil_li5476096274760905021n_assn,
          ( ( Ys = nil_Pr5671120429643327159n_assn )
          & ( Zs3 = nil_Pr5671120429643327159n_assn ) ),
          ? [Xss1: list_l6351802567095793725n_assn,Xs3: list_P8527749157015355191n_assn,Xs5: list_P8527749157015355191n_assn,Xss22: list_l6351802567095793725n_assn] :
            ( ( Xss2 = append2733831349823859410n_assn(Xss1,cons_l2423627976422276333n_assn(aa_lis2797772165808192636n_assn(aa_lis4738365772350288766n_assn(append282499809098378956n_assn,Xs3),Xs5),Xss22)) )
            & ( Ys = aa_lis2797772165808192636n_assn(aa_lis4738365772350288766n_assn(append282499809098378956n_assn,concat6144485081897559666n_assn(Xss1)),Xs3) )
            & ( Zs3 = aa_lis2797772165808192636n_assn(aa_lis4738365772350288766n_assn(append282499809098378956n_assn,Xs5),concat6144485081897559666n_assn(Xss22)) ) ) ) ) ).

% concat_eq_append_conv
tff(fact_1505_concat__eq__append__conv,axiom,
    ! [Xss2: list_list_nat,Ys: list_nat,Zs3: list_nat] :
      ( ( concat_nat(Xss2) = aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Ys),Zs3) )
    <=> $ite(
          Xss2 = nil_list_nat,
          ( ( Ys = nil_nat )
          & ( Zs3 = nil_nat ) ),
          ? [Xss1: list_list_nat,Xs3: list_nat,Xs5: list_nat,Xss22: list_list_nat] :
            ( ( Xss2 = append_list_nat(Xss1,cons_list_nat(aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Xs3),Xs5),Xss22)) )
            & ( Ys = aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,concat_nat(Xss1)),Xs3) )
            & ( Zs3 = aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Xs5),concat_nat(Xss22)) ) ) ) ) ).

% concat_eq_append_conv
tff(fact_1506_concat__eq__append__conv,axiom,
    ! [Xss2: list_list_int,Ys: list_int,Zs3: list_int] :
      ( ( concat_int(Xss2) = aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Ys),Zs3) )
    <=> $ite(
          Xss2 = nil_list_int,
          ( ( Ys = nil_int )
          & ( Zs3 = nil_int ) ),
          ? [Xss1: list_list_int,Xs3: list_int,Xs5: list_int,Xss22: list_list_int] :
            ( ( Xss2 = append_list_int(Xss1,cons_list_int(aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Xs3),Xs5),Xss22)) )
            & ( Ys = aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,concat_int(Xss1)),Xs3) )
            & ( Zs3 = aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Xs5),concat_int(Xss22)) ) ) ) ) ).

% concat_eq_append_conv
tff(fact_1507_rotate1__is__Nil__conv,axiom,
    ! [Xs: list_a] :
      ( ( rotate1_a(Xs) = nil_a )
    <=> ( Xs = nil_a ) ) ).

% rotate1_is_Nil_conv
tff(fact_1508_rotate1__is__Nil__conv,axiom,
    ! [Xs: list_b] :
      ( ( rotate1_b(Xs) = nil_b )
    <=> ( Xs = nil_b ) ) ).

% rotate1_is_Nil_conv
tff(fact_1509_rotate1__is__Nil__conv,axiom,
    ! [Xs: list_P8527749157015355191n_assn] :
      ( ( rotate328796349445179396n_assn(Xs) = nil_Pr5671120429643327159n_assn )
    <=> ( Xs = nil_Pr5671120429643327159n_assn ) ) ).

% rotate1_is_Nil_conv
tff(fact_1510_rotate1__is__Nil__conv,axiom,
    ! [Xs: list_nat] :
      ( ( rotate1_nat(Xs) = nil_nat )
    <=> ( Xs = nil_nat ) ) ).

% rotate1_is_Nil_conv
tff(fact_1511_rotate1__is__Nil__conv,axiom,
    ! [Xs: list_int] :
      ( ( rotate1_int(Xs) = nil_int )
    <=> ( Xs = nil_int ) ) ).

% rotate1_is_Nil_conv
tff(fact_1512_concat__append,axiom,
    ! [Xs: list_list_nat,Ys: list_list_nat] : concat_nat(append_list_nat(Xs,Ys)) = aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,concat_nat(Xs)),concat_nat(Ys)) ).

% concat_append
tff(fact_1513_concat__append,axiom,
    ! [Xs: list_list_int,Ys: list_list_int] : concat_int(append_list_int(Xs,Ys)) = aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,concat_int(Xs)),concat_int(Ys)) ).

% concat_append
tff(fact_1514_assn__aci_I8_J,axiom,
    ! [X: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y)) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y) ).

% assn_aci(8)
tff(fact_1515_assn__aci_I7_J,axiom,
    ! [X: assn,Y: assn,Z: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),Z)) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Z)) ).

% assn_aci(7)
tff(fact_1516_assn__aci_I5_J,axiom,
    ! [X: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),X) ).

% assn_aci(5)
tff(fact_1517_norm__assertion__simps_I32_J,axiom,
    ! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),X) = X ).

% norm_assertion_simps(32)
tff(fact_1518_norm__assertion__simps_I15_J,axiom,
    ! [X: assn,Y: assn,Z: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y)),Z) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),Z)) ).

% norm_assertion_simps(15)
tff(fact_1519_map__concat,axiom,
    ! [F: fun_nat_nat,Xs: list_list_nat] : aa_list_nat_list_nat(aa_fun4736978287202700568st_nat(map_nat_nat,F),concat_nat(Xs)) = concat_nat(map_li7225945977422193158st_nat(aa_fun4736978287202700568st_nat(map_nat_nat,F),Xs)) ).

% map_concat
tff(fact_1520_map__concat,axiom,
    ! [F: fun_Pr3816910165196019820n_assn,Xs: list_l6351802567095793725n_assn] : aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,F),concat6144485081897559666n_assn(Xs)) = concat_assn(map_li562537959812877739t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,F),Xs)) ).

% map_concat
tff(fact_1521_List_Obind__def,axiom,
    ! [Xs: list_a,F: fun_a_681019998106348156n_assn] : bind_a3542047475819770682n_assn(Xs,F) = concat6144485081897559666n_assn(map_a_7516433405034028541n_assn(F,Xs)) ).

% List.bind_def
tff(fact_1522_List_Obind__def,axiom,
    ! [Xs: list_a,F: fun_a_list_nat] : bind_a_nat(Xs,F) = concat_nat(map_a_list_nat(F,Xs)) ).

% List.bind_def
tff(fact_1523_List_Obind__def,axiom,
    ! [Xs: list_a,F: fun_a_list_int] : bind_a_int(Xs,F) = concat_int(map_a_list_int(F,Xs)) ).

% List.bind_def
tff(fact_1524_List_Obind__def,axiom,
    ! [Xs: list_a,F: fun_a_list_b] : bind_a_b(Xs,F) = concat_b(map_a_list_b(F,Xs)) ).

% List.bind_def
tff(fact_1525_List_Obind__def,axiom,
    ! [Xs: list_a,F: fun_a_list_a] : bind_a_a(Xs,F) = concat_a(map_a_list_a(F,Xs)) ).

% List.bind_def
tff(fact_1526_rotate1_Osimps_I1_J,axiom,
    rotate1_a(nil_a) = nil_a ).

% rotate1.simps(1)
tff(fact_1527_rotate1_Osimps_I1_J,axiom,
    rotate1_b(nil_b) = nil_b ).

% rotate1.simps(1)
tff(fact_1528_rotate1_Osimps_I1_J,axiom,
    rotate328796349445179396n_assn(nil_Pr5671120429643327159n_assn) = nil_Pr5671120429643327159n_assn ).

% rotate1.simps(1)
tff(fact_1529_rotate1_Osimps_I1_J,axiom,
    rotate1_nat(nil_nat) = nil_nat ).

% rotate1.simps(1)
tff(fact_1530_rotate1_Osimps_I1_J,axiom,
    rotate1_int(nil_int) = nil_int ).

% rotate1.simps(1)
tff(fact_1531_rotate1__map,axiom,
    ! [F: fun_nat_nat,Xs: list_nat] : rotate1_nat(aa_list_nat_list_nat(aa_fun4736978287202700568st_nat(map_nat_nat,F),Xs)) = aa_list_nat_list_nat(aa_fun4736978287202700568st_nat(map_nat_nat,F),rotate1_nat(Xs)) ).

% rotate1_map
tff(fact_1532_rotate1__map,axiom,
    ! [F: fun_Pr3816910165196019820n_assn,Xs: list_P8527749157015355191n_assn] : rotate1_assn(aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,F),Xs)) = aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,F),rotate328796349445179396n_assn(Xs)) ).

% rotate1_map
tff(fact_1533_concat_Osimps_I1_J,axiom,
    concat_a(nil_list_a) = nil_a ).

% concat.simps(1)
tff(fact_1534_concat_Osimps_I1_J,axiom,
    concat_b(nil_list_b) = nil_b ).

% concat.simps(1)
tff(fact_1535_concat_Osimps_I1_J,axiom,
    concat6144485081897559666n_assn(nil_li5476096274760905021n_assn) = nil_Pr5671120429643327159n_assn ).

% concat.simps(1)
tff(fact_1536_concat_Osimps_I1_J,axiom,
    concat_nat(nil_list_nat) = nil_nat ).

% concat.simps(1)
tff(fact_1537_concat_Osimps_I1_J,axiom,
    concat_int(nil_list_int) = nil_int ).

% concat.simps(1)
tff(fact_1538_concat_Osimps_I2_J,axiom,
    ! [X: list_nat,Xs: list_list_nat] : concat_nat(cons_list_nat(X,Xs)) = aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,X),concat_nat(Xs)) ).

% concat.simps(2)
tff(fact_1539_concat_Osimps_I2_J,axiom,
    ! [X: list_int,Xs: list_list_int] : concat_int(cons_list_int(X,Xs)) = aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,X),concat_int(Xs)) ).

% concat.simps(2)
tff(fact_1540_maps__simps_I2_J,axiom,
    ! [F: fun_a_list_a] : maps_a_a(F,nil_a) = nil_a ).

% maps_simps(2)
tff(fact_1541_maps__simps_I2_J,axiom,
    ! [F: fun_a_list_b] : maps_a_b(F,nil_a) = nil_b ).

% maps_simps(2)
tff(fact_1542_maps__simps_I2_J,axiom,
    ! [F: fun_a_list_nat] : maps_a_nat(F,nil_a) = nil_nat ).

% maps_simps(2)
tff(fact_1543_maps__simps_I2_J,axiom,
    ! [F: fun_a_list_int] : maps_a_int(F,nil_a) = nil_int ).

% maps_simps(2)
tff(fact_1544_maps__simps_I2_J,axiom,
    ! [F: fun_b_list_a] : maps_b_a(F,nil_b) = nil_a ).

% maps_simps(2)
tff(fact_1545_maps__simps_I2_J,axiom,
    ! [F: fun_b_list_b] : maps_b_b(F,nil_b) = nil_b ).

% maps_simps(2)
tff(fact_1546_maps__simps_I2_J,axiom,
    ! [F: fun_b_list_nat] : maps_b_nat(F,nil_b) = nil_nat ).

% maps_simps(2)
tff(fact_1547_maps__simps_I2_J,axiom,
    ! [F: fun_b_list_int] : maps_b_int(F,nil_b) = nil_int ).

% maps_simps(2)
tff(fact_1548_maps__simps_I2_J,axiom,
    ! [F: fun_nat_list_a] : maps_nat_a(F,nil_nat) = nil_a ).

% maps_simps(2)
tff(fact_1549_maps__simps_I2_J,axiom,
    ! [F: fun_nat_list_b] : maps_nat_b(F,nil_nat) = nil_b ).

% maps_simps(2)
tff(fact_1550_merge__list_Osimps_I1_J,axiom,
    merge_list_nat(nil_list_nat,nil_list_nat) = nil_nat ).

% merge_list.simps(1)
tff(fact_1551_merge__list_Osimps_I1_J,axiom,
    merge_list_int(nil_list_int,nil_list_int) = nil_int ).

% merge_list.simps(1)
tff(fact_1552_concat__eq__appendD,axiom,
    ! [Xss2: list_list_nat,Ys: list_nat,Zs3: list_nat] :
      ( ( concat_nat(Xss2) = aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Ys),Zs3) )
     => ( ( Xss2 != nil_list_nat )
       => ? [Xss12: list_list_nat,Xs2: list_nat,Xs4: list_nat,Xss23: list_list_nat] :
            ( ( Xss2 = append_list_nat(Xss12,cons_list_nat(aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Xs2),Xs4),Xss23)) )
            & ( Ys = aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,concat_nat(Xss12)),Xs2) )
            & ( Zs3 = aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Xs4),concat_nat(Xss23)) ) ) ) ) ).

% concat_eq_appendD
tff(fact_1553_concat__eq__appendD,axiom,
    ! [Xss2: list_list_int,Ys: list_int,Zs3: list_int] :
      ( ( concat_int(Xss2) = aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Ys),Zs3) )
     => ( ( Xss2 != nil_list_int )
       => ? [Xss12: list_list_int,Xs2: list_int,Xs4: list_int,Xss23: list_list_int] :
            ( ( Xss2 = append_list_int(Xss12,cons_list_int(aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Xs2),Xs4),Xss23)) )
            & ( Ys = aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,concat_int(Xss12)),Xs2) )
            & ( Zs3 = aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Xs4),concat_int(Xss23)) ) ) ) ) ).

% concat_eq_appendD
tff(fact_1554_arg__min__if__finite_I2_J,axiom,
    ! [S: set_o,F: fun_o_assn] :
      ( aa_set_o_o(finite_finite_o,S)
     => ( ( S != bot_bot_set_o )
       => ~ ? [X4: $o] :
              ( aa_set_o_o(member_o((X4)),S)
              & aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_o_assn(F,(X4))),aa_o_assn(F,aa_set_o_o(lattic7708394068118247271o_assn(F),S))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1555_arg__min__if__finite_I2_J,axiom,
    ! [S: set_nat,F: fun_nat_assn] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( ( S != bot_bot_set_nat )
       => ~ ? [X4: nat] :
              ( aa_set_nat_o(member_nat2(X4),S)
              & aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_nat_assn(F,X4)),aa_nat_assn(F,lattic2187264146484958483t_assn(F,S))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1556_arg__min__if__finite_I2_J,axiom,
    ! [S: set_o,F: fun_o_nat] :
      ( aa_set_o_o(finite_finite_o,S)
     => ( ( S != bot_bot_set_o )
       => ~ ? [X4: $o] :
              ( aa_set_o_o(member_o((X4)),S)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_o_nat(F,(X4))),aa_o_nat(F,aa_set_o_o(lattic2775856028456453135_o_nat(F),S))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1557_arg__min__if__finite_I2_J,axiom,
    ! [S: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( ( S != bot_bot_set_nat )
       => ~ ? [X4: nat] :
              ( aa_set_nat_o(member_nat2(X4),S)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,X4)),aa_nat_nat(F,lattic7446932960582359483at_nat(F,S))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1558_arg__min__if__finite_I2_J,axiom,
    ! [S: set_o,F: fun_o_int] :
      ( aa_set_o_o(finite_finite_o,S)
     => ( ( S != bot_bot_set_o )
       => ~ ? [X4: $o] :
              ( aa_set_o_o(member_o((X4)),S)
              & aa_int_o(aa_int_fun_int_o(ord_less_int,aa_o_int(F,(X4))),aa_o_int(F,aa_set_o_o(lattic2773365557947402859_o_int(F),S))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1559_arg__min__if__finite_I2_J,axiom,
    ! [S: set_nat,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( ( S != bot_bot_set_nat )
       => ~ ? [X4: nat] :
              ( aa_set_nat_o(member_nat2(X4),S)
              & aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,X4)),aa_nat_int(F,lattic7444442490073309207at_int(F,S))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1560_Sup__fin_Ounion,axiom,
    ! [A2: set_assn,B2: set_assn] :
      ( finite_finite_assn(A2)
     => ( ( A2 != bot_bot_set_assn )
       => ( finite_finite_assn(B2)
         => ( ( B2 != bot_bot_set_assn )
           => ( lattic2150320897289308081n_assn(sup_sup_set_assn(A2,B2)) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,lattic2150320897289308081n_assn(A2)),lattic2150320897289308081n_assn(B2)) ) ) ) ) ) ).

% Sup_fin.union
tff(fact_1561_Sup__fin_Ounion,axiom,
    ! [A2: set_set_nat,B2: set_set_nat] :
      ( finite1152437895449049373et_nat(A2)
     => ( ( A2 != bot_bot_set_set_nat )
       => ( finite1152437895449049373et_nat(B2)
         => ( ( B2 != bot_bot_set_set_nat )
           => ( lattic3835124923745554447et_nat(sup_sup_set_set_nat(A2,B2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,lattic3835124923745554447et_nat(A2)),lattic3835124923745554447et_nat(B2)) ) ) ) ) ) ).

% Sup_fin.union
tff(fact_1562_Sup__fin_Ounion,axiom,
    ! [A2: set_o,B2: set_o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ( A2 != bot_bot_set_o )
       => ( aa_set_o_o(finite_finite_o,B2)
         => ( ( B2 != bot_bot_set_o )
           => ( aa_set_o_o(lattic1508158080041050831_fin_o,sup_sup_set_o(A2,B2))
            <=> aa_o_o(aa_o_fun_o_o(sup_sup_o,aa_set_o_o(lattic1508158080041050831_fin_o,A2)),aa_set_o_o(lattic1508158080041050831_fin_o,B2)) ) ) ) ) ) ).

% Sup_fin.union
tff(fact_1563_Sup__fin_Ounion,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( A2 != bot_bot_set_nat )
       => ( aa_set_nat_o(finite_finite_nat,B2)
         => ( ( B2 != bot_bot_set_nat )
           => ( lattic1093996805478795353in_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,lattic1093996805478795353in_nat(A2)),lattic1093996805478795353in_nat(B2)) ) ) ) ) ) ).

% Sup_fin.union
tff(fact_1564_Inf__fin_Ounion,axiom,
    ! [A2: set_assn,B2: set_assn] :
      ( finite_finite_assn(A2)
     => ( ( A2 != bot_bot_set_assn )
       => ( finite_finite_assn(B2)
         => ( ( B2 != bot_bot_set_assn )
           => ( aa_set_assn_assn(lattic47131356835913163n_assn,sup_sup_set_assn(A2,B2)) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_set_assn_assn(lattic47131356835913163n_assn,A2)),aa_set_assn_assn(lattic47131356835913163n_assn,B2)) ) ) ) ) ) ).

% Inf_fin.union
tff(fact_1565_Inf__fin_Ounion,axiom,
    ! [A2: set_set_nat,B2: set_set_nat] :
      ( finite1152437895449049373et_nat(A2)
     => ( ( A2 != bot_bot_set_set_nat )
       => ( finite1152437895449049373et_nat(B2)
         => ( ( B2 != bot_bot_set_set_nat )
           => ( aa_set6364576160234933248et_nat(lattic3014633134055518761et_nat,sup_sup_set_set_nat(A2,B2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set6364576160234933248et_nat(lattic3014633134055518761et_nat,A2)),aa_set6364576160234933248et_nat(lattic3014633134055518761et_nat,B2)) ) ) ) ) ) ).

% Inf_fin.union
tff(fact_1566_Inf__fin_Ounion,axiom,
    ! [A2: set_Product_unit,B2: set_Product_unit] :
      ( finite4290736615968046902t_unit(A2)
     => ( ( A2 != bot_bo3957492148770167129t_unit )
       => ( finite4290736615968046902t_unit(B2)
         => ( ( B2 != bot_bo3957492148770167129t_unit )
           => ( aa_set6350468789499455260t_unit(lattic1263872656861969706t_unit,sup_su793286257634532545t_unit(A2,B2)) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,aa_set6350468789499455260t_unit(lattic1263872656861969706t_unit,A2)),aa_set6350468789499455260t_unit(lattic1263872656861969706t_unit,B2)) ) ) ) ) ) ).

% Inf_fin.union
tff(fact_1567_Inf__fin_Ounion,axiom,
    ! [A2: set_se7855581050983116737at_nat,B2: set_se7855581050983116737at_nat] :
      ( finite9047747110432174090at_nat(A2)
     => ( ( A2 != bot_bo3083307316010499117at_nat )
       => ( finite9047747110432174090at_nat(B2)
         => ( ( B2 != bot_bo3083307316010499117at_nat )
           => ( aa_set3914448398961233820at_nat(lattic30941717366863870at_nat,sup_su3642409539654194069at_nat(A2,B2)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set3914448398961233820at_nat(lattic30941717366863870at_nat,A2)),aa_set3914448398961233820at_nat(lattic30941717366863870at_nat,B2)) ) ) ) ) ) ).

% Inf_fin.union
tff(fact_1568_Inf__fin_Ounion,axiom,
    ! [A2: set_o,B2: set_o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ( A2 != bot_bot_set_o )
       => ( aa_set_o_o(finite_finite_o,B2)
         => ( ( B2 != bot_bot_set_o )
           => ( aa_set_o_o(lattic4107685809792843317_fin_o,sup_sup_set_o(A2,B2))
            <=> aa_o_o(aa_o_fun_o_o(inf_inf_o,aa_set_o_o(lattic4107685809792843317_fin_o,A2)),aa_set_o_o(lattic4107685809792843317_fin_o,B2)) ) ) ) ) ) ).

% Inf_fin.union
tff(fact_1569_Inf__fin_Ounion,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( A2 != bot_bot_set_nat )
       => ( aa_set_nat_o(finite_finite_nat,B2)
         => ( ( B2 != bot_bot_set_nat )
           => ( aa_set_nat_nat(lattic5238388535129920115in_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,aa_set_nat_nat(lattic5238388535129920115in_nat,A2)),aa_set_nat_nat(lattic5238388535129920115in_nat,B2)) ) ) ) ) ) ).

% Inf_fin.union
tff(fact_1570_product_Osimps_I1_J,axiom,
    ! [Uu: list_assn] : product_assn_assn(nil_assn,Uu) = nil_Pr5671120429643327159n_assn ).

% product.simps(1)
tff(fact_1571_butlast__snoc,axiom,
    ! [Xs: list_a,X: a] : butlast_a(aa_list_a_list_a(aa_lis2442066584718795531list_a(append_a,Xs),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X),nil_a))) = Xs ).

% butlast_snoc
tff(fact_1572_butlast__snoc,axiom,
    ! [Xs: list_b,X: b] : butlast_b(aa_list_b_list_b(aa_lis927650220358621516list_b(append_b,Xs),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X),nil_b))) = Xs ).

% butlast_snoc
tff(fact_1573_butlast__snoc,axiom,
    ! [Xs: list_nat,X: nat] : butlast_nat(aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Xs),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X),nil_nat))) = Xs ).

% butlast_snoc
tff(fact_1574_butlast__snoc,axiom,
    ! [Xs: list_int,X: int] : butlast_int(aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Xs),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X),nil_int))) = Xs ).

% butlast_snoc
tff(fact_1575_butlast__snoc,axiom,
    ! [Xs: list_P8527749157015355191n_assn,X: produc6575502325842934193n_assn] : butlas3012047794866324995n_assn(aa_lis2797772165808192636n_assn(aa_lis4738365772350288766n_assn(append282499809098378956n_assn,Xs),aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X),nil_Pr5671120429643327159n_assn))) = Xs ).

% butlast_snoc
tff(fact_1576_last__snoc,axiom,
    ! [Xs: list_a,X: a] : last_a(aa_list_a_list_a(aa_lis2442066584718795531list_a(append_a,Xs),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X),nil_a))) = X ).

% last_snoc
tff(fact_1577_last__snoc,axiom,
    ! [Xs: list_b,X: b] : last_b(aa_list_b_list_b(aa_lis927650220358621516list_b(append_b,Xs),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X),nil_b))) = X ).

% last_snoc
tff(fact_1578_last__snoc,axiom,
    ! [Xs: list_nat,X: nat] : last_nat(aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Xs),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X),nil_nat))) = X ).

% last_snoc
tff(fact_1579_last__snoc,axiom,
    ! [Xs: list_int,X: int] : last_int(aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Xs),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X),nil_int))) = X ).

% last_snoc
tff(fact_1580_last__snoc,axiom,
    ! [Xs: list_P8527749157015355191n_assn,X: produc6575502325842934193n_assn] : last_P8723976779861936080n_assn(aa_lis2797772165808192636n_assn(aa_lis4738365772350288766n_assn(append282499809098378956n_assn,Xs),aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X),nil_Pr5671120429643327159n_assn))) = X ).

% last_snoc
tff(fact_1581_Max__less__iff,axiom,
    ! [A2: set_o,X: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ( A2 != bot_bot_set_o )
       => ( aa_o_o(aa_o_fun_o_o(ord_less_o,aa_set_o_o(lattic1921953407002678535_Max_o,A2)),(X))
        <=> ! [X3: $o] :
              ( aa_set_o_o(member_o((X3)),A2)
             => aa_o_o(aa_o_fun_o_o(ord_less_o,(X3)),(X)) ) ) ) ) ).

% Max_less_iff
tff(fact_1582_Max__less__iff,axiom,
    ! [A2: set_nat,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( A2 != bot_bot_set_nat )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,lattic8265883725875713057ax_nat(A2)),X)
        <=> ! [X3: nat] :
              ( aa_set_nat_o(member_nat2(X3),A2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),X) ) ) ) ) ).

% Max_less_iff
tff(fact_1583_Max__less__iff,axiom,
    ! [A2: set_int,X: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( A2 != bot_bot_set_int )
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,lattic8263393255366662781ax_int(A2)),X)
        <=> ! [X3: int] :
              ( aa_set_int_o(member_int2(X3),A2)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,X3),X) ) ) ) ) ).

% Max_less_iff
tff(fact_1584_concat__conv__foldr,axiom,
    ! [Xss2: list_list_a] : concat_a(Xss2) = foldr_list_a_list_a(append_a,Xss2,nil_a) ).

% concat_conv_foldr
tff(fact_1585_concat__conv__foldr,axiom,
    ! [Xss2: list_list_b] : concat_b(Xss2) = foldr_list_b_list_b(append_b,Xss2,nil_b) ).

% concat_conv_foldr
tff(fact_1586_concat__conv__foldr,axiom,
    ! [Xss2: list_l6351802567095793725n_assn] : concat6144485081897559666n_assn(Xss2) = foldr_4640886882926046823n_assn(append282499809098378956n_assn,Xss2,nil_Pr5671120429643327159n_assn) ).

% concat_conv_foldr
tff(fact_1587_concat__conv__foldr,axiom,
    ! [Xss2: list_list_nat] : concat_nat(Xss2) = foldr_6871341030409798377st_nat(append_nat,Xss2,nil_nat) ).

% concat_conv_foldr
tff(fact_1588_concat__conv__foldr,axiom,
    ! [Xss2: list_list_int] : concat_int(Xss2) = foldr_4541567342299342241st_int(append_int,Xss2,nil_int) ).

% concat_conv_foldr
tff(fact_1589_Min__gr__iff,axiom,
    ! [A2: set_o,X: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ( A2 != bot_bot_set_o )
       => ( aa_o_o(aa_o_fun_o_o(ord_less_o,(X)),aa_set_o_o(lattic1973801136483472281_Min_o,A2))
        <=> ! [X3: $o] :
              ( aa_set_o_o(member_o((X3)),A2)
             => aa_o_o(aa_o_fun_o_o(ord_less_o,(X)),(X3)) ) ) ) ) ).

% Min_gr_iff
tff(fact_1590_Min__gr__iff,axiom,
    ! [A2: set_nat,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( A2 != bot_bot_set_nat )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),lattic8721135487736765967in_nat(A2))
        <=> ! [X3: nat] :
              ( aa_set_nat_o(member_nat2(X3),A2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),X3) ) ) ) ) ).

% Min_gr_iff
tff(fact_1591_Min__gr__iff,axiom,
    ! [A2: set_int,X: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( A2 != bot_bot_set_int )
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),lattic8718645017227715691in_int(A2))
        <=> ! [X3: int] :
              ( aa_set_int_o(member_int2(X3),A2)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,X),X3) ) ) ) ) ).

% Min_gr_iff
tff(fact_1592_SuccI,axiom,
    ! [Kl: list_f5222602229612168821_nat_o,K: fun_Pr6460944699146269935_nat_o,Kl2: set_li6341169313164406869_nat_o] :
      ( member877585773093452190_nat_o(append3077274467817616650_nat_o(Kl,cons_f8040665141538986917_nat_o(K,nil_fu2924554602572761077_nat_o)),Kl2)
     => aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(K),bNF_Gr2876145917579644042_nat_o(Kl2,Kl)) ) ).

% SuccI
tff(fact_1593_SuccI,axiom,
    ! [Kl: list_a,K: a,Kl2: set_list_a] :
      ( member_list_a(aa_list_a_list_a(aa_lis2442066584718795531list_a(append_a,Kl),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,K),nil_a)),Kl2)
     => member_a2(K,bNF_Greatest_Succ_a(Kl2,Kl)) ) ).

% SuccI
tff(fact_1594_SuccI,axiom,
    ! [Kl: list_b,K: b,Kl2: set_list_b] :
      ( member_list_b(aa_list_b_list_b(aa_lis927650220358621516list_b(append_b,Kl),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,K),nil_b)),Kl2)
     => member_b2(K,bNF_Greatest_Succ_b(Kl2,Kl)) ) ).

% SuccI
tff(fact_1595_SuccI,axiom,
    ! [Kl: list_nat,K: nat,Kl2: set_list_nat] :
      ( member_list_nat(aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Kl),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,K),nil_nat)),Kl2)
     => aa_set_nat_o(member_nat2(K),bNF_Gr6352880689984616693cc_nat(Kl2,Kl)) ) ).

% SuccI
tff(fact_1596_SuccI,axiom,
    ! [Kl: list_int,K: int,Kl2: set_list_int] :
      ( member_list_int(aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Kl),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,K),nil_int)),Kl2)
     => aa_set_int_o(member_int2(K),bNF_Gr6350390219475566417cc_int(Kl2,Kl)) ) ).

% SuccI
tff(fact_1597_SuccI,axiom,
    ! [Kl: list_P8527749157015355191n_assn,K: produc6575502325842934193n_assn,Kl2: set_li5131720305576846103n_assn] :
      ( member852475432509897056n_assn(aa_lis2797772165808192636n_assn(aa_lis4738365772350288766n_assn(append282499809098378956n_assn,Kl),aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,K),nil_Pr5671120429643327159n_assn)),Kl2)
     => member7957490590177025114n_assn(K,bNF_Gr3216292445876487756n_assn(Kl2,Kl)) ) ).

% SuccI
tff(fact_1598_foldr__append,axiom,
    ! [F: fun_as3983628580203722482n_assn,Xs: list_assn,Ys: list_assn,A: assn] : aa_assn_assn(foldr_assn_assn(F,append_assn(Xs,Ys)),A) = aa_assn_assn(foldr_assn_assn(F,Xs),aa_assn_assn(foldr_assn_assn(F,Ys),A)) ).

% foldr_append
tff(fact_1599_last__appendL,axiom,
    ! [Ys: list_a,Xs: list_a] :
      ( ( Ys = nil_a )
     => ( last_a(aa_list_a_list_a(aa_lis2442066584718795531list_a(append_a,Xs),Ys)) = last_a(Xs) ) ) ).

% last_appendL
tff(fact_1600_last__appendL,axiom,
    ! [Ys: list_b,Xs: list_b] :
      ( ( Ys = nil_b )
     => ( last_b(aa_list_b_list_b(aa_lis927650220358621516list_b(append_b,Xs),Ys)) = last_b(Xs) ) ) ).

% last_appendL
tff(fact_1601_last__appendL,axiom,
    ! [Ys: list_P8527749157015355191n_assn,Xs: list_P8527749157015355191n_assn] :
      ( ( Ys = nil_Pr5671120429643327159n_assn )
     => ( last_P8723976779861936080n_assn(aa_lis2797772165808192636n_assn(aa_lis4738365772350288766n_assn(append282499809098378956n_assn,Xs),Ys)) = last_P8723976779861936080n_assn(Xs) ) ) ).

% last_appendL
tff(fact_1602_last__appendL,axiom,
    ! [Ys: list_nat,Xs: list_nat] :
      ( ( Ys = nil_nat )
     => ( last_nat(aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Xs),Ys)) = last_nat(Xs) ) ) ).

% last_appendL
tff(fact_1603_last__appendL,axiom,
    ! [Ys: list_int,Xs: list_int] :
      ( ( Ys = nil_int )
     => ( last_int(aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Xs),Ys)) = last_int(Xs) ) ) ).

% last_appendL
tff(fact_1604_last__appendR,axiom,
    ! [Ys: list_a,Xs: list_a] :
      ( ( Ys != nil_a )
     => ( last_a(aa_list_a_list_a(aa_lis2442066584718795531list_a(append_a,Xs),Ys)) = last_a(Ys) ) ) ).

% last_appendR
tff(fact_1605_last__appendR,axiom,
    ! [Ys: list_b,Xs: list_b] :
      ( ( Ys != nil_b )
     => ( last_b(aa_list_b_list_b(aa_lis927650220358621516list_b(append_b,Xs),Ys)) = last_b(Ys) ) ) ).

% last_appendR
tff(fact_1606_last__appendR,axiom,
    ! [Ys: list_P8527749157015355191n_assn,Xs: list_P8527749157015355191n_assn] :
      ( ( Ys != nil_Pr5671120429643327159n_assn )
     => ( last_P8723976779861936080n_assn(aa_lis2797772165808192636n_assn(aa_lis4738365772350288766n_assn(append282499809098378956n_assn,Xs),Ys)) = last_P8723976779861936080n_assn(Ys) ) ) ).

% last_appendR
tff(fact_1607_last__appendR,axiom,
    ! [Ys: list_nat,Xs: list_nat] :
      ( ( Ys != nil_nat )
     => ( last_nat(aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Xs),Ys)) = last_nat(Ys) ) ) ).

% last_appendR
tff(fact_1608_last__appendR,axiom,
    ! [Ys: list_int,Xs: list_int] :
      ( ( Ys != nil_int )
     => ( last_int(aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Xs),Ys)) = last_int(Ys) ) ) ).

% last_appendR
tff(fact_1609_inf__Sup__absorb,axiom,
    ! [A2: set_fu3555266933746557007_nat_o,A: fun_Pr6460944699146269935_nat_o] :
      ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,A2)
     => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A),A2)
       => ( aa_fun2638059832795814268_nat_o(aa_fun6000889628616002998_nat_o(inf_in4424744116413743581_nat_o,A),lattic7388253182610234406_nat_o(A2)) = A ) ) ) ).

% inf_Sup_absorb
tff(fact_1610_inf__Sup__absorb,axiom,
    ! [A2: set_assn,A: assn] :
      ( finite_finite_assn(A2)
     => ( member_assn(A,A2)
       => ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),lattic2150320897289308081n_assn(A2)) = A ) ) ) ).

% inf_Sup_absorb
tff(fact_1611_inf__Sup__absorb,axiom,
    ! [A2: set_nat,A: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(member_nat2(A),A2)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),lattic1093996805478795353in_nat(A2)) = A ) ) ) ).

% inf_Sup_absorb
tff(fact_1612_inf__Sup__absorb,axiom,
    ! [A2: set_set_nat,A: set_nat] :
      ( finite1152437895449049373et_nat(A2)
     => ( member_set_nat(A,A2)
       => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),lattic3835124923745554447et_nat(A2)) = A ) ) ) ).

% inf_Sup_absorb
tff(fact_1613_inf__Sup__absorb,axiom,
    ! [A2: set_Product_unit,A: product_unit] :
      ( finite4290736615968046902t_unit(A2)
     => ( member_Product_unit(A,A2)
       => ( aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,A),lattic5294303975357428420t_unit(A2)) = A ) ) ) ).

% inf_Sup_absorb
tff(fact_1614_inf__Sup__absorb,axiom,
    ! [A2: set_se7855581050983116737at_nat,A: set_Pr1261947904930325089at_nat] :
      ( finite9047747110432174090at_nat(A2)
     => ( member2643936169264416010at_nat(A,A2)
       => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),lattic1541023418247406232at_nat(A2)) = A ) ) ) ).

% inf_Sup_absorb
tff(fact_1615_append__butlast__last__id,axiom,
    ! [Xs: list_a] :
      ( ( Xs != nil_a )
     => ( aa_list_a_list_a(aa_lis2442066584718795531list_a(append_a,butlast_a(Xs)),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,last_a(Xs)),nil_a)) = Xs ) ) ).

% append_butlast_last_id
tff(fact_1616_append__butlast__last__id,axiom,
    ! [Xs: list_b] :
      ( ( Xs != nil_b )
     => ( aa_list_b_list_b(aa_lis927650220358621516list_b(append_b,butlast_b(Xs)),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,last_b(Xs)),nil_b)) = Xs ) ) ).

% append_butlast_last_id
tff(fact_1617_append__butlast__last__id,axiom,
    ! [Xs: list_nat] :
      ( ( Xs != nil_nat )
     => ( aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,butlast_nat(Xs)),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,last_nat(Xs)),nil_nat)) = Xs ) ) ).

% append_butlast_last_id
tff(fact_1618_append__butlast__last__id,axiom,
    ! [Xs: list_int] :
      ( ( Xs != nil_int )
     => ( aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,butlast_int(Xs)),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,last_int(Xs)),nil_int)) = Xs ) ) ).

% append_butlast_last_id
tff(fact_1619_append__butlast__last__id,axiom,
    ! [Xs: list_P8527749157015355191n_assn] :
      ( ( Xs != nil_Pr5671120429643327159n_assn )
     => ( aa_lis2797772165808192636n_assn(aa_lis4738365772350288766n_assn(append282499809098378956n_assn,butlas3012047794866324995n_assn(Xs)),aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,last_P8723976779861936080n_assn(Xs)),nil_Pr5671120429643327159n_assn)) = Xs ) ) ).

% append_butlast_last_id
tff(fact_1620_butlast_Osimps_I1_J,axiom,
    butlast_a(nil_a) = nil_a ).

% butlast.simps(1)
tff(fact_1621_butlast_Osimps_I1_J,axiom,
    butlast_b(nil_b) = nil_b ).

% butlast.simps(1)
tff(fact_1622_butlast_Osimps_I1_J,axiom,
    butlas3012047794866324995n_assn(nil_Pr5671120429643327159n_assn) = nil_Pr5671120429643327159n_assn ).

% butlast.simps(1)
tff(fact_1623_butlast_Osimps_I1_J,axiom,
    butlast_nat(nil_nat) = nil_nat ).

% butlast.simps(1)
tff(fact_1624_butlast_Osimps_I1_J,axiom,
    butlast_int(nil_int) = nil_int ).

% butlast.simps(1)
tff(fact_1625_map__butlast,axiom,
    ! [F: fun_nat_nat,Xs: list_nat] : aa_list_nat_list_nat(aa_fun4736978287202700568st_nat(map_nat_nat,F),butlast_nat(Xs)) = butlast_nat(aa_list_nat_list_nat(aa_fun4736978287202700568st_nat(map_nat_nat,F),Xs)) ).

% map_butlast
tff(fact_1626_map__butlast,axiom,
    ! [F: fun_Pr3816910165196019820n_assn,Xs: list_P8527749157015355191n_assn] : aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,F),butlas3012047794866324995n_assn(Xs)) = butlast_assn(aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,F),Xs)) ).

% map_butlast
tff(fact_1627_snoc__eq__iff__butlast,axiom,
    ! [Xs: list_a,X: a,Ys: list_a] :
      ( ( aa_list_a_list_a(aa_lis2442066584718795531list_a(append_a,Xs),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X),nil_a)) = Ys )
    <=> ( ( Ys != nil_a )
        & ( butlast_a(Ys) = Xs )
        & ( last_a(Ys) = X ) ) ) ).

% snoc_eq_iff_butlast
tff(fact_1628_snoc__eq__iff__butlast,axiom,
    ! [Xs: list_b,X: b,Ys: list_b] :
      ( ( aa_list_b_list_b(aa_lis927650220358621516list_b(append_b,Xs),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X),nil_b)) = Ys )
    <=> ( ( Ys != nil_b )
        & ( butlast_b(Ys) = Xs )
        & ( last_b(Ys) = X ) ) ) ).

% snoc_eq_iff_butlast
tff(fact_1629_snoc__eq__iff__butlast,axiom,
    ! [Xs: list_nat,X: nat,Ys: list_nat] :
      ( ( aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Xs),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X),nil_nat)) = Ys )
    <=> ( ( Ys != nil_nat )
        & ( butlast_nat(Ys) = Xs )
        & ( last_nat(Ys) = X ) ) ) ).

% snoc_eq_iff_butlast
tff(fact_1630_snoc__eq__iff__butlast,axiom,
    ! [Xs: list_int,X: int,Ys: list_int] :
      ( ( aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Xs),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X),nil_int)) = Ys )
    <=> ( ( Ys != nil_int )
        & ( butlast_int(Ys) = Xs )
        & ( last_int(Ys) = X ) ) ) ).

% snoc_eq_iff_butlast
tff(fact_1631_snoc__eq__iff__butlast,axiom,
    ! [Xs: list_P8527749157015355191n_assn,X: produc6575502325842934193n_assn,Ys: list_P8527749157015355191n_assn] :
      ( ( aa_lis2797772165808192636n_assn(aa_lis4738365772350288766n_assn(append282499809098378956n_assn,Xs),aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X),nil_Pr5671120429643327159n_assn)) = Ys )
    <=> ( ( Ys != nil_Pr5671120429643327159n_assn )
        & ( butlas3012047794866324995n_assn(Ys) = Xs )
        & ( last_P8723976779861936080n_assn(Ys) = X ) ) ) ).

% snoc_eq_iff_butlast
tff(fact_1632_snoc__eq__iff__butlast_H,axiom,
    ! [Ys: list_a,Xs: list_a,X: a] :
      ( ( Ys = aa_list_a_list_a(aa_lis2442066584718795531list_a(append_a,Xs),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X),nil_a)) )
    <=> ( ( Ys != nil_a )
        & ( butlast_a(Ys) = Xs )
        & ( last_a(Ys) = X ) ) ) ).

% snoc_eq_iff_butlast'
tff(fact_1633_snoc__eq__iff__butlast_H,axiom,
    ! [Ys: list_b,Xs: list_b,X: b] :
      ( ( Ys = aa_list_b_list_b(aa_lis927650220358621516list_b(append_b,Xs),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X),nil_b)) )
    <=> ( ( Ys != nil_b )
        & ( butlast_b(Ys) = Xs )
        & ( last_b(Ys) = X ) ) ) ).

% snoc_eq_iff_butlast'
tff(fact_1634_snoc__eq__iff__butlast_H,axiom,
    ! [Ys: list_nat,Xs: list_nat,X: nat] :
      ( ( Ys = aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Xs),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X),nil_nat)) )
    <=> ( ( Ys != nil_nat )
        & ( butlast_nat(Ys) = Xs )
        & ( last_nat(Ys) = X ) ) ) ).

% snoc_eq_iff_butlast'
tff(fact_1635_snoc__eq__iff__butlast_H,axiom,
    ! [Ys: list_int,Xs: list_int,X: int] :
      ( ( Ys = aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Xs),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X),nil_int)) )
    <=> ( ( Ys != nil_int )
        & ( butlast_int(Ys) = Xs )
        & ( last_int(Ys) = X ) ) ) ).

% snoc_eq_iff_butlast'
tff(fact_1636_snoc__eq__iff__butlast_H,axiom,
    ! [Ys: list_P8527749157015355191n_assn,Xs: list_P8527749157015355191n_assn,X: produc6575502325842934193n_assn] :
      ( ( Ys = aa_lis2797772165808192636n_assn(aa_lis4738365772350288766n_assn(append282499809098378956n_assn,Xs),aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X),nil_Pr5671120429643327159n_assn)) )
    <=> ( ( Ys != nil_Pr5671120429643327159n_assn )
        & ( butlas3012047794866324995n_assn(Ys) = Xs )
        & ( last_P8723976779861936080n_assn(Ys) = X ) ) ) ).

% snoc_eq_iff_butlast'
tff(fact_1637_Min__in,axiom,
    ! [A2: set_o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ( A2 != bot_bot_set_o )
       => aa_set_o_o(member_o(aa_set_o_o(lattic1973801136483472281_Min_o,A2)),A2) ) ) ).

% Min_in
tff(fact_1638_Min__in,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( A2 != bot_bot_set_nat )
       => aa_set_nat_o(member_nat2(lattic8721135487736765967in_nat(A2)),A2) ) ) ).

% Min_in
tff(fact_1639_Max__in,axiom,
    ! [A2: set_o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ( A2 != bot_bot_set_o )
       => aa_set_o_o(member_o(aa_set_o_o(lattic1921953407002678535_Max_o,A2)),A2) ) ) ).

% Max_in
tff(fact_1640_Max__in,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( A2 != bot_bot_set_nat )
       => aa_set_nat_o(member_nat2(lattic8265883725875713057ax_nat(A2)),A2) ) ) ).

% Max_in
tff(fact_1641_last_Osimps,axiom,
    ! [X: a,Xs: list_a] :
      last_a(aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X),Xs)) = $ite(Xs = nil_a,X,last_a(Xs)) ).

% last.simps
tff(fact_1642_last_Osimps,axiom,
    ! [X: b,Xs: list_b] :
      last_b(aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X),Xs)) = $ite(Xs = nil_b,X,last_b(Xs)) ).

% last.simps
tff(fact_1643_last_Osimps,axiom,
    ! [X: nat,Xs: list_nat] :
      last_nat(aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X),Xs)) = $ite(Xs = nil_nat,X,last_nat(Xs)) ).

% last.simps
tff(fact_1644_last_Osimps,axiom,
    ! [X: int,Xs: list_int] :
      last_int(aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X),Xs)) = $ite(Xs = nil_int,X,last_int(Xs)) ).

% last.simps
tff(fact_1645_last_Osimps,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] :
      last_P8723976779861936080n_assn(aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X),Xs)) = $ite(Xs = nil_Pr5671120429643327159n_assn,X,last_P8723976779861936080n_assn(Xs)) ).

% last.simps
tff(fact_1646_last__ConsL,axiom,
    ! [Xs: list_a,X: a] :
      ( ( Xs = nil_a )
     => ( last_a(aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X),Xs)) = X ) ) ).

% last_ConsL
tff(fact_1647_last__ConsL,axiom,
    ! [Xs: list_b,X: b] :
      ( ( Xs = nil_b )
     => ( last_b(aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X),Xs)) = X ) ) ).

% last_ConsL
tff(fact_1648_last__ConsL,axiom,
    ! [Xs: list_nat,X: nat] :
      ( ( Xs = nil_nat )
     => ( last_nat(aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X),Xs)) = X ) ) ).

% last_ConsL
tff(fact_1649_last__ConsL,axiom,
    ! [Xs: list_int,X: int] :
      ( ( Xs = nil_int )
     => ( last_int(aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X),Xs)) = X ) ) ).

% last_ConsL
tff(fact_1650_last__ConsL,axiom,
    ! [Xs: list_P8527749157015355191n_assn,X: produc6575502325842934193n_assn] :
      ( ( Xs = nil_Pr5671120429643327159n_assn )
     => ( last_P8723976779861936080n_assn(aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X),Xs)) = X ) ) ).

% last_ConsL
tff(fact_1651_last__ConsR,axiom,
    ! [Xs: list_a,X: a] :
      ( ( Xs != nil_a )
     => ( last_a(aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X),Xs)) = last_a(Xs) ) ) ).

% last_ConsR
tff(fact_1652_last__ConsR,axiom,
    ! [Xs: list_b,X: b] :
      ( ( Xs != nil_b )
     => ( last_b(aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X),Xs)) = last_b(Xs) ) ) ).

% last_ConsR
tff(fact_1653_last__ConsR,axiom,
    ! [Xs: list_nat,X: nat] :
      ( ( Xs != nil_nat )
     => ( last_nat(aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X),Xs)) = last_nat(Xs) ) ) ).

% last_ConsR
tff(fact_1654_last__ConsR,axiom,
    ! [Xs: list_int,X: int] :
      ( ( Xs != nil_int )
     => ( last_int(aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X),Xs)) = last_int(Xs) ) ) ).

% last_ConsR
tff(fact_1655_last__ConsR,axiom,
    ! [Xs: list_P8527749157015355191n_assn,X: produc6575502325842934193n_assn] :
      ( ( Xs != nil_Pr5671120429643327159n_assn )
     => ( last_P8723976779861936080n_assn(aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X),Xs)) = last_P8723976779861936080n_assn(Xs) ) ) ).

% last_ConsR
tff(fact_1656_last__map,axiom,
    ! [Xs: list_a,F: fun_a_nat] :
      ( ( Xs != nil_a )
     => ( last_nat(map_a_nat(F,Xs)) = aa_a_nat(F,last_a(Xs)) ) ) ).

% last_map
tff(fact_1657_last__map,axiom,
    ! [Xs: list_b,F: fun_b_nat] :
      ( ( Xs != nil_b )
     => ( last_nat(map_b_nat(F,Xs)) = aa_b_nat(F,last_b(Xs)) ) ) ).

% last_map
tff(fact_1658_last__map,axiom,
    ! [Xs: list_P8527749157015355191n_assn,F: fun_Pr7403558292872094996sn_nat] :
      ( ( Xs != nil_Pr5671120429643327159n_assn )
     => ( last_nat(map_Pr7570552894071451325sn_nat(F,Xs)) = aa_Pro8055486230010607173sn_nat(F,last_P8723976779861936080n_assn(Xs)) ) ) ).

% last_map
tff(fact_1659_last__map,axiom,
    ! [Xs: list_int,F: fun_int_nat] :
      ( ( Xs != nil_int )
     => ( last_nat(map_int_nat(F,Xs)) = aa_int_nat(F,last_int(Xs)) ) ) ).

% last_map
tff(fact_1660_last__map,axiom,
    ! [Xs: list_nat,F: fun_nat_nat] :
      ( ( Xs != nil_nat )
     => ( last_nat(aa_list_nat_list_nat(aa_fun4736978287202700568st_nat(map_nat_nat,F),Xs)) = aa_nat_nat(F,last_nat(Xs)) ) ) ).

% last_map
tff(fact_1661_last__map,axiom,
    ! [Xs: list_P8527749157015355191n_assn,F: fun_Pr3816910165196019820n_assn] :
      ( ( Xs != nil_Pr5671120429643327159n_assn )
     => ( last_assn(aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,F),Xs)) = aa_Pro2979818503705714077n_assn(F,last_P8723976779861936080n_assn(Xs)) ) ) ).

% last_map
tff(fact_1662_last__append,axiom,
    ! [Xs: list_a,Ys: list_a] :
      last_a(aa_list_a_list_a(aa_lis2442066584718795531list_a(append_a,Xs),Ys)) = $ite(Ys = nil_a,last_a(Xs),last_a(Ys)) ).

% last_append
tff(fact_1663_last__append,axiom,
    ! [Xs: list_b,Ys: list_b] :
      last_b(aa_list_b_list_b(aa_lis927650220358621516list_b(append_b,Xs),Ys)) = $ite(Ys = nil_b,last_b(Xs),last_b(Ys)) ).

% last_append
tff(fact_1664_last__append,axiom,
    ! [Xs: list_P8527749157015355191n_assn,Ys: list_P8527749157015355191n_assn] :
      last_P8723976779861936080n_assn(aa_lis2797772165808192636n_assn(aa_lis4738365772350288766n_assn(append282499809098378956n_assn,Xs),Ys)) = $ite(Ys = nil_Pr5671120429643327159n_assn,last_P8723976779861936080n_assn(Xs),last_P8723976779861936080n_assn(Ys)) ).

% last_append
tff(fact_1665_last__append,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      last_nat(aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Xs),Ys)) = $ite(Ys = nil_nat,last_nat(Xs),last_nat(Ys)) ).

% last_append
tff(fact_1666_last__append,axiom,
    ! [Xs: list_int,Ys: list_int] :
      last_int(aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Xs),Ys)) = $ite(Ys = nil_int,last_int(Xs),last_int(Ys)) ).

% last_append
tff(fact_1667_longest__common__suffix,axiom,
    ! [Xs: list_a,Ys: list_a] :
    ? [Ss: list_a,Xs4: list_a,Ys4: list_a] :
      ( ( Xs = aa_list_a_list_a(aa_lis2442066584718795531list_a(append_a,Xs4),Ss) )
      & ( Ys = aa_list_a_list_a(aa_lis2442066584718795531list_a(append_a,Ys4),Ss) )
      & ( ( Xs4 = nil_a )
        | ( Ys4 = nil_a )
        | ( last_a(Xs4) != last_a(Ys4) ) ) ) ).

% longest_common_suffix
tff(fact_1668_longest__common__suffix,axiom,
    ! [Xs: list_b,Ys: list_b] :
    ? [Ss: list_b,Xs4: list_b,Ys4: list_b] :
      ( ( Xs = aa_list_b_list_b(aa_lis927650220358621516list_b(append_b,Xs4),Ss) )
      & ( Ys = aa_list_b_list_b(aa_lis927650220358621516list_b(append_b,Ys4),Ss) )
      & ( ( Xs4 = nil_b )
        | ( Ys4 = nil_b )
        | ( last_b(Xs4) != last_b(Ys4) ) ) ) ).

% longest_common_suffix
tff(fact_1669_longest__common__suffix,axiom,
    ! [Xs: list_P8527749157015355191n_assn,Ys: list_P8527749157015355191n_assn] :
    ? [Ss: list_P8527749157015355191n_assn,Xs4: list_P8527749157015355191n_assn,Ys4: list_P8527749157015355191n_assn] :
      ( ( Xs = aa_lis2797772165808192636n_assn(aa_lis4738365772350288766n_assn(append282499809098378956n_assn,Xs4),Ss) )
      & ( Ys = aa_lis2797772165808192636n_assn(aa_lis4738365772350288766n_assn(append282499809098378956n_assn,Ys4),Ss) )
      & ( ( Xs4 = nil_Pr5671120429643327159n_assn )
        | ( Ys4 = nil_Pr5671120429643327159n_assn )
        | ( last_P8723976779861936080n_assn(Xs4) != last_P8723976779861936080n_assn(Ys4) ) ) ) ).

% longest_common_suffix
tff(fact_1670_longest__common__suffix,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
    ? [Ss: list_nat,Xs4: list_nat,Ys4: list_nat] :
      ( ( Xs = aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Xs4),Ss) )
      & ( Ys = aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Ys4),Ss) )
      & ( ( Xs4 = nil_nat )
        | ( Ys4 = nil_nat )
        | ( last_nat(Xs4) != last_nat(Ys4) ) ) ) ).

% longest_common_suffix
tff(fact_1671_longest__common__suffix,axiom,
    ! [Xs: list_int,Ys: list_int] :
    ? [Ss: list_int,Xs4: list_int,Ys4: list_int] :
      ( ( Xs = aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Xs4),Ss) )
      & ( Ys = aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Ys4),Ss) )
      & ( ( Xs4 = nil_int )
        | ( Ys4 = nil_int )
        | ( last_int(Xs4) != last_int(Ys4) ) ) ) ).

% longest_common_suffix
tff(fact_1672_butlast_Osimps_I2_J,axiom,
    ! [X: a,Xs: list_a] :
      butlast_a(aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X),Xs)) = $ite(Xs = nil_a,nil_a,aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X),butlast_a(Xs))) ).

% butlast.simps(2)
tff(fact_1673_butlast_Osimps_I2_J,axiom,
    ! [X: b,Xs: list_b] :
      butlast_b(aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X),Xs)) = $ite(Xs = nil_b,nil_b,aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X),butlast_b(Xs))) ).

% butlast.simps(2)
tff(fact_1674_butlast_Osimps_I2_J,axiom,
    ! [X: nat,Xs: list_nat] :
      butlast_nat(aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X),Xs)) = $ite(Xs = nil_nat,nil_nat,aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X),butlast_nat(Xs))) ).

% butlast.simps(2)
tff(fact_1675_butlast_Osimps_I2_J,axiom,
    ! [X: int,Xs: list_int] :
      butlast_int(aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X),Xs)) = $ite(Xs = nil_int,nil_int,aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X),butlast_int(Xs))) ).

% butlast.simps(2)
tff(fact_1676_butlast_Osimps_I2_J,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] :
      butlas3012047794866324995n_assn(aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X),Xs)) = $ite(Xs = nil_Pr5671120429643327159n_assn,nil_Pr5671120429643327159n_assn,aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X),butlas3012047794866324995n_assn(Xs))) ).

% butlast.simps(2)
tff(fact_1677_butlast__append,axiom,
    ! [Xs: list_a,Ys: list_a] :
      butlast_a(aa_list_a_list_a(aa_lis2442066584718795531list_a(append_a,Xs),Ys)) = $ite(Ys = nil_a,butlast_a(Xs),aa_list_a_list_a(aa_lis2442066584718795531list_a(append_a,Xs),butlast_a(Ys))) ).

% butlast_append
tff(fact_1678_butlast__append,axiom,
    ! [Xs: list_b,Ys: list_b] :
      butlast_b(aa_list_b_list_b(aa_lis927650220358621516list_b(append_b,Xs),Ys)) = $ite(Ys = nil_b,butlast_b(Xs),aa_list_b_list_b(aa_lis927650220358621516list_b(append_b,Xs),butlast_b(Ys))) ).

% butlast_append
tff(fact_1679_butlast__append,axiom,
    ! [Xs: list_P8527749157015355191n_assn,Ys: list_P8527749157015355191n_assn] :
      butlas3012047794866324995n_assn(aa_lis2797772165808192636n_assn(aa_lis4738365772350288766n_assn(append282499809098378956n_assn,Xs),Ys)) = $ite(Ys = nil_Pr5671120429643327159n_assn,butlas3012047794866324995n_assn(Xs),aa_lis2797772165808192636n_assn(aa_lis4738365772350288766n_assn(append282499809098378956n_assn,Xs),butlas3012047794866324995n_assn(Ys))) ).

% butlast_append
tff(fact_1680_butlast__append,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      butlast_nat(aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Xs),Ys)) = $ite(Ys = nil_nat,butlast_nat(Xs),aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Xs),butlast_nat(Ys))) ).

% butlast_append
tff(fact_1681_butlast__append,axiom,
    ! [Xs: list_int,Ys: list_int] :
      butlast_int(aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Xs),Ys)) = $ite(Ys = nil_int,butlast_int(Xs),aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Xs),butlast_int(Ys))) ).

% butlast_append
tff(fact_1682_Inf__fin_Oin__idem,axiom,
    ! [A2: set_fu3555266933746557007_nat_o,X: fun_Pr6460944699146269935_nat_o] :
      ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,A2)
     => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),A2)
       => ( aa_fun2638059832795814268_nat_o(aa_fun6000889628616002998_nat_o(inf_in4424744116413743581_nat_o,X),lattic6089491107740071820_nat_o(A2)) = lattic6089491107740071820_nat_o(A2) ) ) ) ).

% Inf_fin.in_idem
tff(fact_1683_Inf__fin_Oin__idem,axiom,
    ! [A2: set_assn,X: assn] :
      ( finite_finite_assn(A2)
     => ( member_assn(X,A2)
       => ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),aa_set_assn_assn(lattic47131356835913163n_assn,A2)) = aa_set_assn_assn(lattic47131356835913163n_assn,A2) ) ) ) ).

% Inf_fin.in_idem
tff(fact_1684_Inf__fin_Oin__idem,axiom,
    ! [A2: set_nat,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(member_nat2(X),A2)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),aa_set_nat_nat(lattic5238388535129920115in_nat,A2)) = aa_set_nat_nat(lattic5238388535129920115in_nat,A2) ) ) ) ).

% Inf_fin.in_idem
tff(fact_1685_Inf__fin_Oin__idem,axiom,
    ! [A2: set_set_nat,X: set_nat] :
      ( finite1152437895449049373et_nat(A2)
     => ( member_set_nat(X,A2)
       => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),aa_set6364576160234933248et_nat(lattic3014633134055518761et_nat,A2)) = aa_set6364576160234933248et_nat(lattic3014633134055518761et_nat,A2) ) ) ) ).

% Inf_fin.in_idem
tff(fact_1686_Inf__fin_Oin__idem,axiom,
    ! [A2: set_Product_unit,X: product_unit] :
      ( finite4290736615968046902t_unit(A2)
     => ( member_Product_unit(X,A2)
       => ( aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),aa_set6350468789499455260t_unit(lattic1263872656861969706t_unit,A2)) = aa_set6350468789499455260t_unit(lattic1263872656861969706t_unit,A2) ) ) ) ).

% Inf_fin.in_idem
tff(fact_1687_Inf__fin_Oin__idem,axiom,
    ! [A2: set_se7855581050983116737at_nat,X: set_Pr1261947904930325089at_nat] :
      ( finite9047747110432174090at_nat(A2)
     => ( member2643936169264416010at_nat(X,A2)
       => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),aa_set3914448398961233820at_nat(lattic30941717366863870at_nat,A2)) = aa_set3914448398961233820at_nat(lattic30941717366863870at_nat,A2) ) ) ) ).

% Inf_fin.in_idem
tff(fact_1688_Misc_Ofoldr__Cons,axiom,
    ! [Xs: list_a] : foldr_a_list_a(cons_a,Xs,nil_a) = Xs ).

% Misc.foldr_Cons
tff(fact_1689_Misc_Ofoldr__Cons,axiom,
    ! [Xs: list_b] : foldr_b_list_b(cons_b,Xs,nil_b) = Xs ).

% Misc.foldr_Cons
tff(fact_1690_Misc_Ofoldr__Cons,axiom,
    ! [Xs: list_nat] : foldr_nat_list_nat(cons_nat,Xs,nil_nat) = Xs ).

% Misc.foldr_Cons
tff(fact_1691_Misc_Ofoldr__Cons,axiom,
    ! [Xs: list_int] : foldr_int_list_int(cons_int,Xs,nil_int) = Xs ).

% Misc.foldr_Cons
tff(fact_1692_Misc_Ofoldr__Cons,axiom,
    ! [Xs: list_P8527749157015355191n_assn] : foldr_2213011659765427309n_assn(cons_P2971678138204555879n_assn,Xs,nil_Pr5671120429643327159n_assn) = Xs ).

% Misc.foldr_Cons
tff(fact_1693_Min__less__iff,axiom,
    ! [A2: set_o,X: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ( A2 != bot_bot_set_o )
       => ( aa_o_o(aa_o_fun_o_o(ord_less_o,aa_set_o_o(lattic1973801136483472281_Min_o,A2)),(X))
        <=> ? [X3: $o] :
              ( aa_set_o_o(member_o((X3)),A2)
              & aa_o_o(aa_o_fun_o_o(ord_less_o,(X3)),(X)) ) ) ) ) ).

% Min_less_iff
tff(fact_1694_Min__less__iff,axiom,
    ! [A2: set_nat,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( A2 != bot_bot_set_nat )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,lattic8721135487736765967in_nat(A2)),X)
        <=> ? [X3: nat] :
              ( aa_set_nat_o(member_nat2(X3),A2)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),X) ) ) ) ) ).

% Min_less_iff
tff(fact_1695_Min__less__iff,axiom,
    ! [A2: set_int,X: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( A2 != bot_bot_set_int )
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,lattic8718645017227715691in_int(A2)),X)
        <=> ? [X3: int] :
              ( aa_set_int_o(member_int2(X3),A2)
              & aa_int_o(aa_int_fun_int_o(ord_less_int,X3),X) ) ) ) ) ).

% Min_less_iff
tff(fact_1696_Max__gr__iff,axiom,
    ! [A2: set_o,X: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ( A2 != bot_bot_set_o )
       => ( aa_o_o(aa_o_fun_o_o(ord_less_o,(X)),aa_set_o_o(lattic1921953407002678535_Max_o,A2))
        <=> ? [X3: $o] :
              ( aa_set_o_o(member_o((X3)),A2)
              & aa_o_o(aa_o_fun_o_o(ord_less_o,(X)),(X3)) ) ) ) ) ).

% Max_gr_iff
tff(fact_1697_Max__gr__iff,axiom,
    ! [A2: set_nat,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( A2 != bot_bot_set_nat )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),lattic8265883725875713057ax_nat(A2))
        <=> ? [X3: nat] :
              ( aa_set_nat_o(member_nat2(X3),A2)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),X3) ) ) ) ) ).

% Max_gr_iff
tff(fact_1698_Max__gr__iff,axiom,
    ! [A2: set_int,X: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( A2 != bot_bot_set_int )
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),lattic8263393255366662781ax_int(A2))
        <=> ? [X3: int] :
              ( aa_set_int_o(member_int2(X3),A2)
              & aa_int_o(aa_int_fun_int_o(ord_less_int,X),X3) ) ) ) ) ).

% Max_gr_iff
tff(fact_1699_butlast__eq__consE,axiom,
    ! [La: list_a,X: a,Xs: list_a] :
      ( ( butlast_a(La) = aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X),Xs) )
     => ~ ! [Xl: a] : La != aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X),aa_list_a_list_a(aa_lis2442066584718795531list_a(append_a,Xs),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,Xl),nil_a))) ) ).

% butlast_eq_consE
tff(fact_1700_butlast__eq__consE,axiom,
    ! [La: list_b,X: b,Xs: list_b] :
      ( ( butlast_b(La) = aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X),Xs) )
     => ~ ! [Xl: b] : La != aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X),aa_list_b_list_b(aa_lis927650220358621516list_b(append_b,Xs),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,Xl),nil_b))) ) ).

% butlast_eq_consE
tff(fact_1701_butlast__eq__consE,axiom,
    ! [La: list_nat,X: nat,Xs: list_nat] :
      ( ( butlast_nat(La) = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X),Xs) )
     => ~ ! [Xl: nat] : La != aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X),aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Xs),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Xl),nil_nat))) ) ).

% butlast_eq_consE
tff(fact_1702_butlast__eq__consE,axiom,
    ! [La: list_int,X: int,Xs: list_int] :
      ( ( butlast_int(La) = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X),Xs) )
     => ~ ! [Xl: int] : La != aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X),aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Xs),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Xl),nil_int))) ) ).

% butlast_eq_consE
tff(fact_1703_butlast__eq__consE,axiom,
    ! [La: list_P8527749157015355191n_assn,X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] :
      ( ( butlas3012047794866324995n_assn(La) = aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X),Xs) )
     => ~ ! [Xl: produc6575502325842934193n_assn] : La != aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X),aa_lis2797772165808192636n_assn(aa_lis4738365772350288766n_assn(append282499809098378956n_assn,Xs),aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,Xl),nil_Pr5671120429643327159n_assn))) ) ).

% butlast_eq_consE
tff(fact_1704_butlast__eq__cons__conv,axiom,
    ! [La: list_a,X: a,Xs: list_a] :
      ( ( butlast_a(La) = aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X),Xs) )
    <=> ? [Xl2: a] : La = aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X),aa_list_a_list_a(aa_lis2442066584718795531list_a(append_a,Xs),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,Xl2),nil_a))) ) ).

% butlast_eq_cons_conv
tff(fact_1705_butlast__eq__cons__conv,axiom,
    ! [La: list_b,X: b,Xs: list_b] :
      ( ( butlast_b(La) = aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X),Xs) )
    <=> ? [Xl2: b] : La = aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X),aa_list_b_list_b(aa_lis927650220358621516list_b(append_b,Xs),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,Xl2),nil_b))) ) ).

% butlast_eq_cons_conv
tff(fact_1706_butlast__eq__cons__conv,axiom,
    ! [La: list_nat,X: nat,Xs: list_nat] :
      ( ( butlast_nat(La) = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X),Xs) )
    <=> ? [Xl2: nat] : La = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X),aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Xs),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Xl2),nil_nat))) ) ).

% butlast_eq_cons_conv
tff(fact_1707_butlast__eq__cons__conv,axiom,
    ! [La: list_int,X: int,Xs: list_int] :
      ( ( butlast_int(La) = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X),Xs) )
    <=> ? [Xl2: int] : La = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X),aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Xs),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Xl2),nil_int))) ) ).

% butlast_eq_cons_conv
tff(fact_1708_butlast__eq__cons__conv,axiom,
    ! [La: list_P8527749157015355191n_assn,X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] :
      ( ( butlas3012047794866324995n_assn(La) = aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X),Xs) )
    <=> ? [Xl2: produc6575502325842934193n_assn] : La = aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X),aa_lis2797772165808192636n_assn(aa_lis4738365772350288766n_assn(append282499809098378956n_assn,Xs),aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,Xl2),nil_Pr5671120429643327159n_assn))) ) ).

% butlast_eq_cons_conv
tff(fact_1709_SuccD,axiom,
    ! [K: fun_Pr6460944699146269935_nat_o,Kl2: set_li6341169313164406869_nat_o,Kl: list_f5222602229612168821_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(K),bNF_Gr2876145917579644042_nat_o(Kl2,Kl))
     => member877585773093452190_nat_o(append3077274467817616650_nat_o(Kl,cons_f8040665141538986917_nat_o(K,nil_fu2924554602572761077_nat_o)),Kl2) ) ).

% SuccD
tff(fact_1710_SuccD,axiom,
    ! [K: a,Kl2: set_list_a,Kl: list_a] :
      ( member_a2(K,bNF_Greatest_Succ_a(Kl2,Kl))
     => member_list_a(aa_list_a_list_a(aa_lis2442066584718795531list_a(append_a,Kl),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,K),nil_a)),Kl2) ) ).

% SuccD
tff(fact_1711_SuccD,axiom,
    ! [K: b,Kl2: set_list_b,Kl: list_b] :
      ( member_b2(K,bNF_Greatest_Succ_b(Kl2,Kl))
     => member_list_b(aa_list_b_list_b(aa_lis927650220358621516list_b(append_b,Kl),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,K),nil_b)),Kl2) ) ).

% SuccD
tff(fact_1712_SuccD,axiom,
    ! [K: nat,Kl2: set_list_nat,Kl: list_nat] :
      ( aa_set_nat_o(member_nat2(K),bNF_Gr6352880689984616693cc_nat(Kl2,Kl))
     => member_list_nat(aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Kl),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,K),nil_nat)),Kl2) ) ).

% SuccD
tff(fact_1713_SuccD,axiom,
    ! [K: int,Kl2: set_list_int,Kl: list_int] :
      ( aa_set_int_o(member_int2(K),bNF_Gr6350390219475566417cc_int(Kl2,Kl))
     => member_list_int(aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Kl),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,K),nil_int)),Kl2) ) ).

% SuccD
tff(fact_1714_SuccD,axiom,
    ! [K: produc6575502325842934193n_assn,Kl2: set_li5131720305576846103n_assn,Kl: list_P8527749157015355191n_assn] :
      ( member7957490590177025114n_assn(K,bNF_Gr3216292445876487756n_assn(Kl2,Kl))
     => member852475432509897056n_assn(aa_lis2797772165808192636n_assn(aa_lis4738365772350288766n_assn(append282499809098378956n_assn,Kl),aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,K),nil_Pr5671120429643327159n_assn)),Kl2) ) ).

% SuccD
tff(fact_1715_empty__Shift,axiom,
    ! [Kl2: set_li6341169313164406869_nat_o,K: fun_Pr6460944699146269935_nat_o] :
      ( member877585773093452190_nat_o(nil_fu2924554602572761077_nat_o,Kl2)
     => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(K),bNF_Gr2876145917579644042_nat_o(Kl2,nil_fu2924554602572761077_nat_o))
       => member877585773093452190_nat_o(nil_fu2924554602572761077_nat_o,bNF_Gr2819271973565845774_nat_o(Kl2,K)) ) ) ).

% empty_Shift
tff(fact_1716_empty__Shift,axiom,
    ! [Kl2: set_list_a,K: a] :
      ( member_list_a(nil_a,Kl2)
     => ( member_a2(K,bNF_Greatest_Succ_a(Kl2,nil_a))
       => member_list_a(nil_a,bNF_Greatest_Shift_a(Kl2,K)) ) ) ).

% empty_Shift
tff(fact_1717_empty__Shift,axiom,
    ! [Kl2: set_list_b,K: b] :
      ( member_list_b(nil_b,Kl2)
     => ( member_b2(K,bNF_Greatest_Succ_b(Kl2,nil_b))
       => member_list_b(nil_b,bNF_Greatest_Shift_b(Kl2,K)) ) ) ).

% empty_Shift
tff(fact_1718_empty__Shift,axiom,
    ! [Kl2: set_li5131720305576846103n_assn,K: produc6575502325842934193n_assn] :
      ( member852475432509897056n_assn(nil_Pr5671120429643327159n_assn,Kl2)
     => ( member7957490590177025114n_assn(K,bNF_Gr3216292445876487756n_assn(Kl2,nil_Pr5671120429643327159n_assn))
       => member852475432509897056n_assn(nil_Pr5671120429643327159n_assn,bNF_Gr4113829767105464016n_assn(Kl2,K)) ) ) ).

% empty_Shift
tff(fact_1719_empty__Shift,axiom,
    ! [Kl2: set_list_nat,K: nat] :
      ( member_list_nat(nil_nat,Kl2)
     => ( aa_set_nat_o(member_nat2(K),bNF_Gr6352880689984616693cc_nat(Kl2,nil_nat))
       => member_list_nat(nil_nat,bNF_Gr1872714664788909425ft_nat(Kl2,K)) ) ) ).

% empty_Shift
tff(fact_1720_empty__Shift,axiom,
    ! [Kl2: set_list_int,K: int] :
      ( member_list_int(nil_int,Kl2)
     => ( aa_set_int_o(member_int2(K),bNF_Gr6350390219475566417cc_int(Kl2,nil_int))
       => member_list_int(nil_int,bNF_Gr1870224194279859149ft_int(Kl2,K)) ) ) ).

% empty_Shift
tff(fact_1721_Succ__Shift,axiom,
    ! [Kl2: set_list_nat,K: nat,Kl: list_nat] : bNF_Gr6352880689984616693cc_nat(bNF_Gr1872714664788909425ft_nat(Kl2,K),Kl) = bNF_Gr6352880689984616693cc_nat(Kl2,aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,K),Kl)) ).

% Succ_Shift
tff(fact_1722_Succ__Shift,axiom,
    ! [Kl2: set_list_int,K: int,Kl: list_int] : bNF_Gr6350390219475566417cc_int(bNF_Gr1870224194279859149ft_int(Kl2,K),Kl) = bNF_Gr6350390219475566417cc_int(Kl2,aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,K),Kl)) ).

% Succ_Shift
tff(fact_1723_Succ__Shift,axiom,
    ! [Kl2: set_li5131720305576846103n_assn,K: produc6575502325842934193n_assn,Kl: list_P8527749157015355191n_assn] : bNF_Gr3216292445876487756n_assn(bNF_Gr4113829767105464016n_assn(Kl2,K),Kl) = bNF_Gr3216292445876487756n_assn(Kl2,aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,K),Kl)) ).

% Succ_Shift
tff(fact_1724_Sup__fin_Oinsert,axiom,
    ! [A2: set_assn,X: assn] :
      ( finite_finite_assn(A2)
     => ( ( A2 != bot_bot_set_assn )
       => ( lattic2150320897289308081n_assn(insert_assn(X,A2)) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),lattic2150320897289308081n_assn(A2)) ) ) ) ).

% Sup_fin.insert
tff(fact_1725_Sup__fin_Oinsert,axiom,
    ! [A2: set_set_nat,X: set_nat] :
      ( finite1152437895449049373et_nat(A2)
     => ( ( A2 != bot_bot_set_set_nat )
       => ( lattic3835124923745554447et_nat(insert_set_nat(X,A2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),lattic3835124923745554447et_nat(A2)) ) ) ) ).

% Sup_fin.insert
tff(fact_1726_Sup__fin_Oinsert,axiom,
    ! [A2: set_o,X: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ( A2 != bot_bot_set_o )
       => ( aa_set_o_o(lattic1508158080041050831_fin_o,insert_o((X),A2))
        <=> aa_o_o(aa_o_fun_o_o(sup_sup_o,(X)),aa_set_o_o(lattic1508158080041050831_fin_o,A2)) ) ) ) ).

% Sup_fin.insert
tff(fact_1727_Sup__fin_Oinsert,axiom,
    ! [A2: set_nat,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( A2 != bot_bot_set_nat )
       => ( lattic1093996805478795353in_nat(insert_nat2(X,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),lattic1093996805478795353in_nat(A2)) ) ) ) ).

% Sup_fin.insert
tff(fact_1728_Inf__fin_Oinsert,axiom,
    ! [A2: set_assn,X: assn] :
      ( finite_finite_assn(A2)
     => ( ( A2 != bot_bot_set_assn )
       => ( aa_set_assn_assn(lattic47131356835913163n_assn,insert_assn(X,A2)) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),aa_set_assn_assn(lattic47131356835913163n_assn,A2)) ) ) ) ).

% Inf_fin.insert
tff(fact_1729_Inf__fin_Oinsert,axiom,
    ! [A2: set_set_nat,X: set_nat] :
      ( finite1152437895449049373et_nat(A2)
     => ( ( A2 != bot_bot_set_set_nat )
       => ( aa_set6364576160234933248et_nat(lattic3014633134055518761et_nat,insert_set_nat(X,A2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),aa_set6364576160234933248et_nat(lattic3014633134055518761et_nat,A2)) ) ) ) ).

% Inf_fin.insert
tff(fact_1730_Inf__fin_Oinsert,axiom,
    ! [A2: set_Product_unit,X: product_unit] :
      ( finite4290736615968046902t_unit(A2)
     => ( ( A2 != bot_bo3957492148770167129t_unit )
       => ( aa_set6350468789499455260t_unit(lattic1263872656861969706t_unit,insert_Product_unit(X,A2)) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),aa_set6350468789499455260t_unit(lattic1263872656861969706t_unit,A2)) ) ) ) ).

% Inf_fin.insert
tff(fact_1731_Inf__fin_Oinsert,axiom,
    ! [A2: set_se7855581050983116737at_nat,X: set_Pr1261947904930325089at_nat] :
      ( finite9047747110432174090at_nat(A2)
     => ( ( A2 != bot_bo3083307316010499117at_nat )
       => ( aa_set3914448398961233820at_nat(lattic30941717366863870at_nat,insert9200635055090092081at_nat(X,A2)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),aa_set3914448398961233820at_nat(lattic30941717366863870at_nat,A2)) ) ) ) ).

% Inf_fin.insert
tff(fact_1732_Inf__fin_Oinsert,axiom,
    ! [A2: set_o,X: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ( A2 != bot_bot_set_o )
       => ( aa_set_o_o(lattic4107685809792843317_fin_o,insert_o((X),A2))
        <=> aa_o_o(aa_o_fun_o_o(inf_inf_o,(X)),aa_set_o_o(lattic4107685809792843317_fin_o,A2)) ) ) ) ).

% Inf_fin.insert
tff(fact_1733_Inf__fin_Oinsert,axiom,
    ! [A2: set_nat,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( A2 != bot_bot_set_nat )
       => ( aa_set_nat_nat(lattic5238388535129920115in_nat,insert_nat2(X,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),aa_set_nat_nat(lattic5238388535129920115in_nat,A2)) ) ) ) ).

% Inf_fin.insert
tff(fact_1734_Inf__fin__le__Sup__fin,axiom,
    ! [A2: set_o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ( A2 != bot_bot_set_o )
       => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,aa_set_o_o(lattic4107685809792843317_fin_o,A2)),aa_set_o_o(lattic1508158080041050831_fin_o,A2)) ) ) ).

% Inf_fin_le_Sup_fin
tff(fact_1735_Inf__fin__le__Sup__fin,axiom,
    ! [A2: set_assn] :
      ( finite_finite_assn(A2)
     => ( ( A2 != bot_bot_set_assn )
       => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_set_assn_assn(lattic47131356835913163n_assn,A2)),lattic2150320897289308081n_assn(A2)) ) ) ).

% Inf_fin_le_Sup_fin
tff(fact_1736_Inf__fin__le__Sup__fin,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( A2 != bot_bot_set_nat )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(lattic5238388535129920115in_nat,A2)),lattic1093996805478795353in_nat(A2)) ) ) ).

% Inf_fin_le_Sup_fin
tff(fact_1737_Inf__fin__le__Sup__fin,axiom,
    ! [A2: set_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( A2 != bot_bot_set_int )
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,lattic5235898064620869839in_int(A2)),lattic1091506334969745077in_int(A2)) ) ) ).

% Inf_fin_le_Sup_fin
tff(fact_1738_Inf__fin__le__Sup__fin,axiom,
    ! [A2: set_set_nat] :
      ( finite1152437895449049373et_nat(A2)
     => ( ( A2 != bot_bot_set_set_nat )
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set6364576160234933248et_nat(lattic3014633134055518761et_nat,A2)),lattic3835124923745554447et_nat(A2)) ) ) ).

% Inf_fin_le_Sup_fin
tff(fact_1739_Sup__fin_Osubset,axiom,
    ! [A2: set_assn,B2: set_assn] :
      ( finite_finite_assn(A2)
     => ( ( B2 != bot_bot_set_assn )
       => ( ord_less_eq_set_assn(B2,A2)
         => ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,lattic2150320897289308081n_assn(B2)),lattic2150320897289308081n_assn(A2)) = lattic2150320897289308081n_assn(A2) ) ) ) ) ).

% Sup_fin.subset
tff(fact_1740_Sup__fin_Osubset,axiom,
    ! [A2: set_set_nat,B2: set_set_nat] :
      ( finite1152437895449049373et_nat(A2)
     => ( ( B2 != bot_bot_set_set_nat )
       => ( ord_le6893508408891458716et_nat(B2,A2)
         => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,lattic3835124923745554447et_nat(B2)),lattic3835124923745554447et_nat(A2)) = lattic3835124923745554447et_nat(A2) ) ) ) ) ).

% Sup_fin.subset
tff(fact_1741_Sup__fin_Osubset,axiom,
    ! [A2: set_o,B2: set_o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ( B2 != bot_bot_set_o )
       => ( aa_set_o_o(ord_less_eq_set_o(B2),A2)
         => ( aa_o_o(aa_o_fun_o_o(sup_sup_o,aa_set_o_o(lattic1508158080041050831_fin_o,B2)),aa_set_o_o(lattic1508158080041050831_fin_o,A2))
          <=> aa_set_o_o(lattic1508158080041050831_fin_o,A2) ) ) ) ) ).

% Sup_fin.subset
tff(fact_1742_Sup__fin_Osubset,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( B2 != bot_bot_set_nat )
       => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B2),A2)
         => ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,lattic1093996805478795353in_nat(B2)),lattic1093996805478795353in_nat(A2)) = lattic1093996805478795353in_nat(A2) ) ) ) ) ).

% Sup_fin.subset
tff(fact_1743_Inf__fin_Osubset,axiom,
    ! [A2: set_assn,B2: set_assn] :
      ( finite_finite_assn(A2)
     => ( ( B2 != bot_bot_set_assn )
       => ( ord_less_eq_set_assn(B2,A2)
         => ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_set_assn_assn(lattic47131356835913163n_assn,B2)),aa_set_assn_assn(lattic47131356835913163n_assn,A2)) = aa_set_assn_assn(lattic47131356835913163n_assn,A2) ) ) ) ) ).

% Inf_fin.subset
tff(fact_1744_Inf__fin_Osubset,axiom,
    ! [A2: set_set_nat,B2: set_set_nat] :
      ( finite1152437895449049373et_nat(A2)
     => ( ( B2 != bot_bot_set_set_nat )
       => ( ord_le6893508408891458716et_nat(B2,A2)
         => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set6364576160234933248et_nat(lattic3014633134055518761et_nat,B2)),aa_set6364576160234933248et_nat(lattic3014633134055518761et_nat,A2)) = aa_set6364576160234933248et_nat(lattic3014633134055518761et_nat,A2) ) ) ) ) ).

% Inf_fin.subset
tff(fact_1745_Inf__fin_Osubset,axiom,
    ! [A2: set_Product_unit,B2: set_Product_unit] :
      ( finite4290736615968046902t_unit(A2)
     => ( ( B2 != bot_bo3957492148770167129t_unit )
       => ( ord_le3507040750410214029t_unit(B2,A2)
         => ( aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,aa_set6350468789499455260t_unit(lattic1263872656861969706t_unit,B2)),aa_set6350468789499455260t_unit(lattic1263872656861969706t_unit,A2)) = aa_set6350468789499455260t_unit(lattic1263872656861969706t_unit,A2) ) ) ) ) ).

% Inf_fin.subset
tff(fact_1746_Inf__fin_Osubset,axiom,
    ! [A2: set_se7855581050983116737at_nat,B2: set_se7855581050983116737at_nat] :
      ( finite9047747110432174090at_nat(A2)
     => ( ( B2 != bot_bo3083307316010499117at_nat )
       => ( ord_le2077887516847798113at_nat(B2,A2)
         => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set3914448398961233820at_nat(lattic30941717366863870at_nat,B2)),aa_set3914448398961233820at_nat(lattic30941717366863870at_nat,A2)) = aa_set3914448398961233820at_nat(lattic30941717366863870at_nat,A2) ) ) ) ) ).

% Inf_fin.subset
tff(fact_1747_Inf__fin_Osubset,axiom,
    ! [A2: set_o,B2: set_o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ( B2 != bot_bot_set_o )
       => ( aa_set_o_o(ord_less_eq_set_o(B2),A2)
         => ( aa_o_o(aa_o_fun_o_o(inf_inf_o,aa_set_o_o(lattic4107685809792843317_fin_o,B2)),aa_set_o_o(lattic4107685809792843317_fin_o,A2))
          <=> aa_set_o_o(lattic4107685809792843317_fin_o,A2) ) ) ) ) ).

% Inf_fin.subset
tff(fact_1748_Inf__fin_Osubset,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( B2 != bot_bot_set_nat )
       => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B2),A2)
         => ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,aa_set_nat_nat(lattic5238388535129920115in_nat,B2)),aa_set_nat_nat(lattic5238388535129920115in_nat,A2)) = aa_set_nat_nat(lattic5238388535129920115in_nat,A2) ) ) ) ) ).

% Inf_fin.subset
tff(fact_1749_Sup__fin_Oinsert__not__elem,axiom,
    ! [A2: set_fu3555266933746557007_nat_o,X: fun_Pr6460944699146269935_nat_o] :
      ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,A2)
     => ( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),A2)
       => ( ( A2 != bot_bo8632255151191677115_nat_o )
         => ( lattic7388253182610234406_nat_o(insert4923026906511862463_nat_o(X,A2)) = aa_fun2638059832795814268_nat_o(aa_fun6000889628616002998_nat_o(sup_su6210301778384944323_nat_o,X),lattic7388253182610234406_nat_o(A2)) ) ) ) ) ).

% Sup_fin.insert_not_elem
tff(fact_1750_Sup__fin_Oinsert__not__elem,axiom,
    ! [A2: set_assn,X: assn] :
      ( finite_finite_assn(A2)
     => ( ~ member_assn(X,A2)
       => ( ( A2 != bot_bot_set_assn )
         => ( lattic2150320897289308081n_assn(insert_assn(X,A2)) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),lattic2150320897289308081n_assn(A2)) ) ) ) ) ).

% Sup_fin.insert_not_elem
tff(fact_1751_Sup__fin_Oinsert__not__elem,axiom,
    ! [A2: set_set_nat,X: set_nat] :
      ( finite1152437895449049373et_nat(A2)
     => ( ~ member_set_nat(X,A2)
       => ( ( A2 != bot_bot_set_set_nat )
         => ( lattic3835124923745554447et_nat(insert_set_nat(X,A2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),lattic3835124923745554447et_nat(A2)) ) ) ) ) ).

% Sup_fin.insert_not_elem
tff(fact_1752_Sup__fin_Oinsert__not__elem,axiom,
    ! [A2: set_o,X: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ~ aa_set_o_o(member_o((X)),A2)
       => ( ( A2 != bot_bot_set_o )
         => ( aa_set_o_o(lattic1508158080041050831_fin_o,insert_o((X),A2))
          <=> aa_o_o(aa_o_fun_o_o(sup_sup_o,(X)),aa_set_o_o(lattic1508158080041050831_fin_o,A2)) ) ) ) ) ).

% Sup_fin.insert_not_elem
tff(fact_1753_Sup__fin_Oinsert__not__elem,axiom,
    ! [A2: set_nat,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ~ aa_set_nat_o(member_nat2(X),A2)
       => ( ( A2 != bot_bot_set_nat )
         => ( lattic1093996805478795353in_nat(insert_nat2(X,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),lattic1093996805478795353in_nat(A2)) ) ) ) ) ).

% Sup_fin.insert_not_elem
tff(fact_1754_Sup__fin_Oclosed,axiom,
    ! [A2: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,A2)
     => ( ( A2 != bot_bo8632255151191677115_nat_o )
       => ( ! [X2: fun_Pr6460944699146269935_nat_o,Y2: fun_Pr6460944699146269935_nat_o] : aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(aa_fun2638059832795814268_nat_o(aa_fun6000889628616002998_nat_o(sup_su6210301778384944323_nat_o,X2),Y2)),insert4923026906511862463_nat_o(X2,insert4923026906511862463_nat_o(Y2,bot_bo8632255151191677115_nat_o)))
         => aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(lattic7388253182610234406_nat_o(A2)),A2) ) ) ) ).

% Sup_fin.closed
tff(fact_1755_Sup__fin_Oclosed,axiom,
    ! [A2: set_assn] :
      ( finite_finite_assn(A2)
     => ( ( A2 != bot_bot_set_assn )
       => ( ! [X2: assn,Y2: assn] : member_assn(aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X2),Y2),insert_assn(X2,insert_assn(Y2,bot_bot_set_assn)))
         => member_assn(lattic2150320897289308081n_assn(A2),A2) ) ) ) ).

% Sup_fin.closed
tff(fact_1756_Sup__fin_Oclosed,axiom,
    ! [A2: set_set_nat] :
      ( finite1152437895449049373et_nat(A2)
     => ( ( A2 != bot_bot_set_set_nat )
       => ( ! [X2: set_nat,Y2: set_nat] : member_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X2),Y2),insert_set_nat(X2,insert_set_nat(Y2,bot_bot_set_set_nat)))
         => member_set_nat(lattic3835124923745554447et_nat(A2),A2) ) ) ) ).

% Sup_fin.closed
tff(fact_1757_Sup__fin_Oclosed,axiom,
    ! [A2: set_o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ( A2 != bot_bot_set_o )
       => ( ! [X2: $o,Y2: $o] : aa_set_o_o(member_o(aa_o_o(aa_o_fun_o_o(sup_sup_o,(X2)),(Y2))),insert_o((X2),insert_o((Y2),bot_bot_set_o)))
         => aa_set_o_o(member_o(aa_set_o_o(lattic1508158080041050831_fin_o,A2)),A2) ) ) ) ).

% Sup_fin.closed
tff(fact_1758_Sup__fin_Oclosed,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( A2 != bot_bot_set_nat )
       => ( ! [X2: nat,Y2: nat] : aa_set_nat_o(member_nat2(aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X2),Y2)),insert_nat2(X2,insert_nat2(Y2,bot_bot_set_nat)))
         => aa_set_nat_o(member_nat2(lattic1093996805478795353in_nat(A2)),A2) ) ) ) ).

% Sup_fin.closed
tff(fact_1759_order__refl,axiom,
    ! [X: assn] : aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),X) ).

% order_refl
tff(fact_1760_order__refl,axiom,
    ! [X: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),X) ).

% order_refl
tff(fact_1761_order__refl,axiom,
    ! [X: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),X) ).

% order_refl
tff(fact_1762_order__refl,axiom,
    ! [X: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),X) ).

% order_refl
tff(fact_1763_dual__order_Orefl,axiom,
    ! [A: assn] : aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),A) ).

% dual_order.refl
tff(fact_1764_dual__order_Orefl,axiom,
    ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),A) ).

% dual_order.refl
tff(fact_1765_dual__order_Orefl,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),A) ).

% dual_order.refl
tff(fact_1766_dual__order_Orefl,axiom,
    ! [A: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),A) ).

% dual_order.refl
tff(fact_1767_subsetI,axiom,
    ! [A2: set_fu3555266933746557007_nat_o,B2: set_fu3555266933746557007_nat_o] :
      ( ! [X2: fun_Pr6460944699146269935_nat_o] :
          ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X2),A2)
         => aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X2),B2) )
     => aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(A2),B2) ) ).

% subsetI
tff(fact_1768_subsetI,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(member_nat2(X2),A2)
         => aa_set_nat_o(member_nat2(X2),B2) )
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B2) ) ).

% subsetI
tff(fact_1769_subset__antisym,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B2),A2)
       => ( A2 = B2 ) ) ) ).

% subset_antisym
tff(fact_1770_insertCI,axiom,
    ! [A: $o,B2: set_o,B: $o] :
      ( ( ~ aa_set_o_o(member_o((A)),B2)
       => ( (A)
        <=> (B) ) )
     => aa_set_o_o(member_o((A)),insert_o((B),B2)) ) ).

% insertCI
tff(fact_1771_insertCI,axiom,
    ! [A: fun_Pr6460944699146269935_nat_o,B2: set_fu3555266933746557007_nat_o,B: fun_Pr6460944699146269935_nat_o] :
      ( ( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A),B2)
       => ( A = B ) )
     => aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A),insert4923026906511862463_nat_o(B,B2)) ) ).

% insertCI
tff(fact_1772_insert__iff,axiom,
    ! [A: $o,B: $o,A2: set_o] :
      ( aa_set_o_o(member_o((A)),insert_o((B),A2))
    <=> ( ( (A)
        <=> (B) )
        | aa_set_o_o(member_o((A)),A2) ) ) ).

% insert_iff
tff(fact_1773_insert__iff,axiom,
    ! [A: fun_Pr6460944699146269935_nat_o,B: fun_Pr6460944699146269935_nat_o,A2: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A),insert4923026906511862463_nat_o(B,A2))
    <=> ( ( A = B )
        | aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A),A2) ) ) ).

% insert_iff
tff(fact_1774_insert__absorb2,axiom,
    ! [X: $o,A2: set_o] : insert_o((X),insert_o((X),A2)) = insert_o((X),A2) ).

% insert_absorb2
tff(fact_1775_inf_Obounded__iff,axiom,
    ! [A: product_unit,B: product_unit,C2: product_unit] :
      ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,A),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,B),C2))
    <=> ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,A),B)
        & aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,A),C2) ) ) ).

% inf.bounded_iff
tff(fact_1776_inf_Obounded__iff,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B),C2))
    <=> ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),B)
        & aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),C2) ) ) ).

% inf.bounded_iff
tff(fact_1777_inf_Obounded__iff,axiom,
    ! [A: assn,B: assn,C2: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,B),C2))
    <=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
        & aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),C2) ) ) ).

% inf.bounded_iff
tff(fact_1778_inf_Obounded__iff,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,B),C2))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C2) ) ) ).

% inf.bounded_iff
tff(fact_1779_inf_Obounded__iff,axiom,
    ! [A: int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(inf_inf_int,B),C2))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C2) ) ) ).

% inf.bounded_iff
tff(fact_1780_inf_Obounded__iff,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B),C2))
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B)
        & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),C2) ) ) ).

% inf.bounded_iff
tff(fact_1781_le__inf__iff,axiom,
    ! [X: product_unit,Y: product_unit,Z: product_unit] :
      ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,X),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,Y),Z))
    <=> ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,X),Y)
        & aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,X),Z) ) ) ).

% le_inf_iff
tff(fact_1782_le__inf__iff,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y),Z))
    <=> ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X),Y)
        & aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X),Z) ) ) ).

% le_inf_iff
tff(fact_1783_le__inf__iff,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),Z))
    <=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),Y)
        & aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),Z) ) ) ).

% le_inf_iff
tff(fact_1784_le__inf__iff,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Y),Z))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Z) ) ) ).

% le_inf_iff
tff(fact_1785_le__inf__iff,axiom,
    ! [X: int,Y: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),aa_int_int(aa_int_fun_int_int(inf_inf_int,Y),Z))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Z) ) ) ).

% le_inf_iff
tff(fact_1786_le__inf__iff,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),Z))
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Y)
        & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Z) ) ) ).

% le_inf_iff
tff(fact_1787_le__sup__iff,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y)),Z)
    <=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),Z)
        & aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,Y),Z) ) ) ).

% le_sup_iff
tff(fact_1788_le__sup__iff,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y)),Z)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Z)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Z) ) ) ).

% le_sup_iff
tff(fact_1789_le__sup__iff,axiom,
    ! [X: int,Y: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y)),Z)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Z)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Z) ) ) ).

% le_sup_iff
tff(fact_1790_le__sup__iff,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y)),Z)
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Z)
        & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Y),Z) ) ) ).

% le_sup_iff
tff(fact_1791_sup_Obounded__iff,axiom,
    ! [B: assn,C2: assn,A: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,B),C2)),A)
    <=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),A)
        & aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,C2),A) ) ) ).

% sup.bounded_iff
tff(fact_1792_sup_Obounded__iff,axiom,
    ! [B: nat,C2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,B),C2)),A)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),A) ) ) ).

% sup.bounded_iff
tff(fact_1793_sup_Obounded__iff,axiom,
    ! [B: int,C2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,B),C2)),A)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),A) ) ) ).

% sup.bounded_iff
tff(fact_1794_sup_Obounded__iff,axiom,
    ! [B: set_nat,C2: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B),C2)),A)
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A)
        & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C2),A) ) ) ).

% sup.bounded_iff
tff(fact_1795_empty__subsetI,axiom,
    ! [A2: set_o] : aa_set_o_o(ord_less_eq_set_o(bot_bot_set_o),A2) ).

% empty_subsetI
tff(fact_1796_empty__subsetI,axiom,
    ! [A2: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,bot_bot_set_nat),A2) ).

% empty_subsetI
tff(fact_1797_subset__empty,axiom,
    ! [A2: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(A2),bot_bot_set_o)
    <=> ( A2 = bot_bot_set_o ) ) ).

% subset_empty
tff(fact_1798_subset__empty,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),bot_bot_set_nat)
    <=> ( A2 = bot_bot_set_nat ) ) ).

% subset_empty
tff(fact_1799_singletonI,axiom,
    ! [A: fun_Pr6460944699146269935_nat_o] : aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A),insert4923026906511862463_nat_o(A,bot_bo8632255151191677115_nat_o)) ).

% singletonI
tff(fact_1800_singletonI,axiom,
    ! [A: $o] : aa_set_o_o(member_o((A)),insert_o((A),bot_bot_set_o)) ).

% singletonI
tff(fact_1801_singletonI,axiom,
    ! [A: nat] : aa_set_nat_o(member_nat2(A),insert_nat2(A,bot_bot_set_nat)) ).

% singletonI
tff(fact_1802_insert__subset,axiom,
    ! [X: $o,A2: set_o,B2: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(insert_o((X),A2)),B2)
    <=> ( aa_set_o_o(member_o((X)),B2)
        & aa_set_o_o(ord_less_eq_set_o(A2),B2) ) ) ).

% insert_subset
tff(fact_1803_insert__subset,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,A2: set_fu3555266933746557007_nat_o,B2: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(insert4923026906511862463_nat_o(X,A2)),B2)
    <=> ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),B2)
        & aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(A2),B2) ) ) ).

% insert_subset
tff(fact_1804_insert__subset,axiom,
    ! [X: nat,A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,insert_nat2(X,A2)),B2)
    <=> ( aa_set_nat_o(member_nat2(X),B2)
        & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B2) ) ) ).

% insert_subset
tff(fact_1805_Int__subset__iff,axiom,
    ! [C3: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,C3),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2))
    <=> ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,C3),A2)
        & aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,C3),B2) ) ) ).

% Int_subset_iff
tff(fact_1806_Int__subset__iff,axiom,
    ! [C3: set_nat,A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2))
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C3),A2)
        & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C3),B2) ) ) ).

% Int_subset_iff
tff(fact_1807_Int__insert__right__if1,axiom,
    ! [A: $o,A2: set_o,B2: set_o] :
      ( aa_set_o_o(member_o((A)),A2)
     => ( inf_inf_set_o(A2,insert_o((A),B2)) = insert_o((A),inf_inf_set_o(A2,B2)) ) ) ).

% Int_insert_right_if1
tff(fact_1808_Int__insert__right__if1,axiom,
    ! [A: fun_Pr6460944699146269935_nat_o,A2: set_fu3555266933746557007_nat_o,B2: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A),A2)
     => ( inf_in3376754634429691197_nat_o(A2,insert4923026906511862463_nat_o(A,B2)) = insert4923026906511862463_nat_o(A,inf_in3376754634429691197_nat_o(A2,B2)) ) ) ).

% Int_insert_right_if1
tff(fact_1809_Int__insert__right__if1,axiom,
    ! [A: nat,A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(member_nat2(A),A2)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),insert_nat2(A,B2)) = insert_nat2(A,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2)) ) ) ).

% Int_insert_right_if1
tff(fact_1810_Int__insert__right__if1,axiom,
    ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( member8440522571783428010at_nat(A,A2)
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),insert8211810215607154385at_nat(A,B2)) = insert8211810215607154385at_nat(A,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2)) ) ) ).

% Int_insert_right_if1
tff(fact_1811_Int__insert__right__if0,axiom,
    ! [A: $o,A2: set_o,B2: set_o] :
      ( ~ aa_set_o_o(member_o((A)),A2)
     => ( inf_inf_set_o(A2,insert_o((A),B2)) = inf_inf_set_o(A2,B2) ) ) ).

% Int_insert_right_if0
tff(fact_1812_Int__insert__right__if0,axiom,
    ! [A: fun_Pr6460944699146269935_nat_o,A2: set_fu3555266933746557007_nat_o,B2: set_fu3555266933746557007_nat_o] :
      ( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A),A2)
     => ( inf_in3376754634429691197_nat_o(A2,insert4923026906511862463_nat_o(A,B2)) = inf_in3376754634429691197_nat_o(A2,B2) ) ) ).

% Int_insert_right_if0
tff(fact_1813_Int__insert__right__if0,axiom,
    ! [A: nat,A2: set_nat,B2: set_nat] :
      ( ~ aa_set_nat_o(member_nat2(A),A2)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),insert_nat2(A,B2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) ) ) ).

% Int_insert_right_if0
tff(fact_1814_Int__insert__right__if0,axiom,
    ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ~ member8440522571783428010at_nat(A,A2)
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),insert8211810215607154385at_nat(A,B2)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2) ) ) ).

% Int_insert_right_if0
tff(fact_1815_insert__inter__insert,axiom,
    ! [A: $o,A2: set_o,B2: set_o] : inf_inf_set_o(insert_o((A),A2),insert_o((A),B2)) = insert_o((A),inf_inf_set_o(A2,B2)) ).

% insert_inter_insert
tff(fact_1816_insert__inter__insert,axiom,
    ! [A: nat,A2: set_nat,B2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,insert_nat2(A,A2)),insert_nat2(A,B2)) = insert_nat2(A,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2)) ).

% insert_inter_insert
tff(fact_1817_insert__inter__insert,axiom,
    ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,insert8211810215607154385at_nat(A,A2)),insert8211810215607154385at_nat(A,B2)) = insert8211810215607154385at_nat(A,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2)) ).

% insert_inter_insert
tff(fact_1818_Int__insert__left__if1,axiom,
    ! [A: $o,C3: set_o,B2: set_o] :
      ( aa_set_o_o(member_o((A)),C3)
     => ( inf_inf_set_o(insert_o((A),B2),C3) = insert_o((A),inf_inf_set_o(B2,C3)) ) ) ).

% Int_insert_left_if1
tff(fact_1819_Int__insert__left__if1,axiom,
    ! [A: fun_Pr6460944699146269935_nat_o,C3: set_fu3555266933746557007_nat_o,B2: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A),C3)
     => ( inf_in3376754634429691197_nat_o(insert4923026906511862463_nat_o(A,B2),C3) = insert4923026906511862463_nat_o(A,inf_in3376754634429691197_nat_o(B2,C3)) ) ) ).

% Int_insert_left_if1
tff(fact_1820_Int__insert__left__if1,axiom,
    ! [A: nat,C3: set_nat,B2: set_nat] :
      ( aa_set_nat_o(member_nat2(A),C3)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,insert_nat2(A,B2)),C3) = insert_nat2(A,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),C3)) ) ) ).

% Int_insert_left_if1
tff(fact_1821_Int__insert__left__if1,axiom,
    ! [A: product_prod_nat_nat,C3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( member8440522571783428010at_nat(A,C3)
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,insert8211810215607154385at_nat(A,B2)),C3) = insert8211810215607154385at_nat(A,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B2),C3)) ) ) ).

% Int_insert_left_if1
tff(fact_1822_Int__insert__left__if0,axiom,
    ! [A: $o,C3: set_o,B2: set_o] :
      ( ~ aa_set_o_o(member_o((A)),C3)
     => ( inf_inf_set_o(insert_o((A),B2),C3) = inf_inf_set_o(B2,C3) ) ) ).

% Int_insert_left_if0
tff(fact_1823_Int__insert__left__if0,axiom,
    ! [A: fun_Pr6460944699146269935_nat_o,C3: set_fu3555266933746557007_nat_o,B2: set_fu3555266933746557007_nat_o] :
      ( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A),C3)
     => ( inf_in3376754634429691197_nat_o(insert4923026906511862463_nat_o(A,B2),C3) = inf_in3376754634429691197_nat_o(B2,C3) ) ) ).

% Int_insert_left_if0
tff(fact_1824_Int__insert__left__if0,axiom,
    ! [A: nat,C3: set_nat,B2: set_nat] :
      ( ~ aa_set_nat_o(member_nat2(A),C3)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,insert_nat2(A,B2)),C3) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),C3) ) ) ).

% Int_insert_left_if0
tff(fact_1825_Int__insert__left__if0,axiom,
    ! [A: product_prod_nat_nat,C3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ~ member8440522571783428010at_nat(A,C3)
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,insert8211810215607154385at_nat(A,B2)),C3) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B2),C3) ) ) ).

% Int_insert_left_if0
tff(fact_1826_Un__subset__iff,axiom,
    ! [A2: set_nat,B2: set_nat,C3: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)),C3)
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),C3)
        & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B2),C3) ) ) ).

% Un_subset_iff
tff(fact_1827_Un__insert__left,axiom,
    ! [A: $o,B2: set_o,C3: set_o] : sup_sup_set_o(insert_o((A),B2),C3) = insert_o((A),sup_sup_set_o(B2,C3)) ).

% Un_insert_left
tff(fact_1828_Un__insert__left,axiom,
    ! [A: nat,B2: set_nat,C3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,insert_nat2(A,B2)),C3) = insert_nat2(A,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B2),C3)) ).

% Un_insert_left
tff(fact_1829_Un__insert__right,axiom,
    ! [A2: set_o,A: $o,B2: set_o] : sup_sup_set_o(A2,insert_o((A),B2)) = insert_o((A),sup_sup_set_o(A2,B2)) ).

% Un_insert_right
tff(fact_1830_Un__insert__right,axiom,
    ! [A2: set_nat,A: nat,B2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),insert_nat2(A,B2)) = insert_nat2(A,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)) ).

% Un_insert_right
tff(fact_1831_psubsetI,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B2)
     => ( ( A2 != B2 )
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A2),B2) ) ) ).

% psubsetI
tff(fact_1832_singleton__insert__inj__eq,axiom,
    ! [B: $o,A: $o,A2: set_o] :
      ( ( insert_o((B),bot_bot_set_o) = insert_o((A),A2) )
    <=> ( ( (A)
        <=> (B) )
        & aa_set_o_o(ord_less_eq_set_o(A2),insert_o((B),bot_bot_set_o)) ) ) ).

% singleton_insert_inj_eq
tff(fact_1833_singleton__insert__inj__eq,axiom,
    ! [B: nat,A: nat,A2: set_nat] :
      ( ( insert_nat2(B,bot_bot_set_nat) = insert_nat2(A,A2) )
    <=> ( ( A = B )
        & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),insert_nat2(B,bot_bot_set_nat)) ) ) ).

% singleton_insert_inj_eq
tff(fact_1834_singleton__insert__inj__eq_H,axiom,
    ! [A: $o,A2: set_o,B: $o] :
      ( ( insert_o((A),A2) = insert_o((B),bot_bot_set_o) )
    <=> ( ( (A)
        <=> (B) )
        & aa_set_o_o(ord_less_eq_set_o(A2),insert_o((B),bot_bot_set_o)) ) ) ).

% singleton_insert_inj_eq'
tff(fact_1835_singleton__insert__inj__eq_H,axiom,
    ! [A: nat,A2: set_nat,B: nat] :
      ( ( insert_nat2(A,A2) = insert_nat2(B,bot_bot_set_nat) )
    <=> ( ( A = B )
        & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),insert_nat2(B,bot_bot_set_nat)) ) ) ).

% singleton_insert_inj_eq'
tff(fact_1836_disjoint__insert_I2_J,axiom,
    ! [A2: set_fu3555266933746557007_nat_o,B: fun_Pr6460944699146269935_nat_o,B2: set_fu3555266933746557007_nat_o] :
      ( ( bot_bo8632255151191677115_nat_o = inf_in3376754634429691197_nat_o(A2,insert4923026906511862463_nat_o(B,B2)) )
    <=> ( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(B),A2)
        & ( bot_bo8632255151191677115_nat_o = inf_in3376754634429691197_nat_o(A2,B2) ) ) ) ).

% disjoint_insert(2)
tff(fact_1837_disjoint__insert_I2_J,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( bot_bo2099793752762293965at_nat = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),insert8211810215607154385at_nat(B,B2)) )
    <=> ( ~ member8440522571783428010at_nat(B,A2)
        & ( bot_bo2099793752762293965at_nat = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2) ) ) ) ).

% disjoint_insert(2)
tff(fact_1838_disjoint__insert_I2_J,axiom,
    ! [A2: set_o,B: $o,B2: set_o] :
      ( ( bot_bot_set_o = inf_inf_set_o(A2,insert_o((B),B2)) )
    <=> ( ~ aa_set_o_o(member_o((B)),A2)
        & ( bot_bot_set_o = inf_inf_set_o(A2,B2) ) ) ) ).

% disjoint_insert(2)
tff(fact_1839_disjoint__insert_I2_J,axiom,
    ! [A2: set_nat,B: nat,B2: set_nat] :
      ( ( bot_bot_set_nat = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),insert_nat2(B,B2)) )
    <=> ( ~ aa_set_nat_o(member_nat2(B),A2)
        & ( bot_bot_set_nat = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) ) ) ) ).

% disjoint_insert(2)
tff(fact_1840_disjoint__insert_I1_J,axiom,
    ! [B2: set_fu3555266933746557007_nat_o,A: fun_Pr6460944699146269935_nat_o,A2: set_fu3555266933746557007_nat_o] :
      ( ( inf_in3376754634429691197_nat_o(B2,insert4923026906511862463_nat_o(A,A2)) = bot_bo8632255151191677115_nat_o )
    <=> ( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A),B2)
        & ( inf_in3376754634429691197_nat_o(B2,A2) = bot_bo8632255151191677115_nat_o ) ) ) ).

% disjoint_insert(1)
tff(fact_1841_disjoint__insert_I1_J,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B2),insert8211810215607154385at_nat(A,A2)) = bot_bo2099793752762293965at_nat )
    <=> ( ~ member8440522571783428010at_nat(A,B2)
        & ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B2),A2) = bot_bo2099793752762293965at_nat ) ) ) ).

% disjoint_insert(1)
tff(fact_1842_disjoint__insert_I1_J,axiom,
    ! [B2: set_o,A: $o,A2: set_o] :
      ( ( inf_inf_set_o(B2,insert_o((A),A2)) = bot_bot_set_o )
    <=> ( ~ aa_set_o_o(member_o((A)),B2)
        & ( inf_inf_set_o(B2,A2) = bot_bot_set_o ) ) ) ).

% disjoint_insert(1)
tff(fact_1843_disjoint__insert_I1_J,axiom,
    ! [B2: set_nat,A: nat,A2: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),insert_nat2(A,A2)) = bot_bot_set_nat )
    <=> ( ~ aa_set_nat_o(member_nat2(A),B2)
        & ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),A2) = bot_bot_set_nat ) ) ) ).

% disjoint_insert(1)
tff(fact_1844_insert__disjoint_I2_J,axiom,
    ! [A: fun_Pr6460944699146269935_nat_o,A2: set_fu3555266933746557007_nat_o,B2: set_fu3555266933746557007_nat_o] :
      ( ( bot_bo8632255151191677115_nat_o = inf_in3376754634429691197_nat_o(insert4923026906511862463_nat_o(A,A2),B2) )
    <=> ( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A),B2)
        & ( bot_bo8632255151191677115_nat_o = inf_in3376754634429691197_nat_o(A2,B2) ) ) ) ).

% insert_disjoint(2)
tff(fact_1845_insert__disjoint_I2_J,axiom,
    ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( bot_bo2099793752762293965at_nat = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,insert8211810215607154385at_nat(A,A2)),B2) )
    <=> ( ~ member8440522571783428010at_nat(A,B2)
        & ( bot_bo2099793752762293965at_nat = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2) ) ) ) ).

% insert_disjoint(2)
tff(fact_1846_insert__disjoint_I2_J,axiom,
    ! [A: $o,A2: set_o,B2: set_o] :
      ( ( bot_bot_set_o = inf_inf_set_o(insert_o((A),A2),B2) )
    <=> ( ~ aa_set_o_o(member_o((A)),B2)
        & ( bot_bot_set_o = inf_inf_set_o(A2,B2) ) ) ) ).

% insert_disjoint(2)
tff(fact_1847_insert__disjoint_I2_J,axiom,
    ! [A: nat,A2: set_nat,B2: set_nat] :
      ( ( bot_bot_set_nat = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,insert_nat2(A,A2)),B2) )
    <=> ( ~ aa_set_nat_o(member_nat2(A),B2)
        & ( bot_bot_set_nat = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) ) ) ) ).

% insert_disjoint(2)
tff(fact_1848_insert__disjoint_I1_J,axiom,
    ! [A: fun_Pr6460944699146269935_nat_o,A2: set_fu3555266933746557007_nat_o,B2: set_fu3555266933746557007_nat_o] :
      ( ( inf_in3376754634429691197_nat_o(insert4923026906511862463_nat_o(A,A2),B2) = bot_bo8632255151191677115_nat_o )
    <=> ( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A),B2)
        & ( inf_in3376754634429691197_nat_o(A2,B2) = bot_bo8632255151191677115_nat_o ) ) ) ).

% insert_disjoint(1)
tff(fact_1849_insert__disjoint_I1_J,axiom,
    ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,insert8211810215607154385at_nat(A,A2)),B2) = bot_bo2099793752762293965at_nat )
    <=> ( ~ member8440522571783428010at_nat(A,B2)
        & ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2) = bot_bo2099793752762293965at_nat ) ) ) ).

% insert_disjoint(1)
tff(fact_1850_insert__disjoint_I1_J,axiom,
    ! [A: $o,A2: set_o,B2: set_o] :
      ( ( inf_inf_set_o(insert_o((A),A2),B2) = bot_bot_set_o )
    <=> ( ~ aa_set_o_o(member_o((A)),B2)
        & ( inf_inf_set_o(A2,B2) = bot_bot_set_o ) ) ) ).

% insert_disjoint(1)
tff(fact_1851_insert__disjoint_I1_J,axiom,
    ! [A: nat,A2: set_nat,B2: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,insert_nat2(A,A2)),B2) = bot_bot_set_nat )
    <=> ( ~ aa_set_nat_o(member_nat2(A),B2)
        & ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) = bot_bot_set_nat ) ) ) ).

% insert_disjoint(1)
tff(fact_1852_Max__singleton,axiom,
    ! [X: $o] :
      ( aa_set_o_o(lattic1921953407002678535_Max_o,insert_o((X),bot_bot_set_o))
    <=> (X) ) ).

% Max_singleton
tff(fact_1853_Max__singleton,axiom,
    ! [X: nat] : lattic8265883725875713057ax_nat(insert_nat2(X,bot_bot_set_nat)) = X ).

% Max_singleton
tff(fact_1854_Min__singleton,axiom,
    ! [X: $o] :
      ( aa_set_o_o(lattic1973801136483472281_Min_o,insert_o((X),bot_bot_set_o))
    <=> (X) ) ).

% Min_singleton
tff(fact_1855_Min__singleton,axiom,
    ! [X: nat] : lattic8721135487736765967in_nat(insert_nat2(X,bot_bot_set_nat)) = X ).

% Min_singleton
tff(fact_1856_Sup__fin_Osingleton,axiom,
    ! [X: $o] :
      ( aa_set_o_o(lattic1508158080041050831_fin_o,insert_o((X),bot_bot_set_o))
    <=> (X) ) ).

% Sup_fin.singleton
tff(fact_1857_Sup__fin_Osingleton,axiom,
    ! [X: nat] : lattic1093996805478795353in_nat(insert_nat2(X,bot_bot_set_nat)) = X ).

% Sup_fin.singleton
tff(fact_1858_Inf__fin_Osingleton,axiom,
    ! [X: $o] :
      ( aa_set_o_o(lattic4107685809792843317_fin_o,insert_o((X),bot_bot_set_o))
    <=> (X) ) ).

% Inf_fin.singleton
tff(fact_1859_Inf__fin_Osingleton,axiom,
    ! [X: nat] : aa_set_nat_nat(lattic5238388535129920115in_nat,insert_nat2(X,bot_bot_set_nat)) = X ).

% Inf_fin.singleton
tff(fact_1860_Max_Obounded__iff,axiom,
    ! [A2: set_o,X: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ( A2 != bot_bot_set_o )
       => ( aa_o_o(aa_o_fun_o_o(ord_less_eq_o,aa_set_o_o(lattic1921953407002678535_Max_o,A2)),(X))
        <=> ! [X3: $o] :
              ( aa_set_o_o(member_o((X3)),A2)
             => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(X3)),(X)) ) ) ) ) ).

% Max.bounded_iff
tff(fact_1861_Max_Obounded__iff,axiom,
    ! [A2: set_nat,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( A2 != bot_bot_set_nat )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,lattic8265883725875713057ax_nat(A2)),X)
        <=> ! [X3: nat] :
              ( aa_set_nat_o(member_nat2(X3),A2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),X) ) ) ) ) ).

% Max.bounded_iff
tff(fact_1862_Max_Obounded__iff,axiom,
    ! [A2: set_int,X: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( A2 != bot_bot_set_int )
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,lattic8263393255366662781ax_int(A2)),X)
        <=> ! [X3: int] :
              ( aa_set_int_o(member_int2(X3),A2)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),X) ) ) ) ) ).

% Max.bounded_iff
tff(fact_1863_Min_Obounded__iff,axiom,
    ! [A2: set_o,X: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ( A2 != bot_bot_set_o )
       => ( aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(X)),aa_set_o_o(lattic1973801136483472281_Min_o,A2))
        <=> ! [X3: $o] :
              ( aa_set_o_o(member_o((X3)),A2)
             => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(X)),(X3)) ) ) ) ) ).

% Min.bounded_iff
tff(fact_1864_Min_Obounded__iff,axiom,
    ! [A2: set_nat,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( A2 != bot_bot_set_nat )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),lattic8721135487736765967in_nat(A2))
        <=> ! [X3: nat] :
              ( aa_set_nat_o(member_nat2(X3),A2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),X3) ) ) ) ) ).

% Min.bounded_iff
tff(fact_1865_Min_Obounded__iff,axiom,
    ! [A2: set_int,X: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( A2 != bot_bot_set_int )
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),lattic8718645017227715691in_int(A2))
        <=> ! [X3: int] :
              ( aa_set_int_o(member_int2(X3),A2)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),X3) ) ) ) ) ).

% Min.bounded_iff
tff(fact_1866_in__mono,axiom,
    ! [A2: set_fu3555266933746557007_nat_o,B2: set_fu3555266933746557007_nat_o,X: fun_Pr6460944699146269935_nat_o] :
      ( aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(A2),B2)
     => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),A2)
       => aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),B2) ) ) ).

% in_mono
tff(fact_1867_in__mono,axiom,
    ! [A2: set_nat,B2: set_nat,X: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B2)
     => ( aa_set_nat_o(member_nat2(X),A2)
       => aa_set_nat_o(member_nat2(X),B2) ) ) ).

% in_mono
tff(fact_1868_insertE,axiom,
    ! [A: $o,B: $o,A2: set_o] :
      ( aa_set_o_o(member_o((A)),insert_o((B),A2))
     => ( ( (A)
        <=> ~ (B) )
       => aa_set_o_o(member_o((A)),A2) ) ) ).

% insertE
tff(fact_1869_insertE,axiom,
    ! [A: fun_Pr6460944699146269935_nat_o,B: fun_Pr6460944699146269935_nat_o,A2: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A),insert4923026906511862463_nat_o(B,A2))
     => ( ( A != B )
       => aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A),A2) ) ) ).

% insertE
tff(fact_1870_subsetD,axiom,
    ! [A2: set_fu3555266933746557007_nat_o,B2: set_fu3555266933746557007_nat_o,C2: fun_Pr6460944699146269935_nat_o] :
      ( aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(A2),B2)
     => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C2),A2)
       => aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C2),B2) ) ) ).

% subsetD
tff(fact_1871_subsetD,axiom,
    ! [A2: set_nat,B2: set_nat,C2: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B2)
     => ( aa_set_nat_o(member_nat2(C2),A2)
       => aa_set_nat_o(member_nat2(C2),B2) ) ) ).

% subsetD
tff(fact_1872_insertI1,axiom,
    ! [A: $o,B2: set_o] : aa_set_o_o(member_o((A)),insert_o((A),B2)) ).

% insertI1
tff(fact_1873_insertI1,axiom,
    ! [A: fun_Pr6460944699146269935_nat_o,B2: set_fu3555266933746557007_nat_o] : aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A),insert4923026906511862463_nat_o(A,B2)) ).

% insertI1
tff(fact_1874_insertI2,axiom,
    ! [A: $o,B2: set_o,B: $o] :
      ( aa_set_o_o(member_o((A)),B2)
     => aa_set_o_o(member_o((A)),insert_o((B),B2)) ) ).

% insertI2
tff(fact_1875_insertI2,axiom,
    ! [A: fun_Pr6460944699146269935_nat_o,B2: set_fu3555266933746557007_nat_o,B: fun_Pr6460944699146269935_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A),B2)
     => aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A),insert4923026906511862463_nat_o(B,B2)) ) ).

% insertI2
tff(fact_1876_equalityE,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( A2 = B2 )
     => ~ ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B2)
         => ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B2),A2) ) ) ).

% equalityE
tff(fact_1877_subset__eq,axiom,
    ! [A2: set_fu3555266933746557007_nat_o,B2: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(A2),B2)
    <=> ! [X3: fun_Pr6460944699146269935_nat_o] :
          ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X3),A2)
         => aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X3),B2) ) ) ).

% subset_eq
tff(fact_1878_subset__eq,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B2)
    <=> ! [X3: nat] :
          ( aa_set_nat_o(member_nat2(X3),A2)
         => aa_set_nat_o(member_nat2(X3),B2) ) ) ).

% subset_eq
tff(fact_1879_equalityD1,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( A2 = B2 )
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B2) ) ).

% equalityD1
tff(fact_1880_equalityD2,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( A2 = B2 )
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B2),A2) ) ).

% equalityD2
tff(fact_1881_Set_Oset__insert,axiom,
    ! [X: $o,A2: set_o] :
      ( aa_set_o_o(member_o((X)),A2)
     => ~ ! [B5: set_o] :
            ( ( A2 = insert_o((X),B5) )
           => aa_set_o_o(member_o((X)),B5) ) ) ).

% Set.set_insert
tff(fact_1882_Set_Oset__insert,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,A2: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),A2)
     => ~ ! [B5: set_fu3555266933746557007_nat_o] :
            ( ( A2 = insert4923026906511862463_nat_o(X,B5) )
           => aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),B5) ) ) ).

% Set.set_insert
tff(fact_1883_subset__iff,axiom,
    ! [A2: set_fu3555266933746557007_nat_o,B2: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(A2),B2)
    <=> ! [T2: fun_Pr6460944699146269935_nat_o] :
          ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(T2),A2)
         => aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(T2),B2) ) ) ).

% subset_iff
tff(fact_1884_subset__iff,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B2)
    <=> ! [T2: nat] :
          ( aa_set_nat_o(member_nat2(T2),A2)
         => aa_set_nat_o(member_nat2(T2),B2) ) ) ).

% subset_iff
tff(fact_1885_insert__mono,axiom,
    ! [C3: set_o,D: set_o,A: $o] :
      ( aa_set_o_o(ord_less_eq_set_o(C3),D)
     => aa_set_o_o(ord_less_eq_set_o(insert_o((A),C3)),insert_o((A),D)) ) ).

% insert_mono
tff(fact_1886_insert__mono,axiom,
    ! [C3: set_nat,D: set_nat,A: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C3),D)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,insert_nat2(A,C3)),insert_nat2(A,D)) ) ).

% insert_mono
tff(fact_1887_subset__refl,axiom,
    ! [A2: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),A2) ).

% subset_refl
tff(fact_1888_Collect__mono,axiom,
    ! [Pa: fun_fu5135969500543783852at_o_o,Q: fun_fu5135969500543783852at_o_o] :
      ( ! [X2: fun_Pr6460944699146269935_nat_o] :
          ( aa_fun1675921005893230629at_o_o(Pa,X2)
         => aa_fun1675921005893230629at_o_o(Q,X2) )
     => aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(collec6407654713642994138_nat_o(Pa)),collec6407654713642994138_nat_o(Q)) ) ).

% Collect_mono
tff(fact_1889_Collect__mono,axiom,
    ! [Pa: fun_Pr4796943543331071514_nat_o,Q: fun_Pr4796943543331071514_nat_o] :
      ( ! [X2: product_prod_nat_nat] :
          ( aa_Pro6150637604445964947_nat_o(Pa,X2)
         => aa_Pro6150637604445964947_nat_o(Q,X2) )
     => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,collec3392354462482085612at_nat(Pa)),collec3392354462482085612at_nat(Q)) ) ).

% Collect_mono
tff(fact_1890_Collect__mono,axiom,
    ! [Pa: fun_nat_o,Q: fun_nat_o] :
      ( ! [X2: nat] :
          ( aa_nat_o(Pa,X2)
         => aa_nat_o(Q,X2) )
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,collect_nat(Pa)),collect_nat(Q)) ) ).

% Collect_mono
tff(fact_1891_insert__ident,axiom,
    ! [X: $o,A2: set_o,B2: set_o] :
      ( ~ aa_set_o_o(member_o((X)),A2)
     => ( ~ aa_set_o_o(member_o((X)),B2)
       => ( ( insert_o((X),A2) = insert_o((X),B2) )
        <=> ( A2 = B2 ) ) ) ) ).

% insert_ident
tff(fact_1892_insert__ident,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,A2: set_fu3555266933746557007_nat_o,B2: set_fu3555266933746557007_nat_o] :
      ( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),A2)
     => ( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),B2)
       => ( ( insert4923026906511862463_nat_o(X,A2) = insert4923026906511862463_nat_o(X,B2) )
        <=> ( A2 = B2 ) ) ) ) ).

% insert_ident
tff(fact_1893_subset__trans,axiom,
    ! [A2: set_nat,B2: set_nat,C3: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B2),C3)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),C3) ) ) ).

% subset_trans
tff(fact_1894_insert__absorb,axiom,
    ! [A: $o,A2: set_o] :
      ( aa_set_o_o(member_o((A)),A2)
     => ( insert_o((A),A2) = A2 ) ) ).

% insert_absorb
tff(fact_1895_insert__absorb,axiom,
    ! [A: fun_Pr6460944699146269935_nat_o,A2: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A),A2)
     => ( insert4923026906511862463_nat_o(A,A2) = A2 ) ) ).

% insert_absorb
tff(fact_1896_insert__eq__iff,axiom,
    ! [A: $o,A2: set_o,B: $o,B2: set_o] :
      ( ~ aa_set_o_o(member_o((A)),A2)
     => ( ~ aa_set_o_o(member_o((B)),B2)
       => ( ( insert_o((A),A2) = insert_o((B),B2) )
        <=> $ite(
              (A) = (B),
              A2 = B2,
              ? [C4: set_o] :
                ( ( A2 = insert_o((B),C4) )
                & ~ aa_set_o_o(member_o((B)),C4)
                & ( B2 = insert_o((A),C4) )
                & ~ aa_set_o_o(member_o((A)),C4) ) ) ) ) ) ).

% insert_eq_iff
tff(fact_1897_insert__eq__iff,axiom,
    ! [A: fun_Pr6460944699146269935_nat_o,A2: set_fu3555266933746557007_nat_o,B: fun_Pr6460944699146269935_nat_o,B2: set_fu3555266933746557007_nat_o] :
      ( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A),A2)
     => ( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(B),B2)
       => ( ( insert4923026906511862463_nat_o(A,A2) = insert4923026906511862463_nat_o(B,B2) )
        <=> $ite(
              A = B,
              A2 = B2,
              ? [C4: set_fu3555266933746557007_nat_o] :
                ( ( A2 = insert4923026906511862463_nat_o(B,C4) )
                & ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(B),C4)
                & ( B2 = insert4923026906511862463_nat_o(A,C4) )
                & ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A),C4) ) ) ) ) ) ).

% insert_eq_iff
tff(fact_1898_set__eq__subset,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( A2 = B2 )
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B2)
        & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B2),A2) ) ) ).

% set_eq_subset
tff(fact_1899_subset__insert,axiom,
    ! [X: $o,A2: set_o,B2: set_o] :
      ( ~ aa_set_o_o(member_o((X)),A2)
     => ( aa_set_o_o(ord_less_eq_set_o(A2),insert_o((X),B2))
      <=> aa_set_o_o(ord_less_eq_set_o(A2),B2) ) ) ).

% subset_insert
tff(fact_1900_subset__insert,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,A2: set_fu3555266933746557007_nat_o,B2: set_fu3555266933746557007_nat_o] :
      ( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),A2)
     => ( aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(A2),insert4923026906511862463_nat_o(X,B2))
      <=> aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(A2),B2) ) ) ).

% subset_insert
tff(fact_1901_subset__insert,axiom,
    ! [X: nat,A2: set_nat,B2: set_nat] :
      ( ~ aa_set_nat_o(member_nat2(X),A2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),insert_nat2(X,B2))
      <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B2) ) ) ).

% subset_insert
tff(fact_1902_insert__commute,axiom,
    ! [X: $o,Y: $o,A2: set_o] : insert_o((X),insert_o((Y),A2)) = insert_o((Y),insert_o((X),A2)) ).

% insert_commute
tff(fact_1903_subset__insertI,axiom,
    ! [B2: set_o,A: $o] : aa_set_o_o(ord_less_eq_set_o(B2),insert_o((A),B2)) ).

% subset_insertI
tff(fact_1904_subset__insertI,axiom,
    ! [B2: set_nat,A: nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B2),insert_nat2(A,B2)) ).

% subset_insertI
tff(fact_1905_subset__insertI2,axiom,
    ! [A2: set_o,B2: set_o,B: $o] :
      ( aa_set_o_o(ord_less_eq_set_o(A2),B2)
     => aa_set_o_o(ord_less_eq_set_o(A2),insert_o((B),B2)) ) ).

% subset_insertI2
tff(fact_1906_subset__insertI2,axiom,
    ! [A2: set_nat,B2: set_nat,B: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B2)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),insert_nat2(B,B2)) ) ).

% subset_insertI2
tff(fact_1907_Collect__mono__iff,axiom,
    ! [Pa: fun_fu5135969500543783852at_o_o,Q: fun_fu5135969500543783852at_o_o] :
      ( aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(collec6407654713642994138_nat_o(Pa)),collec6407654713642994138_nat_o(Q))
    <=> ! [X3: fun_Pr6460944699146269935_nat_o] :
          ( aa_fun1675921005893230629at_o_o(Pa,X3)
         => aa_fun1675921005893230629at_o_o(Q,X3) ) ) ).

% Collect_mono_iff
tff(fact_1908_Collect__mono__iff,axiom,
    ! [Pa: fun_Pr4796943543331071514_nat_o,Q: fun_Pr4796943543331071514_nat_o] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,collec3392354462482085612at_nat(Pa)),collec3392354462482085612at_nat(Q))
    <=> ! [X3: product_prod_nat_nat] :
          ( aa_Pro6150637604445964947_nat_o(Pa,X3)
         => aa_Pro6150637604445964947_nat_o(Q,X3) ) ) ).

% Collect_mono_iff
tff(fact_1909_Collect__mono__iff,axiom,
    ! [Pa: fun_nat_o,Q: fun_nat_o] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,collect_nat(Pa)),collect_nat(Q))
    <=> ! [X3: nat] :
          ( aa_nat_o(Pa,X3)
         => aa_nat_o(Q,X3) ) ) ).

% Collect_mono_iff
tff(fact_1910_mk__disjoint__insert,axiom,
    ! [A: $o,A2: set_o] :
      ( aa_set_o_o(member_o((A)),A2)
     => ? [B5: set_o] :
          ( ( A2 = insert_o((A),B5) )
          & ~ aa_set_o_o(member_o((A)),B5) ) ) ).

% mk_disjoint_insert
tff(fact_1911_mk__disjoint__insert,axiom,
    ! [A: fun_Pr6460944699146269935_nat_o,A2: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A),A2)
     => ? [B5: set_fu3555266933746557007_nat_o] :
          ( ( A2 = insert4923026906511862463_nat_o(A,B5) )
          & ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A),B5) ) ) ).

% mk_disjoint_insert
tff(fact_1912_nle__le,axiom,
    ! [A: nat,B: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
        & ( B != A ) ) ) ).

% nle_le
tff(fact_1913_nle__le,axiom,
    ! [A: int,B: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
        & ( B != A ) ) ) ).

% nle_le
tff(fact_1914_le__cases3,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
       => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Z) )
     => ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Z) )
       => ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Z)
           => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),Y) )
         => ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),Y)
             => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X) )
           => ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Z)
               => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),X) )
             => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),X)
                 => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y) ) ) ) ) ) ) ).

% le_cases3
tff(fact_1915_le__cases3,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
       => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Z) )
     => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),X)
         => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Z) )
       => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Z)
           => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),Y) )
         => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),Y)
             => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),X) )
           => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Z)
               => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),X) )
             => ~ ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),X)
                 => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y) ) ) ) ) ) ) ).

% le_cases3
tff(fact_1916_order__class_Oorder__eq__iff,axiom,
    ! [X: assn,Y: assn] :
      ( ( X = Y )
    <=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),Y)
        & aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,Y),X) ) ) ).

% order_class.order_eq_iff
tff(fact_1917_order__class_Oorder__eq__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( X = Y )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X) ) ) ).

% order_class.order_eq_iff
tff(fact_1918_order__class_Oorder__eq__iff,axiom,
    ! [X: int,Y: int] :
      ( ( X = Y )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),X) ) ) ).

% order_class.order_eq_iff
tff(fact_1919_order__class_Oorder__eq__iff,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( X = Y )
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Y)
        & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Y),X) ) ) ).

% order_class.order_eq_iff
tff(fact_1920_ord__eq__le__trans,axiom,
    ! [A: assn,B: assn,C2: assn] :
      ( ( A = B )
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),C2)
       => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),C2) ) ) ).

% ord_eq_le_trans
tff(fact_1921_ord__eq__le__trans,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( A = B )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C2) ) ) ).

% ord_eq_le_trans
tff(fact_1922_ord__eq__le__trans,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( A = B )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C2) ) ) ).

% ord_eq_le_trans
tff(fact_1923_ord__eq__le__trans,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat] :
      ( ( A = B )
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),C2)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),C2) ) ) ).

% ord_eq_le_trans
tff(fact_1924_ord__le__eq__trans,axiom,
    ! [A: assn,B: assn,C2: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
     => ( ( B = C2 )
       => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),C2) ) ) ).

% ord_le_eq_trans
tff(fact_1925_ord__le__eq__trans,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( ( B = C2 )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C2) ) ) ).

% ord_le_eq_trans
tff(fact_1926_ord__le__eq__trans,axiom,
    ! [A: int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( ( B = C2 )
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C2) ) ) ).

% ord_le_eq_trans
tff(fact_1927_ord__le__eq__trans,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B)
     => ( ( B = C2 )
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),C2) ) ) ).

% ord_le_eq_trans
tff(fact_1928_order__antisym,axiom,
    ! [X: assn,Y: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),Y)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,Y),X)
       => ( X = Y ) ) ) ).

% order_antisym
tff(fact_1929_order__antisym,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X)
       => ( X = Y ) ) ) ).

% order_antisym
tff(fact_1930_order__antisym,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),X)
       => ( X = Y ) ) ) ).

% order_antisym
tff(fact_1931_order__antisym,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Y)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Y),X)
       => ( X = Y ) ) ) ).

% order_antisym
tff(fact_1932_order_Otrans,axiom,
    ! [A: assn,B: assn,C2: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),C2)
       => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),C2) ) ) ).

% order.trans
tff(fact_1933_order_Otrans,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C2) ) ) ).

% order.trans
tff(fact_1934_order_Otrans,axiom,
    ! [A: int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C2) ) ) ).

% order.trans
tff(fact_1935_order_Otrans,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),C2)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),C2) ) ) ).

% order.trans
tff(fact_1936_order__trans,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),Y)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,Y),Z)
       => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),Z) ) ) ).

% order_trans
tff(fact_1937_order__trans,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Z)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Z) ) ) ).

% order_trans
tff(fact_1938_order__trans,axiom,
    ! [X: int,Y: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Z) ) ) ).

% order_trans
tff(fact_1939_order__trans,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Y)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Y),Z)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Z) ) ) ).

% order_trans
tff(fact_1940_linorder__wlog,axiom,
    ! [Pa: fun_nat_fun_nat_o,A: nat,B: nat] :
      ( ! [A3: nat,B3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A3),B3)
         => aa_nat_o(aa_nat_fun_nat_o(Pa,A3),B3) )
     => ( ! [A3: nat,B3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(Pa,B3),A3)
           => aa_nat_o(aa_nat_fun_nat_o(Pa,A3),B3) )
       => aa_nat_o(aa_nat_fun_nat_o(Pa,A),B) ) ) ).

% linorder_wlog
tff(fact_1941_linorder__wlog,axiom,
    ! [Pa: fun_int_fun_int_o,A: int,B: int] :
      ( ! [A3: int,B3: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A3),B3)
         => aa_int_o(aa_int_fun_int_o(Pa,A3),B3) )
     => ( ! [A3: int,B3: int] :
            ( aa_int_o(aa_int_fun_int_o(Pa,B3),A3)
           => aa_int_o(aa_int_fun_int_o(Pa,A3),B3) )
       => aa_int_o(aa_int_fun_int_o(Pa,A),B) ) ) ).

% linorder_wlog
tff(fact_1942_dual__order_Oeq__iff,axiom,
    ! [A: assn,B: assn] :
      ( ( A = B )
    <=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),A)
        & aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B) ) ) ).

% dual_order.eq_iff
tff(fact_1943_dual__order_Oeq__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ) ).

% dual_order.eq_iff
tff(fact_1944_dual__order_Oeq__iff,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).

% dual_order.eq_iff
tff(fact_1945_dual__order_Oeq__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( A = B )
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A)
        & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B) ) ) ).

% dual_order.eq_iff
tff(fact_1946_dual__order_Oantisym,axiom,
    ! [B: assn,A: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),A)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
       => ( A = B ) ) ) ).

% dual_order.antisym
tff(fact_1947_dual__order_Oantisym,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
       => ( A = B ) ) ) ).

% dual_order.antisym
tff(fact_1948_dual__order_Oantisym,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
       => ( A = B ) ) ) ).

% dual_order.antisym
tff(fact_1949_dual__order_Oantisym,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B)
       => ( A = B ) ) ) ).

% dual_order.antisym
tff(fact_1950_dual__order_Otrans,axiom,
    ! [B: assn,A: assn,C2: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),A)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,C2),B)
       => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,C2),A) ) ) ).

% dual_order.trans
tff(fact_1951_dual__order_Otrans,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),A) ) ) ).

% dual_order.trans
tff(fact_1952_dual__order_Otrans,axiom,
    ! [B: int,A: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),A) ) ) ).

% dual_order.trans
tff(fact_1953_dual__order_Otrans,axiom,
    ! [B: set_nat,A: set_nat,C2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C2),B)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C2),A) ) ) ).

% dual_order.trans
tff(fact_1954_antisym,axiom,
    ! [A: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),A)
       => ( A = B ) ) ) ).

% antisym
tff(fact_1955_antisym,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
       => ( A = B ) ) ) ).

% antisym
tff(fact_1956_antisym,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
       => ( A = B ) ) ) ).

% antisym
tff(fact_1957_antisym,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A)
       => ( A = B ) ) ) ).

% antisym
tff(fact_1958_Orderings_Oorder__eq__iff,axiom,
    ! [A: assn,B: assn] :
      ( ( A = B )
    <=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
        & aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),A) ) ) ).

% Orderings.order_eq_iff
tff(fact_1959_Orderings_Oorder__eq__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A) ) ) ).

% Orderings.order_eq_iff
tff(fact_1960_Orderings_Oorder__eq__iff,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ) ).

% Orderings.order_eq_iff
tff(fact_1961_Orderings_Oorder__eq__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( A = B )
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B)
        & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A) ) ) ).

% Orderings.order_eq_iff
tff(fact_1962_order__subst1,axiom,
    ! [A: assn,F: fun_assn_assn,B: assn,C2: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),aa_assn_assn(F,B))
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),C2)
       => ( ! [X2: assn,Y2: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X2),Y2)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(F,X2)),aa_assn_assn(F,Y2)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),aa_assn_assn(F,C2)) ) ) ) ).

% order_subst1
tff(fact_1963_order__subst1,axiom,
    ! [A: assn,F: fun_nat_assn,B: nat,C2: nat] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),aa_nat_assn(F,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C2)
       => ( ! [X2: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Y2)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_nat_assn(F,X2)),aa_nat_assn(F,Y2)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),aa_nat_assn(F,C2)) ) ) ) ).

% order_subst1
tff(fact_1964_order__subst1,axiom,
    ! [A: assn,F: fun_int_assn,B: int,C2: int] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),aa_int_assn(F,B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C2)
       => ( ! [X2: int,Y2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X2),Y2)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_int_assn(F,X2)),aa_int_assn(F,Y2)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),aa_int_assn(F,C2)) ) ) ) ).

% order_subst1
tff(fact_1965_order__subst1,axiom,
    ! [A: nat,F: fun_assn_nat,B: assn,C2: assn] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_assn_nat(F,B))
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),C2)
       => ( ! [X2: assn,Y2: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X2),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_assn_nat(F,X2)),aa_assn_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_assn_nat(F,C2)) ) ) ) ).

% order_subst1
tff(fact_1966_order__subst1,axiom,
    ! [A: nat,F: fun_nat_nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(F,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C2)
       => ( ! [X2: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,X2)),aa_nat_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(F,C2)) ) ) ) ).

% order_subst1
tff(fact_1967_order__subst1,axiom,
    ! [A: nat,F: fun_int_nat,B: int,C2: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_int_nat(F,B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C2)
       => ( ! [X2: int,Y2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X2),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,X2)),aa_int_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_int_nat(F,C2)) ) ) ) ).

% order_subst1
tff(fact_1968_order__subst1,axiom,
    ! [A: int,F: fun_assn_int,B: assn,C2: assn] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_assn_int(F,B))
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),C2)
       => ( ! [X2: assn,Y2: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X2),Y2)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_assn_int(F,X2)),aa_assn_int(F,Y2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_assn_int(F,C2)) ) ) ) ).

% order_subst1
tff(fact_1969_order__subst1,axiom,
    ! [A: int,F: fun_nat_int,B: nat,C2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(F,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C2)
       => ( ! [X2: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Y2)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,X2)),aa_nat_int(F,Y2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(F,C2)) ) ) ) ).

% order_subst1
tff(fact_1970_order__subst1,axiom,
    ! [A: int,F: fun_int_int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(F,B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C2)
       => ( ! [X2: int,Y2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X2),Y2)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(F,X2)),aa_int_int(F,Y2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(F,C2)) ) ) ) ).

% order_subst1
tff(fact_1971_order__subst1,axiom,
    ! [A: assn,F: fun_set_nat_assn,B: set_nat,C2: set_nat] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),aa_set_nat_assn(F,B))
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),C2)
       => ( ! [X2: set_nat,Y2: set_nat] :
              ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X2),Y2)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_set_nat_assn(F,X2)),aa_set_nat_assn(F,Y2)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),aa_set_nat_assn(F,C2)) ) ) ) ).

% order_subst1
tff(fact_1972_order__subst2,axiom,
    ! [A: assn,B: assn,F: fun_assn_assn,C2: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(F,B)),C2)
       => ( ! [X2: assn,Y2: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X2),Y2)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(F,X2)),aa_assn_assn(F,Y2)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(F,A)),C2) ) ) ) ).

% order_subst2
tff(fact_1973_order__subst2,axiom,
    ! [A: assn,B: assn,F: fun_assn_nat,C2: nat] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_assn_nat(F,B)),C2)
       => ( ! [X2: assn,Y2: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X2),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_assn_nat(F,X2)),aa_assn_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_assn_nat(F,A)),C2) ) ) ) ).

% order_subst2
tff(fact_1974_order__subst2,axiom,
    ! [A: assn,B: assn,F: fun_assn_int,C2: int] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_assn_int(F,B)),C2)
       => ( ! [X2: assn,Y2: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X2),Y2)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_assn_int(F,X2)),aa_assn_int(F,Y2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_assn_int(F,A)),C2) ) ) ) ).

% order_subst2
tff(fact_1975_order__subst2,axiom,
    ! [A: nat,B: nat,F: fun_nat_assn,C2: assn] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_nat_assn(F,B)),C2)
       => ( ! [X2: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Y2)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_nat_assn(F,X2)),aa_nat_assn(F,Y2)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_nat_assn(F,A)),C2) ) ) ) ).

% order_subst2
tff(fact_1976_order__subst2,axiom,
    ! [A: nat,B: nat,F: fun_nat_nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,B)),C2)
       => ( ! [X2: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,X2)),aa_nat_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,A)),C2) ) ) ) ).

% order_subst2
tff(fact_1977_order__subst2,axiom,
    ! [A: nat,B: nat,F: fun_nat_int,C2: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,B)),C2)
       => ( ! [X2: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Y2)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,X2)),aa_nat_int(F,Y2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,A)),C2) ) ) ) ).

% order_subst2
tff(fact_1978_order__subst2,axiom,
    ! [A: int,B: int,F: fun_int_assn,C2: assn] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_int_assn(F,B)),C2)
       => ( ! [X2: int,Y2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X2),Y2)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_int_assn(F,X2)),aa_int_assn(F,Y2)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_int_assn(F,A)),C2) ) ) ) ).

% order_subst2
tff(fact_1979_order__subst2,axiom,
    ! [A: int,B: int,F: fun_int_nat,C2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,B)),C2)
       => ( ! [X2: int,Y2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X2),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,X2)),aa_int_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,A)),C2) ) ) ) ).

% order_subst2
tff(fact_1980_order__subst2,axiom,
    ! [A: int,B: int,F: fun_int_int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(F,B)),C2)
       => ( ! [X2: int,Y2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X2),Y2)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(F,X2)),aa_int_int(F,Y2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(F,A)),C2) ) ) ) ).

% order_subst2
tff(fact_1981_order__subst2,axiom,
    ! [A: assn,B: assn,F: fun_assn_set_nat,C2: set_nat] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_assn_set_nat(F,B)),C2)
       => ( ! [X2: assn,Y2: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X2),Y2)
             => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_assn_set_nat(F,X2)),aa_assn_set_nat(F,Y2)) )
         => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_assn_set_nat(F,A)),C2) ) ) ) ).

% order_subst2
tff(fact_1982_ord__eq__le__eq__trans,axiom,
    ! [A: assn,B: assn,C2: assn,D2: assn] :
      ( ( A = B )
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),C2)
       => ( ( C2 = D2 )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),D2) ) ) ) ).

% ord_eq_le_eq_trans
tff(fact_1983_ord__eq__le__eq__trans,axiom,
    ! [A: nat,B: nat,C2: nat,D2: nat] :
      ( ( A = B )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C2)
       => ( ( C2 = D2 )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),D2) ) ) ) ).

% ord_eq_le_eq_trans
tff(fact_1984_ord__eq__le__eq__trans,axiom,
    ! [A: int,B: int,C2: int,D2: int] :
      ( ( A = B )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C2)
       => ( ( C2 = D2 )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),D2) ) ) ) ).

% ord_eq_le_eq_trans
tff(fact_1985_ord__eq__le__eq__trans,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat,D2: set_nat] :
      ( ( A = B )
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),C2)
       => ( ( C2 = D2 )
         => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),D2) ) ) ) ).

% ord_eq_le_eq_trans
tff(fact_1986_order__eq__refl,axiom,
    ! [X: assn,Y: assn] :
      ( ( X = Y )
     => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),Y) ) ).

% order_eq_refl
tff(fact_1987_order__eq__refl,axiom,
    ! [X: nat,Y: nat] :
      ( ( X = Y )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y) ) ).

% order_eq_refl
tff(fact_1988_order__eq__refl,axiom,
    ! [X: int,Y: int] :
      ( ( X = Y )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y) ) ).

% order_eq_refl
tff(fact_1989_order__eq__refl,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( X = Y )
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Y) ) ).

% order_eq_refl
tff(fact_1990_subset__Collect__conv,axiom,
    ! [S: set_fu3555266933746557007_nat_o,Pa: fun_fu5135969500543783852at_o_o] :
      ( aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(S),collec6407654713642994138_nat_o(Pa))
    <=> ! [X3: fun_Pr6460944699146269935_nat_o] :
          ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X3),S)
         => aa_fun1675921005893230629at_o_o(Pa,X3) ) ) ).

% subset_Collect_conv
tff(fact_1991_subset__Collect__conv,axiom,
    ! [S: set_Pr1261947904930325089at_nat,Pa: fun_Pr4796943543331071514_nat_o] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,S),collec3392354462482085612at_nat(Pa))
    <=> ! [X3: product_prod_nat_nat] :
          ( member8440522571783428010at_nat(X3,S)
         => aa_Pro6150637604445964947_nat_o(Pa,X3) ) ) ).

% subset_Collect_conv
tff(fact_1992_subset__Collect__conv,axiom,
    ! [S: set_nat,Pa: fun_nat_o] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S),collect_nat(Pa))
    <=> ! [X3: nat] :
          ( aa_set_nat_o(member_nat2(X3),S)
         => aa_nat_o(Pa,X3) ) ) ).

% subset_Collect_conv
tff(fact_1993_linorder__linear,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
      | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X) ) ).

% linorder_linear
tff(fact_1994_linorder__linear,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
      | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),X) ) ).

% linorder_linear
tff(fact_1995_ord__eq__le__subst,axiom,
    ! [A: assn,F: fun_assn_assn,B: assn,C2: assn] :
      ( ( A = aa_assn_assn(F,B) )
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),C2)
       => ( ! [X2: assn,Y2: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X2),Y2)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(F,X2)),aa_assn_assn(F,Y2)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),aa_assn_assn(F,C2)) ) ) ) ).

% ord_eq_le_subst
tff(fact_1996_ord__eq__le__subst,axiom,
    ! [A: nat,F: fun_assn_nat,B: assn,C2: assn] :
      ( ( A = aa_assn_nat(F,B) )
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),C2)
       => ( ! [X2: assn,Y2: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X2),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_assn_nat(F,X2)),aa_assn_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_assn_nat(F,C2)) ) ) ) ).

% ord_eq_le_subst
tff(fact_1997_ord__eq__le__subst,axiom,
    ! [A: int,F: fun_assn_int,B: assn,C2: assn] :
      ( ( A = aa_assn_int(F,B) )
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),C2)
       => ( ! [X2: assn,Y2: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X2),Y2)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_assn_int(F,X2)),aa_assn_int(F,Y2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_assn_int(F,C2)) ) ) ) ).

% ord_eq_le_subst
tff(fact_1998_ord__eq__le__subst,axiom,
    ! [A: assn,F: fun_nat_assn,B: nat,C2: nat] :
      ( ( A = aa_nat_assn(F,B) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C2)
       => ( ! [X2: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Y2)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_nat_assn(F,X2)),aa_nat_assn(F,Y2)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),aa_nat_assn(F,C2)) ) ) ) ).

% ord_eq_le_subst
tff(fact_1999_ord__eq__le__subst,axiom,
    ! [A: nat,F: fun_nat_nat,B: nat,C2: nat] :
      ( ( A = aa_nat_nat(F,B) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C2)
       => ( ! [X2: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,X2)),aa_nat_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(F,C2)) ) ) ) ).

% ord_eq_le_subst
tff(fact_2000_ord__eq__le__subst,axiom,
    ! [A: int,F: fun_nat_int,B: nat,C2: nat] :
      ( ( A = aa_nat_int(F,B) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C2)
       => ( ! [X2: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Y2)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,X2)),aa_nat_int(F,Y2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(F,C2)) ) ) ) ).

% ord_eq_le_subst
tff(fact_2001_ord__eq__le__subst,axiom,
    ! [A: assn,F: fun_int_assn,B: int,C2: int] :
      ( ( A = aa_int_assn(F,B) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C2)
       => ( ! [X2: int,Y2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X2),Y2)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_int_assn(F,X2)),aa_int_assn(F,Y2)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),aa_int_assn(F,C2)) ) ) ) ).

% ord_eq_le_subst
tff(fact_2002_ord__eq__le__subst,axiom,
    ! [A: nat,F: fun_int_nat,B: int,C2: int] :
      ( ( A = aa_int_nat(F,B) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C2)
       => ( ! [X2: int,Y2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X2),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,X2)),aa_int_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_int_nat(F,C2)) ) ) ) ).

% ord_eq_le_subst
tff(fact_2003_ord__eq__le__subst,axiom,
    ! [A: int,F: fun_int_int,B: int,C2: int] :
      ( ( A = aa_int_int(F,B) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C2)
       => ( ! [X2: int,Y2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X2),Y2)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(F,X2)),aa_int_int(F,Y2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(F,C2)) ) ) ) ).

% ord_eq_le_subst
tff(fact_2004_ord__eq__le__subst,axiom,
    ! [A: set_nat,F: fun_assn_set_nat,B: assn,C2: assn] :
      ( ( A = aa_assn_set_nat(F,B) )
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),C2)
       => ( ! [X2: assn,Y2: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X2),Y2)
             => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_assn_set_nat(F,X2)),aa_assn_set_nat(F,Y2)) )
         => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),aa_assn_set_nat(F,C2)) ) ) ) ).

% ord_eq_le_subst
tff(fact_2005_ord__le__eq__subst,axiom,
    ! [A: assn,B: assn,F: fun_assn_assn,C2: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
     => ( ( aa_assn_assn(F,B) = C2 )
       => ( ! [X2: assn,Y2: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X2),Y2)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(F,X2)),aa_assn_assn(F,Y2)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(F,A)),C2) ) ) ) ).

% ord_le_eq_subst
tff(fact_2006_ord__le__eq__subst,axiom,
    ! [A: assn,B: assn,F: fun_assn_nat,C2: nat] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
     => ( ( aa_assn_nat(F,B) = C2 )
       => ( ! [X2: assn,Y2: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X2),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_assn_nat(F,X2)),aa_assn_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_assn_nat(F,A)),C2) ) ) ) ).

% ord_le_eq_subst
tff(fact_2007_ord__le__eq__subst,axiom,
    ! [A: assn,B: assn,F: fun_assn_int,C2: int] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
     => ( ( aa_assn_int(F,B) = C2 )
       => ( ! [X2: assn,Y2: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X2),Y2)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_assn_int(F,X2)),aa_assn_int(F,Y2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_assn_int(F,A)),C2) ) ) ) ).

% ord_le_eq_subst
tff(fact_2008_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F: fun_nat_assn,C2: assn] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( ( aa_nat_assn(F,B) = C2 )
       => ( ! [X2: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Y2)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_nat_assn(F,X2)),aa_nat_assn(F,Y2)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_nat_assn(F,A)),C2) ) ) ) ).

% ord_le_eq_subst
tff(fact_2009_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F: fun_nat_nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( ( aa_nat_nat(F,B) = C2 )
       => ( ! [X2: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,X2)),aa_nat_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,A)),C2) ) ) ) ).

% ord_le_eq_subst
tff(fact_2010_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F: fun_nat_int,C2: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( ( aa_nat_int(F,B) = C2 )
       => ( ! [X2: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Y2)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,X2)),aa_nat_int(F,Y2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,A)),C2) ) ) ) ).

% ord_le_eq_subst
tff(fact_2011_ord__le__eq__subst,axiom,
    ! [A: int,B: int,F: fun_int_assn,C2: assn] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( ( aa_int_assn(F,B) = C2 )
       => ( ! [X2: int,Y2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X2),Y2)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_int_assn(F,X2)),aa_int_assn(F,Y2)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_int_assn(F,A)),C2) ) ) ) ).

% ord_le_eq_subst
tff(fact_2012_ord__le__eq__subst,axiom,
    ! [A: int,B: int,F: fun_int_nat,C2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( ( aa_int_nat(F,B) = C2 )
       => ( ! [X2: int,Y2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X2),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,X2)),aa_int_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,A)),C2) ) ) ) ).

% ord_le_eq_subst
tff(fact_2013_ord__le__eq__subst,axiom,
    ! [A: int,B: int,F: fun_int_int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( ( aa_int_int(F,B) = C2 )
       => ( ! [X2: int,Y2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X2),Y2)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(F,X2)),aa_int_int(F,Y2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(F,A)),C2) ) ) ) ).

% ord_le_eq_subst
tff(fact_2014_ord__le__eq__subst,axiom,
    ! [A: assn,B: assn,F: fun_assn_set_nat,C2: set_nat] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
     => ( ( aa_assn_set_nat(F,B) = C2 )
       => ( ! [X2: assn,Y2: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X2),Y2)
             => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_assn_set_nat(F,X2)),aa_assn_set_nat(F,Y2)) )
         => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_assn_set_nat(F,A)),C2) ) ) ) ).

% ord_le_eq_subst
tff(fact_2015_linorder__le__cases,axiom,
    ! [X: nat,Y: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X) ) ).

% linorder_le_cases
tff(fact_2016_linorder__le__cases,axiom,
    ! [X: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),X) ) ).

% linorder_le_cases
tff(fact_2017_order__antisym__conv,axiom,
    ! [Y: assn,X: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,Y),X)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),Y)
      <=> ( X = Y ) ) ) ).

% order_antisym_conv
tff(fact_2018_order__antisym__conv,axiom,
    ! [Y: nat,X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
      <=> ( X = Y ) ) ) ).

% order_antisym_conv
tff(fact_2019_order__antisym__conv,axiom,
    ! [Y: int,X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),X)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
      <=> ( X = Y ) ) ) ).

% order_antisym_conv
tff(fact_2020_order__antisym__conv,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Y),X)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Y)
      <=> ( X = Y ) ) ) ).

% order_antisym_conv
tff(fact_2021_subset__singletonD,axiom,
    ! [A2: set_o,X: $o] :
      ( aa_set_o_o(ord_less_eq_set_o(A2),insert_o((X),bot_bot_set_o))
     => ( ( A2 = bot_bot_set_o )
        | ( A2 = insert_o((X),bot_bot_set_o) ) ) ) ).

% subset_singletonD
tff(fact_2022_subset__singletonD,axiom,
    ! [A2: set_nat,X: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),insert_nat2(X,bot_bot_set_nat))
     => ( ( A2 = bot_bot_set_nat )
        | ( A2 = insert_nat2(X,bot_bot_set_nat) ) ) ) ).

% subset_singletonD
tff(fact_2023_subset__singleton__iff,axiom,
    ! [X5: set_o,A: $o] :
      ( aa_set_o_o(ord_less_eq_set_o(X5),insert_o((A),bot_bot_set_o))
    <=> ( ( X5 = bot_bot_set_o )
        | ( X5 = insert_o((A),bot_bot_set_o) ) ) ) ).

% subset_singleton_iff
tff(fact_2024_subset__singleton__iff,axiom,
    ! [X5: set_nat,A: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X5),insert_nat2(A,bot_bot_set_nat))
    <=> ( ( X5 = bot_bot_set_nat )
        | ( X5 = insert_nat2(A,bot_bot_set_nat) ) ) ) ).

% subset_singleton_iff
tff(fact_2025_finite__ranking__induct,axiom,
    ! [S: set_fu3555266933746557007_nat_o,Pa: fun_se7459991829846211660at_o_o,F: fun_fu3504661237974713170_o_nat] :
      ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,S)
     => ( aa_set5091322549519198405at_o_o(Pa,bot_bo8632255151191677115_nat_o)
       => ( ! [X2: fun_Pr6460944699146269935_nat_o,S2: set_fu3555266933746557007_nat_o] :
              ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,S2)
             => ( ! [Y4: fun_Pr6460944699146269935_nat_o] :
                    ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Y4),S2)
                   => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_fun767828674014743683_o_nat(F,Y4)),aa_fun767828674014743683_o_nat(F,X2)) )
               => ( aa_set5091322549519198405at_o_o(Pa,S2)
                 => aa_set5091322549519198405at_o_o(Pa,insert4923026906511862463_nat_o(X2,S2)) ) ) )
         => aa_set5091322549519198405at_o_o(Pa,S) ) ) ) ).

% finite_ranking_induct
tff(fact_2026_finite__ranking__induct,axiom,
    ! [S: set_o,Pa: fun_set_o_o,F: fun_o_nat] :
      ( aa_set_o_o(finite_finite_o,S)
     => ( aa_set_o_o(Pa,bot_bot_set_o)
       => ( ! [X2: $o,S2: set_o] :
              ( aa_set_o_o(finite_finite_o,S2)
             => ( ! [Y4: $o] :
                    ( aa_set_o_o(member_o((Y4)),S2)
                   => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_o_nat(F,(Y4))),aa_o_nat(F,(X2))) )
               => ( aa_set_o_o(Pa,S2)
                 => aa_set_o_o(Pa,insert_o((X2),S2)) ) ) )
         => aa_set_o_o(Pa,S) ) ) ) ).

% finite_ranking_induct
tff(fact_2027_finite__ranking__induct,axiom,
    ! [S: set_nat,Pa: fun_set_nat_o,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( aa_set_nat_o(Pa,bot_bot_set_nat)
       => ( ! [X2: nat,S2: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,S2)
             => ( ! [Y4: nat] :
                    ( aa_set_nat_o(member_nat2(Y4),S2)
                   => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,Y4)),aa_nat_nat(F,X2)) )
               => ( aa_set_nat_o(Pa,S2)
                 => aa_set_nat_o(Pa,insert_nat2(X2,S2)) ) ) )
         => aa_set_nat_o(Pa,S) ) ) ) ).

% finite_ranking_induct
tff(fact_2028_finite__ranking__induct,axiom,
    ! [S: set_fu3555266933746557007_nat_o,Pa: fun_se7459991829846211660at_o_o,F: fun_fu8550182255320292270_o_int] :
      ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,S)
     => ( aa_set5091322549519198405at_o_o(Pa,bot_bo8632255151191677115_nat_o)
       => ( ! [X2: fun_Pr6460944699146269935_nat_o,S2: set_fu3555266933746557007_nat_o] :
              ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,S2)
             => ( ! [Y4: fun_Pr6460944699146269935_nat_o] :
                    ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Y4),S2)
                   => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_fun765338203505693407_o_int(F,Y4)),aa_fun765338203505693407_o_int(F,X2)) )
               => ( aa_set5091322549519198405at_o_o(Pa,S2)
                 => aa_set5091322549519198405at_o_o(Pa,insert4923026906511862463_nat_o(X2,S2)) ) ) )
         => aa_set5091322549519198405at_o_o(Pa,S) ) ) ) ).

% finite_ranking_induct
tff(fact_2029_finite__ranking__induct,axiom,
    ! [S: set_o,Pa: fun_set_o_o,F: fun_o_int] :
      ( aa_set_o_o(finite_finite_o,S)
     => ( aa_set_o_o(Pa,bot_bot_set_o)
       => ( ! [X2: $o,S2: set_o] :
              ( aa_set_o_o(finite_finite_o,S2)
             => ( ! [Y4: $o] :
                    ( aa_set_o_o(member_o((Y4)),S2)
                   => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_o_int(F,(Y4))),aa_o_int(F,(X2))) )
               => ( aa_set_o_o(Pa,S2)
                 => aa_set_o_o(Pa,insert_o((X2),S2)) ) ) )
         => aa_set_o_o(Pa,S) ) ) ) ).

% finite_ranking_induct
tff(fact_2030_finite__ranking__induct,axiom,
    ! [S: set_nat,Pa: fun_set_nat_o,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( aa_set_nat_o(Pa,bot_bot_set_nat)
       => ( ! [X2: nat,S2: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,S2)
             => ( ! [Y4: nat] :
                    ( aa_set_nat_o(member_nat2(Y4),S2)
                   => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,Y4)),aa_nat_int(F,X2)) )
               => ( aa_set_nat_o(Pa,S2)
                 => aa_set_nat_o(Pa,insert_nat2(X2,S2)) ) ) )
         => aa_set_nat_o(Pa,S) ) ) ) ).

% finite_ranking_induct
tff(fact_2031_finite__subset__induct_H,axiom,
    ! [F2: set_fu3555266933746557007_nat_o,A2: set_fu3555266933746557007_nat_o,Pa: fun_se7459991829846211660at_o_o] :
      ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,F2)
     => ( aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(F2),A2)
       => ( aa_set5091322549519198405at_o_o(Pa,bot_bo8632255151191677115_nat_o)
         => ( ! [A3: fun_Pr6460944699146269935_nat_o,F3: set_fu3555266933746557007_nat_o] :
                ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,F3)
               => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A3),A2)
                 => ( aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(F3),A2)
                   => ( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A3),F3)
                     => ( aa_set5091322549519198405at_o_o(Pa,F3)
                       => aa_set5091322549519198405at_o_o(Pa,insert4923026906511862463_nat_o(A3,F3)) ) ) ) ) )
           => aa_set5091322549519198405at_o_o(Pa,F2) ) ) ) ) ).

% finite_subset_induct'
tff(fact_2032_finite__subset__induct_H,axiom,
    ! [F2: set_o,A2: set_o,Pa: fun_set_o_o] :
      ( aa_set_o_o(finite_finite_o,F2)
     => ( aa_set_o_o(ord_less_eq_set_o(F2),A2)
       => ( aa_set_o_o(Pa,bot_bot_set_o)
         => ( ! [A3: $o,F3: set_o] :
                ( aa_set_o_o(finite_finite_o,F3)
               => ( aa_set_o_o(member_o((A3)),A2)
                 => ( aa_set_o_o(ord_less_eq_set_o(F3),A2)
                   => ( ~ aa_set_o_o(member_o((A3)),F3)
                     => ( aa_set_o_o(Pa,F3)
                       => aa_set_o_o(Pa,insert_o((A3),F3)) ) ) ) ) )
           => aa_set_o_o(Pa,F2) ) ) ) ) ).

% finite_subset_induct'
tff(fact_2033_finite__subset__induct_H,axiom,
    ! [F2: set_nat,A2: set_nat,Pa: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,F2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,F2),A2)
       => ( aa_set_nat_o(Pa,bot_bot_set_nat)
         => ( ! [A3: nat,F3: set_nat] :
                ( aa_set_nat_o(finite_finite_nat,F3)
               => ( aa_set_nat_o(member_nat2(A3),A2)
                 => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,F3),A2)
                   => ( ~ aa_set_nat_o(member_nat2(A3),F3)
                     => ( aa_set_nat_o(Pa,F3)
                       => aa_set_nat_o(Pa,insert_nat2(A3,F3)) ) ) ) ) )
           => aa_set_nat_o(Pa,F2) ) ) ) ) ).

% finite_subset_induct'
tff(fact_2034_finite__subset__induct,axiom,
    ! [F2: set_fu3555266933746557007_nat_o,A2: set_fu3555266933746557007_nat_o,Pa: fun_se7459991829846211660at_o_o] :
      ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,F2)
     => ( aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(F2),A2)
       => ( aa_set5091322549519198405at_o_o(Pa,bot_bo8632255151191677115_nat_o)
         => ( ! [A3: fun_Pr6460944699146269935_nat_o,F3: set_fu3555266933746557007_nat_o] :
                ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,F3)
               => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A3),A2)
                 => ( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A3),F3)
                   => ( aa_set5091322549519198405at_o_o(Pa,F3)
                     => aa_set5091322549519198405at_o_o(Pa,insert4923026906511862463_nat_o(A3,F3)) ) ) ) )
           => aa_set5091322549519198405at_o_o(Pa,F2) ) ) ) ) ).

% finite_subset_induct
tff(fact_2035_finite__subset__induct,axiom,
    ! [F2: set_o,A2: set_o,Pa: fun_set_o_o] :
      ( aa_set_o_o(finite_finite_o,F2)
     => ( aa_set_o_o(ord_less_eq_set_o(F2),A2)
       => ( aa_set_o_o(Pa,bot_bot_set_o)
         => ( ! [A3: $o,F3: set_o] :
                ( aa_set_o_o(finite_finite_o,F3)
               => ( aa_set_o_o(member_o((A3)),A2)
                 => ( ~ aa_set_o_o(member_o((A3)),F3)
                   => ( aa_set_o_o(Pa,F3)
                     => aa_set_o_o(Pa,insert_o((A3),F3)) ) ) ) )
           => aa_set_o_o(Pa,F2) ) ) ) ) ).

% finite_subset_induct
tff(fact_2036_finite__subset__induct,axiom,
    ! [F2: set_nat,A2: set_nat,Pa: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,F2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,F2),A2)
       => ( aa_set_nat_o(Pa,bot_bot_set_nat)
         => ( ! [A3: nat,F3: set_nat] :
                ( aa_set_nat_o(finite_finite_nat,F3)
               => ( aa_set_nat_o(member_nat2(A3),A2)
                 => ( ~ aa_set_nat_o(member_nat2(A3),F3)
                   => ( aa_set_nat_o(Pa,F3)
                     => aa_set_nat_o(Pa,insert_nat2(A3,F3)) ) ) ) )
           => aa_set_nat_o(Pa,F2) ) ) ) ) ).

% finite_subset_induct
tff(fact_2037_singleton__inject,axiom,
    ! [A: $o,B: $o] :
      ( ( insert_o((A),bot_bot_set_o) = insert_o((B),bot_bot_set_o) )
     => ( (A)
      <=> (B) ) ) ).

% singleton_inject
tff(fact_2038_singleton__inject,axiom,
    ! [A: nat,B: nat] :
      ( ( insert_nat2(A,bot_bot_set_nat) = insert_nat2(B,bot_bot_set_nat) )
     => ( A = B ) ) ).

% singleton_inject
tff(fact_2039_insert__not__empty,axiom,
    ! [A: $o,A2: set_o] : insert_o((A),A2) != bot_bot_set_o ).

% insert_not_empty
tff(fact_2040_insert__not__empty,axiom,
    ! [A: nat,A2: set_nat] : insert_nat2(A,A2) != bot_bot_set_nat ).

% insert_not_empty
tff(fact_2041_doubleton__eq__iff,axiom,
    ! [A: $o,B: $o,C2: $o,D2: $o] :
      ( ( insert_o((A),insert_o((B),bot_bot_set_o)) = insert_o((C2),insert_o((D2),bot_bot_set_o)) )
    <=> ( ( ( (A)
          <=> (C2) )
          & ( (B)
          <=> (D2) ) )
        | ( ( (A)
          <=> (D2) )
          & ( (B)
          <=> (C2) ) ) ) ) ).

% doubleton_eq_iff
tff(fact_2042_doubleton__eq__iff,axiom,
    ! [A: nat,B: nat,C2: nat,D2: nat] :
      ( ( insert_nat2(A,insert_nat2(B,bot_bot_set_nat)) = insert_nat2(C2,insert_nat2(D2,bot_bot_set_nat)) )
    <=> ( ( ( A = C2 )
          & ( B = D2 ) )
        | ( ( A = D2 )
          & ( B = C2 ) ) ) ) ).

% doubleton_eq_iff
tff(fact_2043_singleton__iff,axiom,
    ! [B: fun_Pr6460944699146269935_nat_o,A: fun_Pr6460944699146269935_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(B),insert4923026906511862463_nat_o(A,bot_bo8632255151191677115_nat_o))
    <=> ( B = A ) ) ).

% singleton_iff
tff(fact_2044_singleton__iff,axiom,
    ! [B: $o,A: $o] :
      ( aa_set_o_o(member_o((B)),insert_o((A),bot_bot_set_o))
    <=> ( (B)
      <=> (A) ) ) ).

% singleton_iff
tff(fact_2045_singleton__iff,axiom,
    ! [B: nat,A: nat] :
      ( aa_set_nat_o(member_nat2(B),insert_nat2(A,bot_bot_set_nat))
    <=> ( B = A ) ) ).

% singleton_iff
tff(fact_2046_singletonD,axiom,
    ! [B: fun_Pr6460944699146269935_nat_o,A: fun_Pr6460944699146269935_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(B),insert4923026906511862463_nat_o(A,bot_bo8632255151191677115_nat_o))
     => ( B = A ) ) ).

% singletonD
tff(fact_2047_singletonD,axiom,
    ! [B: $o,A: $o] :
      ( aa_set_o_o(member_o((B)),insert_o((A),bot_bot_set_o))
     => ( (B)
      <=> (A) ) ) ).

% singletonD
tff(fact_2048_singletonD,axiom,
    ! [B: nat,A: nat] :
      ( aa_set_nat_o(member_nat2(B),insert_nat2(A,bot_bot_set_nat))
     => ( B = A ) ) ).

% singletonD
tff(fact_2049_leD,axiom,
    ! [Y: assn,X: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,Y),X)
     => ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),Y) ) ).

% leD
tff(fact_2050_leD,axiom,
    ! [Y: nat,X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y) ) ).

% leD
tff(fact_2051_leD,axiom,
    ! [Y: int,X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),X)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y) ) ).

% leD
tff(fact_2052_leD,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Y),X)
     => ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,X),Y) ) ).

% leD
tff(fact_2053_leI,axiom,
    ! [X: nat,Y: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X) ) ).

% leI
tff(fact_2054_leI,axiom,
    ! [X: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),X) ) ).

% leI
tff(fact_2055_nless__le,axiom,
    ! [A: assn,B: assn] :
      ( ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
    <=> ( ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
        | ( A = B ) ) ) ).

% nless_le
tff(fact_2056_nless__le,axiom,
    ! [A: nat,B: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
        | ( A = B ) ) ) ).

% nless_le
tff(fact_2057_nless__le,axiom,
    ! [A: int,B: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
    <=> ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
        | ( A = B ) ) ) ).

% nless_le
tff(fact_2058_nless__le,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),B)
    <=> ( ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B)
        | ( A = B ) ) ) ).

% nless_le
tff(fact_2059_antisym__conv1,axiom,
    ! [X: assn,Y: assn] :
      ( ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),Y)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),Y)
      <=> ( X = Y ) ) ) ).

% antisym_conv1
tff(fact_2060_antisym__conv1,axiom,
    ! [X: nat,Y: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
      <=> ( X = Y ) ) ) ).

% antisym_conv1
tff(fact_2061_antisym__conv1,axiom,
    ! [X: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
      <=> ( X = Y ) ) ) ).

% antisym_conv1
tff(fact_2062_antisym__conv1,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,X),Y)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Y)
      <=> ( X = Y ) ) ) ).

% antisym_conv1
tff(fact_2063_antisym__conv2,axiom,
    ! [X: assn,Y: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),Y)
     => ( ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),Y)
      <=> ( X = Y ) ) ) ).

% antisym_conv2
tff(fact_2064_antisym__conv2,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
      <=> ( X = Y ) ) ) ).

% antisym_conv2
tff(fact_2065_antisym__conv2,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
     => ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
      <=> ( X = Y ) ) ) ).

% antisym_conv2
tff(fact_2066_antisym__conv2,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Y)
     => ( ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,X),Y)
      <=> ( X = Y ) ) ) ).

% antisym_conv2
tff(fact_2067_less__le__not__le,axiom,
    ! [X: assn,Y: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),Y)
    <=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),Y)
        & ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,Y),X) ) ) ).

% less_le_not_le
tff(fact_2068_less__le__not__le,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
        & ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X) ) ) ).

% less_le_not_le
tff(fact_2069_less__le__not__le,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
        & ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),X) ) ) ).

% less_le_not_le
tff(fact_2070_less__le__not__le,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,X),Y)
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Y)
        & ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Y),X) ) ) ).

% less_le_not_le
tff(fact_2071_not__le__imp__less,axiom,
    ! [Y: nat,X: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y) ) ).

% not_le_imp_less
tff(fact_2072_not__le__imp__less,axiom,
    ! [Y: int,X: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),X)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y) ) ).

% not_le_imp_less
tff(fact_2073_order_Oorder__iff__strict,axiom,
    ! [A: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
    <=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_2074_order_Oorder__iff__strict,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_2075_order_Oorder__iff__strict,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_2076_order_Oorder__iff__strict,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B)
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),B)
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_2077_order_Ostrict__iff__order,axiom,
    ! [A: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
    <=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_2078_order_Ostrict__iff__order,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_2079_order_Ostrict__iff__order,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_2080_order_Ostrict__iff__order,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),B)
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B)
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_2081_order_Ostrict__trans1,axiom,
    ! [A: assn,B: assn,C2: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),C2)
       => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),C2) ) ) ).

% order.strict_trans1
tff(fact_2082_order_Ostrict__trans1,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),C2) ) ) ).

% order.strict_trans1
tff(fact_2083_order_Ostrict__trans1,axiom,
    ! [A: int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),C2) ) ) ).

% order.strict_trans1
tff(fact_2084_order_Ostrict__trans1,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,B),C2)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),C2) ) ) ).

% order.strict_trans1
tff(fact_2085_order_Ostrict__trans2,axiom,
    ! [A: assn,B: assn,C2: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),C2)
       => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),C2) ) ) ).

% order.strict_trans2
tff(fact_2086_order_Ostrict__trans2,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),C2) ) ) ).

% order.strict_trans2
tff(fact_2087_order_Ostrict__trans2,axiom,
    ! [A: int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),C2) ) ) ).

% order.strict_trans2
tff(fact_2088_order_Ostrict__trans2,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),B)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),C2)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),C2) ) ) ).

% order.strict_trans2
tff(fact_2089_order_Ostrict__iff__not,axiom,
    ! [A: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
    <=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
        & ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),A) ) ) ).

% order.strict_iff_not
tff(fact_2090_order_Ostrict__iff__not,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
        & ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A) ) ) ).

% order.strict_iff_not
tff(fact_2091_order_Ostrict__iff__not,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
        & ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ) ).

% order.strict_iff_not
tff(fact_2092_order_Ostrict__iff__not,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),B)
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B)
        & ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A) ) ) ).

% order.strict_iff_not
tff(fact_2093_dual__order_Oorder__iff__strict,axiom,
    ! [B: assn,A: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),A)
    <=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),A)
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_2094_dual__order_Oorder__iff__strict,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_2095_dual__order_Oorder__iff__strict,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_2096_dual__order_Oorder__iff__strict,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A)
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,B),A)
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_2097_dual__order_Ostrict__iff__order,axiom,
    ! [B: assn,A: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),A)
    <=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),A)
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_2098_dual__order_Ostrict__iff__order,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_2099_dual__order_Ostrict__iff__order,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_2100_dual__order_Ostrict__iff__order,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,B),A)
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A)
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_2101_dual__order_Ostrict__trans1,axiom,
    ! [B: assn,A: assn,C2: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),A)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,C2),B)
       => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,C2),A) ) ) ).

% dual_order.strict_trans1
tff(fact_2102_dual__order_Ostrict__trans1,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),A) ) ) ).

% dual_order.strict_trans1
tff(fact_2103_dual__order_Ostrict__trans1,axiom,
    ! [B: int,A: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,C2),A) ) ) ).

% dual_order.strict_trans1
tff(fact_2104_dual__order_Ostrict__trans1,axiom,
    ! [B: set_nat,A: set_nat,C2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,C2),B)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,C2),A) ) ) ).

% dual_order.strict_trans1
tff(fact_2105_dual__order_Ostrict__trans2,axiom,
    ! [B: assn,A: assn,C2: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),A)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,C2),B)
       => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,C2),A) ) ) ).

% dual_order.strict_trans2
tff(fact_2106_dual__order_Ostrict__trans2,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),A) ) ) ).

% dual_order.strict_trans2
tff(fact_2107_dual__order_Ostrict__trans2,axiom,
    ! [B: int,A: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,C2),A) ) ) ).

% dual_order.strict_trans2
tff(fact_2108_dual__order_Ostrict__trans2,axiom,
    ! [B: set_nat,A: set_nat,C2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,B),A)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C2),B)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,C2),A) ) ) ).

% dual_order.strict_trans2
tff(fact_2109_dual__order_Ostrict__iff__not,axiom,
    ! [B: assn,A: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),A)
    <=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),A)
        & ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B) ) ) ).

% dual_order.strict_iff_not
tff(fact_2110_dual__order_Ostrict__iff__not,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
        & ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ) ).

% dual_order.strict_iff_not
tff(fact_2111_dual__order_Ostrict__iff__not,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
        & ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).

% dual_order.strict_iff_not
tff(fact_2112_dual__order_Ostrict__iff__not,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,B),A)
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A)
        & ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B) ) ) ).

% dual_order.strict_iff_not
tff(fact_2113_order_Ostrict__implies__order,axiom,
    ! [A: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
     => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B) ) ).

% order.strict_implies_order
tff(fact_2114_order_Ostrict__implies__order,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% order.strict_implies_order
tff(fact_2115_order_Ostrict__implies__order,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% order.strict_implies_order
tff(fact_2116_order_Ostrict__implies__order,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),B)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B) ) ).

% order.strict_implies_order
tff(fact_2117_dual__order_Ostrict__implies__order,axiom,
    ! [B: assn,A: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),A)
     => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),A) ) ).

% dual_order.strict_implies_order
tff(fact_2118_dual__order_Ostrict__implies__order,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A) ) ).

% dual_order.strict_implies_order
tff(fact_2119_dual__order_Ostrict__implies__order,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ).

% dual_order.strict_implies_order
tff(fact_2120_dual__order_Ostrict__implies__order,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,B),A)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A) ) ).

% dual_order.strict_implies_order
tff(fact_2121_order__le__less,axiom,
    ! [X: assn,Y: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),Y)
    <=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),Y)
        | ( X = Y ) ) ) ).

% order_le_less
tff(fact_2122_order__le__less,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
        | ( X = Y ) ) ) ).

% order_le_less
tff(fact_2123_order__le__less,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
        | ( X = Y ) ) ) ).

% order_le_less
tff(fact_2124_order__le__less,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Y)
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,X),Y)
        | ( X = Y ) ) ) ).

% order_le_less
tff(fact_2125_order__less__le,axiom,
    ! [X: assn,Y: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),Y)
    <=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),Y)
        & ( X != Y ) ) ) ).

% order_less_le
tff(fact_2126_order__less__le,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
        & ( X != Y ) ) ) ).

% order_less_le
tff(fact_2127_order__less__le,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
        & ( X != Y ) ) ) ).

% order_less_le
tff(fact_2128_order__less__le,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,X),Y)
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Y)
        & ( X != Y ) ) ) ).

% order_less_le
tff(fact_2129_linorder__not__le,axiom,
    ! [X: nat,Y: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),X) ) ).

% linorder_not_le
tff(fact_2130_linorder__not__le,axiom,
    ! [X: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Y),X) ) ).

% linorder_not_le
tff(fact_2131_linorder__not__less,axiom,
    ! [X: nat,Y: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X) ) ).

% linorder_not_less
tff(fact_2132_linorder__not__less,axiom,
    ! [X: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),X) ) ).

% linorder_not_less
tff(fact_2133_order__less__imp__le,axiom,
    ! [X: assn,Y: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),Y)
     => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),Y) ) ).

% order_less_imp_le
tff(fact_2134_order__less__imp__le,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y) ) ).

% order_less_imp_le
tff(fact_2135_order__less__imp__le,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y) ) ).

% order_less_imp_le
tff(fact_2136_order__less__imp__le,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,X),Y)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Y) ) ).

% order_less_imp_le
tff(fact_2137_order__le__neq__trans,axiom,
    ! [A: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
     => ( ( A != B )
       => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B) ) ) ).

% order_le_neq_trans
tff(fact_2138_order__le__neq__trans,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( ( A != B )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ) ).

% order_le_neq_trans
tff(fact_2139_order__le__neq__trans,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( ( A != B )
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).

% order_le_neq_trans
tff(fact_2140_order__le__neq__trans,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B)
     => ( ( A != B )
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),B) ) ) ).

% order_le_neq_trans
tff(fact_2141_order__neq__le__trans,axiom,
    ! [A: assn,B: assn] :
      ( ( A != B )
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
       => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B) ) ) ).

% order_neq_le_trans
tff(fact_2142_order__neq__le__trans,axiom,
    ! [A: nat,B: nat] :
      ( ( A != B )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ) ).

% order_neq_le_trans
tff(fact_2143_order__neq__le__trans,axiom,
    ! [A: int,B: int] :
      ( ( A != B )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).

% order_neq_le_trans
tff(fact_2144_order__neq__le__trans,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( A != B )
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),B) ) ) ).

% order_neq_le_trans
tff(fact_2145_order__le__less__trans,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),Y)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,Y),Z)
       => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),Z) ) ) ).

% order_le_less_trans
tff(fact_2146_order__le__less__trans,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Z)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z) ) ) ).

% order_le_less_trans
tff(fact_2147_order__le__less__trans,axiom,
    ! [X: int,Y: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z) ) ) ).

% order_le_less_trans
tff(fact_2148_order__le__less__trans,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Y)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,Y),Z)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,X),Z) ) ) ).

% order_le_less_trans
tff(fact_2149_order__less__le__trans,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),Y)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,Y),Z)
       => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),Z) ) ) ).

% order_less_le_trans
tff(fact_2150_order__less__le__trans,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Z)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z) ) ) ).

% order_less_le_trans
tff(fact_2151_order__less__le__trans,axiom,
    ! [X: int,Y: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z) ) ) ).

% order_less_le_trans
tff(fact_2152_order__less__le__trans,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,X),Y)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Y),Z)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,X),Z) ) ) ).

% order_less_le_trans
tff(fact_2153_order__le__less__subst1,axiom,
    ! [A: assn,F: fun_assn_assn,B: assn,C2: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),aa_assn_assn(F,B))
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),C2)
       => ( ! [X2: assn,Y2: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X2),Y2)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_assn_assn(F,X2)),aa_assn_assn(F,Y2)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),aa_assn_assn(F,C2)) ) ) ) ).

% order_le_less_subst1
tff(fact_2154_order__le__less__subst1,axiom,
    ! [A: assn,F: fun_nat_assn,B: nat,C2: nat] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),aa_nat_assn(F,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C2)
       => ( ! [X2: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Y2)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_nat_assn(F,X2)),aa_nat_assn(F,Y2)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),aa_nat_assn(F,C2)) ) ) ) ).

% order_le_less_subst1
tff(fact_2155_order__le__less__subst1,axiom,
    ! [A: assn,F: fun_int_assn,B: int,C2: int] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),aa_int_assn(F,B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C2)
       => ( ! [X2: int,Y2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Y2)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_int_assn(F,X2)),aa_int_assn(F,Y2)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),aa_int_assn(F,C2)) ) ) ) ).

% order_le_less_subst1
tff(fact_2156_order__le__less__subst1,axiom,
    ! [A: nat,F: fun_assn_nat,B: assn,C2: assn] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_assn_nat(F,B))
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),C2)
       => ( ! [X2: assn,Y2: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X2),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_assn_nat(F,X2)),aa_assn_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_assn_nat(F,C2)) ) ) ) ).

% order_le_less_subst1
tff(fact_2157_order__le__less__subst1,axiom,
    ! [A: nat,F: fun_nat_nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(F,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C2)
       => ( ! [X2: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,X2)),aa_nat_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_nat_nat(F,C2)) ) ) ) ).

% order_le_less_subst1
tff(fact_2158_order__le__less__subst1,axiom,
    ! [A: nat,F: fun_int_nat,B: int,C2: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_int_nat(F,B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C2)
       => ( ! [X2: int,Y2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(F,X2)),aa_int_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_int_nat(F,C2)) ) ) ) ).

% order_le_less_subst1
tff(fact_2159_order__le__less__subst1,axiom,
    ! [A: int,F: fun_assn_int,B: assn,C2: assn] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_assn_int(F,B))
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),C2)
       => ( ! [X2: assn,Y2: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X2),Y2)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_assn_int(F,X2)),aa_assn_int(F,Y2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_assn_int(F,C2)) ) ) ) ).

% order_le_less_subst1
tff(fact_2160_order__le__less__subst1,axiom,
    ! [A: int,F: fun_nat_int,B: nat,C2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(F,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C2)
       => ( ! [X2: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Y2)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,X2)),aa_nat_int(F,Y2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(F,C2)) ) ) ) ).

% order_le_less_subst1
tff(fact_2161_order__le__less__subst1,axiom,
    ! [A: int,F: fun_int_int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(F,B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C2)
       => ( ! [X2: int,Y2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Y2)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(F,X2)),aa_int_int(F,Y2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(F,C2)) ) ) ) ).

% order_le_less_subst1
tff(fact_2162_order__le__less__subst1,axiom,
    ! [A: set_nat,F: fun_assn_set_nat,B: assn,C2: assn] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),aa_assn_set_nat(F,B))
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),C2)
       => ( ! [X2: assn,Y2: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X2),Y2)
             => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,aa_assn_set_nat(F,X2)),aa_assn_set_nat(F,Y2)) )
         => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),aa_assn_set_nat(F,C2)) ) ) ) ).

% order_le_less_subst1
tff(fact_2163_order__le__less__subst2,axiom,
    ! [A: assn,B: assn,F: fun_assn_assn,C2: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_assn_assn(F,B)),C2)
       => ( ! [X2: assn,Y2: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X2),Y2)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(F,X2)),aa_assn_assn(F,Y2)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_assn_assn(F,A)),C2) ) ) ) ).

% order_le_less_subst2
tff(fact_2164_order__le__less__subst2,axiom,
    ! [A: assn,B: assn,F: fun_assn_nat,C2: nat] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_assn_nat(F,B)),C2)
       => ( ! [X2: assn,Y2: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X2),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_assn_nat(F,X2)),aa_assn_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_assn_nat(F,A)),C2) ) ) ) ).

% order_le_less_subst2
tff(fact_2165_order__le__less__subst2,axiom,
    ! [A: assn,B: assn,F: fun_assn_int,C2: int] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_assn_int(F,B)),C2)
       => ( ! [X2: assn,Y2: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X2),Y2)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_assn_int(F,X2)),aa_assn_int(F,Y2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_assn_int(F,A)),C2) ) ) ) ).

% order_le_less_subst2
tff(fact_2166_order__le__less__subst2,axiom,
    ! [A: nat,B: nat,F: fun_nat_assn,C2: assn] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_nat_assn(F,B)),C2)
       => ( ! [X2: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Y2)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_nat_assn(F,X2)),aa_nat_assn(F,Y2)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_nat_assn(F,A)),C2) ) ) ) ).

% order_le_less_subst2
tff(fact_2167_order__le__less__subst2,axiom,
    ! [A: nat,B: nat,F: fun_nat_nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,B)),C2)
       => ( ! [X2: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,X2)),aa_nat_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,A)),C2) ) ) ) ).

% order_le_less_subst2
tff(fact_2168_order__le__less__subst2,axiom,
    ! [A: nat,B: nat,F: fun_nat_int,C2: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,B)),C2)
       => ( ! [X2: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Y2)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,X2)),aa_nat_int(F,Y2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,A)),C2) ) ) ) ).

% order_le_less_subst2
tff(fact_2169_order__le__less__subst2,axiom,
    ! [A: int,B: int,F: fun_int_assn,C2: assn] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_int_assn(F,B)),C2)
       => ( ! [X2: int,Y2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X2),Y2)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_int_assn(F,X2)),aa_int_assn(F,Y2)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_int_assn(F,A)),C2) ) ) ) ).

% order_le_less_subst2
tff(fact_2170_order__le__less__subst2,axiom,
    ! [A: int,B: int,F: fun_int_nat,C2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(F,B)),C2)
       => ( ! [X2: int,Y2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X2),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,X2)),aa_int_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(F,A)),C2) ) ) ) ).

% order_le_less_subst2
tff(fact_2171_order__le__less__subst2,axiom,
    ! [A: int,B: int,F: fun_int_int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(F,B)),C2)
       => ( ! [X2: int,Y2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X2),Y2)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(F,X2)),aa_int_int(F,Y2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(F,A)),C2) ) ) ) ).

% order_le_less_subst2
tff(fact_2172_order__le__less__subst2,axiom,
    ! [A: assn,B: assn,F: fun_assn_set_nat,C2: set_nat] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,aa_assn_set_nat(F,B)),C2)
       => ( ! [X2: assn,Y2: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X2),Y2)
             => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_assn_set_nat(F,X2)),aa_assn_set_nat(F,Y2)) )
         => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,aa_assn_set_nat(F,A)),C2) ) ) ) ).

% order_le_less_subst2
tff(fact_2173_order__less__le__subst1,axiom,
    ! [A: assn,F: fun_assn_assn,B: assn,C2: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),aa_assn_assn(F,B))
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),C2)
       => ( ! [X2: assn,Y2: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X2),Y2)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(F,X2)),aa_assn_assn(F,Y2)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),aa_assn_assn(F,C2)) ) ) ) ).

% order_less_le_subst1
tff(fact_2174_order__less__le__subst1,axiom,
    ! [A: nat,F: fun_assn_nat,B: assn,C2: assn] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_assn_nat(F,B))
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),C2)
       => ( ! [X2: assn,Y2: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X2),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_assn_nat(F,X2)),aa_assn_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_assn_nat(F,C2)) ) ) ) ).

% order_less_le_subst1
tff(fact_2175_order__less__le__subst1,axiom,
    ! [A: int,F: fun_assn_int,B: assn,C2: assn] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_assn_int(F,B))
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),C2)
       => ( ! [X2: assn,Y2: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X2),Y2)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_assn_int(F,X2)),aa_assn_int(F,Y2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_assn_int(F,C2)) ) ) ) ).

% order_less_le_subst1
tff(fact_2176_order__less__le__subst1,axiom,
    ! [A: assn,F: fun_nat_assn,B: nat,C2: nat] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),aa_nat_assn(F,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C2)
       => ( ! [X2: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Y2)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_nat_assn(F,X2)),aa_nat_assn(F,Y2)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),aa_nat_assn(F,C2)) ) ) ) ).

% order_less_le_subst1
tff(fact_2177_order__less__le__subst1,axiom,
    ! [A: nat,F: fun_nat_nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_nat_nat(F,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C2)
       => ( ! [X2: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,X2)),aa_nat_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_nat_nat(F,C2)) ) ) ) ).

% order_less_le_subst1
tff(fact_2178_order__less__le__subst1,axiom,
    ! [A: int,F: fun_nat_int,B: nat,C2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(F,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C2)
       => ( ! [X2: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Y2)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,X2)),aa_nat_int(F,Y2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(F,C2)) ) ) ) ).

% order_less_le_subst1
tff(fact_2179_order__less__le__subst1,axiom,
    ! [A: assn,F: fun_int_assn,B: int,C2: int] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),aa_int_assn(F,B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C2)
       => ( ! [X2: int,Y2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X2),Y2)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_int_assn(F,X2)),aa_int_assn(F,Y2)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),aa_int_assn(F,C2)) ) ) ) ).

% order_less_le_subst1
tff(fact_2180_order__less__le__subst1,axiom,
    ! [A: nat,F: fun_int_nat,B: int,C2: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_int_nat(F,B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C2)
       => ( ! [X2: int,Y2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X2),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,X2)),aa_int_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_int_nat(F,C2)) ) ) ) ).

% order_less_le_subst1
tff(fact_2181_order__less__le__subst1,axiom,
    ! [A: int,F: fun_int_int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(F,B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C2)
       => ( ! [X2: int,Y2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X2),Y2)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(F,X2)),aa_int_int(F,Y2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(F,C2)) ) ) ) ).

% order_less_le_subst1
tff(fact_2182_order__less__le__subst1,axiom,
    ! [A: set_nat,F: fun_assn_set_nat,B: assn,C2: assn] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),aa_assn_set_nat(F,B))
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),C2)
       => ( ! [X2: assn,Y2: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X2),Y2)
             => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_assn_set_nat(F,X2)),aa_assn_set_nat(F,Y2)) )
         => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),aa_assn_set_nat(F,C2)) ) ) ) ).

% order_less_le_subst1
tff(fact_2183_order__less__le__subst2,axiom,
    ! [A: assn,B: assn,F: fun_assn_assn,C2: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(F,B)),C2)
       => ( ! [X2: assn,Y2: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X2),Y2)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_assn_assn(F,X2)),aa_assn_assn(F,Y2)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_assn_assn(F,A)),C2) ) ) ) ).

% order_less_le_subst2
tff(fact_2184_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F: fun_nat_assn,C2: assn] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_nat_assn(F,B)),C2)
       => ( ! [X2: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Y2)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_nat_assn(F,X2)),aa_nat_assn(F,Y2)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_nat_assn(F,A)),C2) ) ) ) ).

% order_less_le_subst2
tff(fact_2185_order__less__le__subst2,axiom,
    ! [A: int,B: int,F: fun_int_assn,C2: assn] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_int_assn(F,B)),C2)
       => ( ! [X2: int,Y2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Y2)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_int_assn(F,X2)),aa_int_assn(F,Y2)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_int_assn(F,A)),C2) ) ) ) ).

% order_less_le_subst2
tff(fact_2186_order__less__le__subst2,axiom,
    ! [A: assn,B: assn,F: fun_assn_nat,C2: nat] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_assn_nat(F,B)),C2)
       => ( ! [X2: assn,Y2: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X2),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_assn_nat(F,X2)),aa_assn_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_assn_nat(F,A)),C2) ) ) ) ).

% order_less_le_subst2
tff(fact_2187_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F: fun_nat_nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,B)),C2)
       => ( ! [X2: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,X2)),aa_nat_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,A)),C2) ) ) ) ).

% order_less_le_subst2
tff(fact_2188_order__less__le__subst2,axiom,
    ! [A: int,B: int,F: fun_int_nat,C2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,B)),C2)
       => ( ! [X2: int,Y2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(F,X2)),aa_int_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(F,A)),C2) ) ) ) ).

% order_less_le_subst2
tff(fact_2189_order__less__le__subst2,axiom,
    ! [A: assn,B: assn,F: fun_assn_int,C2: int] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_assn_int(F,B)),C2)
       => ( ! [X2: assn,Y2: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X2),Y2)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_assn_int(F,X2)),aa_assn_int(F,Y2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_assn_int(F,A)),C2) ) ) ) ).

% order_less_le_subst2
tff(fact_2190_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F: fun_nat_int,C2: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,B)),C2)
       => ( ! [X2: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Y2)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,X2)),aa_nat_int(F,Y2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,A)),C2) ) ) ) ).

% order_less_le_subst2
tff(fact_2191_order__less__le__subst2,axiom,
    ! [A: int,B: int,F: fun_int_int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(F,B)),C2)
       => ( ! [X2: int,Y2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Y2)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(F,X2)),aa_int_int(F,Y2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(F,A)),C2) ) ) ) ).

% order_less_le_subst2
tff(fact_2192_order__less__le__subst2,axiom,
    ! [A: assn,B: assn,F: fun_assn_set_nat,C2: set_nat] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_assn_set_nat(F,B)),C2)
       => ( ! [X2: assn,Y2: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X2),Y2)
             => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,aa_assn_set_nat(F,X2)),aa_assn_set_nat(F,Y2)) )
         => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,aa_assn_set_nat(F,A)),C2) ) ) ) ).

% order_less_le_subst2
tff(fact_2193_linorder__le__less__linear,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
      | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),X) ) ).

% linorder_le_less_linear
tff(fact_2194_linorder__le__less__linear,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
      | aa_int_o(aa_int_fun_int_o(ord_less_int,Y),X) ) ).

% linorder_le_less_linear
tff(fact_2195_order__le__imp__less__or__eq,axiom,
    ! [X: assn,Y: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),Y)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),Y)
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_2196_order__le__imp__less__or__eq,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_2197_order__le__imp__less__or__eq,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_2198_order__le__imp__less__or__eq,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Y)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,X),Y)
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_2199_Int__insert__right,axiom,
    ! [A2: set_o,A: $o,B2: set_o] :
      inf_inf_set_o(A2,insert_o((A),B2)) = $ite(aa_set_o_o(member_o((A)),A2),insert_o((A),inf_inf_set_o(A2,B2)),inf_inf_set_o(A2,B2)) ).

% Int_insert_right
tff(fact_2200_Int__insert__right,axiom,
    ! [A2: set_fu3555266933746557007_nat_o,A: fun_Pr6460944699146269935_nat_o,B2: set_fu3555266933746557007_nat_o] :
      inf_in3376754634429691197_nat_o(A2,insert4923026906511862463_nat_o(A,B2)) = $ite(aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A),A2),insert4923026906511862463_nat_o(A,inf_in3376754634429691197_nat_o(A2,B2)),inf_in3376754634429691197_nat_o(A2,B2)) ).

% Int_insert_right
tff(fact_2201_Int__insert__right,axiom,
    ! [A2: set_nat,A: nat,B2: set_nat] :
      aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),insert_nat2(A,B2)) = $ite(aa_set_nat_o(member_nat2(A),A2),insert_nat2(A,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2)) ).

% Int_insert_right
tff(fact_2202_Int__insert__right,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
      aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),insert8211810215607154385at_nat(A,B2)) = $ite(member8440522571783428010at_nat(A,A2),insert8211810215607154385at_nat(A,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2)) ).

% Int_insert_right
tff(fact_2203_Int__insert__left,axiom,
    ! [A: $o,B2: set_o,C3: set_o] :
      inf_inf_set_o(insert_o((A),B2),C3) = $ite(aa_set_o_o(member_o((A)),C3),insert_o((A),inf_inf_set_o(B2,C3)),inf_inf_set_o(B2,C3)) ).

% Int_insert_left
tff(fact_2204_Int__insert__left,axiom,
    ! [A: fun_Pr6460944699146269935_nat_o,B2: set_fu3555266933746557007_nat_o,C3: set_fu3555266933746557007_nat_o] :
      inf_in3376754634429691197_nat_o(insert4923026906511862463_nat_o(A,B2),C3) = $ite(aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A),C3),insert4923026906511862463_nat_o(A,inf_in3376754634429691197_nat_o(B2,C3)),inf_in3376754634429691197_nat_o(B2,C3)) ).

% Int_insert_left
tff(fact_2205_Int__insert__left,axiom,
    ! [A: nat,B2: set_nat,C3: set_nat] :
      aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,insert_nat2(A,B2)),C3) = $ite(aa_set_nat_o(member_nat2(A),C3),insert_nat2(A,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),C3)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),C3)) ).

% Int_insert_left
tff(fact_2206_Int__insert__left,axiom,
    ! [A: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat,C3: set_Pr1261947904930325089at_nat] :
      aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,insert8211810215607154385at_nat(A,B2)),C3) = $ite(member8440522571783428010at_nat(A,C3),insert8211810215607154385at_nat(A,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B2),C3)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B2),C3)) ).

% Int_insert_left
tff(fact_2207_le__numeral__extra_I4_J,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),one_one_nat) ).

% le_numeral_extra(4)
tff(fact_2208_le__numeral__extra_I4_J,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),one_one_int) ).

% le_numeral_extra(4)
tff(fact_2209_bot_Oextremum__uniqueI,axiom,
    ! [A: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(A),bot_bot_set_o)
     => ( A = bot_bot_set_o ) ) ).

% bot.extremum_uniqueI
tff(fact_2210_bot_Oextremum__uniqueI,axiom,
    ! [A: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),bot_bot_assn)
     => ( A = bot_bot_assn ) ) ).

% bot.extremum_uniqueI
tff(fact_2211_bot_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),bot_bot_nat)
     => ( A = bot_bot_nat ) ) ).

% bot.extremum_uniqueI
tff(fact_2212_bot_Oextremum__uniqueI,axiom,
    ! [A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),bot_bot_set_nat)
     => ( A = bot_bot_set_nat ) ) ).

% bot.extremum_uniqueI
tff(fact_2213_bot_Oextremum__unique,axiom,
    ! [A: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(A),bot_bot_set_o)
    <=> ( A = bot_bot_set_o ) ) ).

% bot.extremum_unique
tff(fact_2214_bot_Oextremum__unique,axiom,
    ! [A: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),bot_bot_assn)
    <=> ( A = bot_bot_assn ) ) ).

% bot.extremum_unique
tff(fact_2215_bot_Oextremum__unique,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),bot_bot_nat)
    <=> ( A = bot_bot_nat ) ) ).

% bot.extremum_unique
tff(fact_2216_bot_Oextremum__unique,axiom,
    ! [A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),bot_bot_set_nat)
    <=> ( A = bot_bot_set_nat ) ) ).

% bot.extremum_unique
tff(fact_2217_bot_Oextremum,axiom,
    ! [A: set_o] : aa_set_o_o(ord_less_eq_set_o(bot_bot_set_o),A) ).

% bot.extremum
tff(fact_2218_bot_Oextremum,axiom,
    ! [A: assn] : aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,bot_bot_assn),A) ).

% bot.extremum
tff(fact_2219_bot_Oextremum,axiom,
    ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,bot_bot_nat),A) ).

% bot.extremum
tff(fact_2220_bot_Oextremum,axiom,
    ! [A: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,bot_bot_set_nat),A) ).

% bot.extremum
tff(fact_2221_inf_OcoboundedI2,axiom,
    ! [B: product_unit,C2: product_unit,A: product_unit] :
      ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,B),C2)
     => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,A),B)),C2) ) ).

% inf.coboundedI2
tff(fact_2222_inf_OcoboundedI2,axiom,
    ! [B: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,B),C2)
     => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)),C2) ) ).

% inf.coboundedI2
tff(fact_2223_inf_OcoboundedI2,axiom,
    ! [B: assn,C2: assn,A: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),C2)
     => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B)),C2) ) ).

% inf.coboundedI2
tff(fact_2224_inf_OcoboundedI2,axiom,
    ! [B: nat,C2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)),C2) ) ).

% inf.coboundedI2
tff(fact_2225_inf_OcoboundedI2,axiom,
    ! [B: int,C2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C2)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B)),C2) ) ).

% inf.coboundedI2
tff(fact_2226_inf_OcoboundedI2,axiom,
    ! [B: set_nat,C2: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),C2)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)),C2) ) ).

% inf.coboundedI2
tff(fact_2227_inf_OcoboundedI1,axiom,
    ! [A: product_unit,C2: product_unit,B: product_unit] :
      ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,A),C2)
     => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,A),B)),C2) ) ).

% inf.coboundedI1
tff(fact_2228_inf_OcoboundedI1,axiom,
    ! [A: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),C2)
     => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)),C2) ) ).

% inf.coboundedI1
tff(fact_2229_inf_OcoboundedI1,axiom,
    ! [A: assn,C2: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),C2)
     => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B)),C2) ) ).

% inf.coboundedI1
tff(fact_2230_inf_OcoboundedI1,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)),C2) ) ).

% inf.coboundedI1
tff(fact_2231_inf_OcoboundedI1,axiom,
    ! [A: int,C2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C2)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B)),C2) ) ).

% inf.coboundedI1
tff(fact_2232_inf_OcoboundedI1,axiom,
    ! [A: set_nat,C2: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),C2)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)),C2) ) ).

% inf.coboundedI1
tff(fact_2233_inf_Oabsorb__iff2,axiom,
    ! [B: product_unit,A: product_unit] :
      ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,B),A)
    <=> ( aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,A),B) = B ) ) ).

% inf.absorb_iff2
tff(fact_2234_inf_Oabsorb__iff2,axiom,
    ! [B: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,B),A)
    <=> ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) = B ) ) ).

% inf.absorb_iff2
tff(fact_2235_inf_Oabsorb__iff2,axiom,
    ! [B: assn,A: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),A)
    <=> ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B) = B ) ) ).

% inf.absorb_iff2
tff(fact_2236_inf_Oabsorb__iff2,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B) = B ) ) ).

% inf.absorb_iff2
tff(fact_2237_inf_Oabsorb__iff2,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
    <=> ( aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B) = B ) ) ).

% inf.absorb_iff2
tff(fact_2238_inf_Oabsorb__iff2,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A)
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) = B ) ) ).

% inf.absorb_iff2
tff(fact_2239_inf_Oabsorb__iff1,axiom,
    ! [A: product_unit,B: product_unit] :
      ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,A),B)
    <=> ( aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,A),B) = A ) ) ).

% inf.absorb_iff1
tff(fact_2240_inf_Oabsorb__iff1,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),B)
    <=> ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) = A ) ) ).

% inf.absorb_iff1
tff(fact_2241_inf_Oabsorb__iff1,axiom,
    ! [A: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
    <=> ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B) = A ) ) ).

% inf.absorb_iff1
tff(fact_2242_inf_Oabsorb__iff1,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B) = A ) ) ).

% inf.absorb_iff1
tff(fact_2243_inf_Oabsorb__iff1,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
    <=> ( aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B) = A ) ) ).

% inf.absorb_iff1
tff(fact_2244_inf_Oabsorb__iff1,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B)
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) = A ) ) ).

% inf.absorb_iff1
tff(fact_2245_inf_Ocobounded2,axiom,
    ! [A: product_unit,B: product_unit] : aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,A),B)),B) ).

% inf.cobounded2
tff(fact_2246_inf_Ocobounded2,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] : aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)),B) ).

% inf.cobounded2
tff(fact_2247_inf_Ocobounded2,axiom,
    ! [A: assn,B: assn] : aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B)),B) ).

% inf.cobounded2
tff(fact_2248_inf_Ocobounded2,axiom,
    ! [A: nat,B: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)),B) ).

% inf.cobounded2
tff(fact_2249_inf_Ocobounded2,axiom,
    ! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B)),B) ).

% inf.cobounded2
tff(fact_2250_inf_Ocobounded2,axiom,
    ! [A: set_nat,B: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)),B) ).

% inf.cobounded2
tff(fact_2251_inf_Ocobounded1,axiom,
    ! [A: product_unit,B: product_unit] : aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,A),B)),A) ).

% inf.cobounded1
tff(fact_2252_inf_Ocobounded1,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] : aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)),A) ).

% inf.cobounded1
tff(fact_2253_inf_Ocobounded1,axiom,
    ! [A: assn,B: assn] : aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B)),A) ).

% inf.cobounded1
tff(fact_2254_inf_Ocobounded1,axiom,
    ! [A: nat,B: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)),A) ).

% inf.cobounded1
tff(fact_2255_inf_Ocobounded1,axiom,
    ! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B)),A) ).

% inf.cobounded1
tff(fact_2256_inf_Ocobounded1,axiom,
    ! [A: set_nat,B: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)),A) ).

% inf.cobounded1
tff(fact_2257_inf_Oorder__iff,axiom,
    ! [A: product_unit,B: product_unit] :
      ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,A),B)
    <=> ( A = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,A),B) ) ) ).

% inf.order_iff
tff(fact_2258_inf_Oorder__iff,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),B)
    <=> ( A = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) ) ) ).

% inf.order_iff
tff(fact_2259_inf_Oorder__iff,axiom,
    ! [A: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
    <=> ( A = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B) ) ) ).

% inf.order_iff
tff(fact_2260_inf_Oorder__iff,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
    <=> ( A = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B) ) ) ).

% inf.order_iff
tff(fact_2261_inf_Oorder__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
    <=> ( A = aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B) ) ) ).

% inf.order_iff
tff(fact_2262_inf_Oorder__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B)
    <=> ( A = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) ) ) ).

% inf.order_iff
tff(fact_2263_inf__greatest,axiom,
    ! [X: product_unit,Y: product_unit,Z: product_unit] :
      ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,X),Y)
     => ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,X),Z)
       => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,X),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,Y),Z)) ) ) ).

% inf_greatest
tff(fact_2264_inf__greatest,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X),Y)
     => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X),Z)
       => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y),Z)) ) ) ).

% inf_greatest
tff(fact_2265_inf__greatest,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),Y)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),Z)
       => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),Z)) ) ) ).

% inf_greatest
tff(fact_2266_inf__greatest,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Z)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Y),Z)) ) ) ).

% inf_greatest
tff(fact_2267_inf__greatest,axiom,
    ! [X: int,Y: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),aa_int_int(aa_int_fun_int_int(inf_inf_int,Y),Z)) ) ) ).

% inf_greatest
tff(fact_2268_inf__greatest,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Y)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Z)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),Z)) ) ) ).

% inf_greatest
tff(fact_2269_inf_OboundedI,axiom,
    ! [A: product_unit,B: product_unit,C2: product_unit] :
      ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,A),B)
     => ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,A),C2)
       => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,A),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,B),C2)) ) ) ).

% inf.boundedI
tff(fact_2270_inf_OboundedI,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),B)
     => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),C2)
       => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B),C2)) ) ) ).

% inf.boundedI
tff(fact_2271_inf_OboundedI,axiom,
    ! [A: assn,B: assn,C2: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),C2)
       => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,B),C2)) ) ) ).

% inf.boundedI
tff(fact_2272_inf_OboundedI,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,B),C2)) ) ) ).

% inf.boundedI
tff(fact_2273_inf_OboundedI,axiom,
    ! [A: int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(inf_inf_int,B),C2)) ) ) ).

% inf.boundedI
tff(fact_2274_inf_OboundedI,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),C2)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B),C2)) ) ) ).

% inf.boundedI
tff(fact_2275_inf_OboundedE,axiom,
    ! [A: product_unit,B: product_unit,C2: product_unit] :
      ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,A),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,B),C2))
     => ~ ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,A),B)
         => ~ aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,A),C2) ) ) ).

% inf.boundedE
tff(fact_2276_inf_OboundedE,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B),C2))
     => ~ ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),B)
         => ~ aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),C2) ) ) ).

% inf.boundedE
tff(fact_2277_inf_OboundedE,axiom,
    ! [A: assn,B: assn,C2: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,B),C2))
     => ~ ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
         => ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),C2) ) ) ).

% inf.boundedE
tff(fact_2278_inf_OboundedE,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,B),C2))
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C2) ) ) ).

% inf.boundedE
tff(fact_2279_inf_OboundedE,axiom,
    ! [A: int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(inf_inf_int,B),C2))
     => ~ ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
         => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C2) ) ) ).

% inf.boundedE
tff(fact_2280_inf_OboundedE,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B),C2))
     => ~ ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B)
         => ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),C2) ) ) ).

% inf.boundedE
tff(fact_2281_inf__absorb2,axiom,
    ! [Y: product_unit,X: product_unit] :
      ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,Y),X)
     => ( aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),Y) = Y ) ) ).

% inf_absorb2
tff(fact_2282_inf__absorb2,axiom,
    ! [Y: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,Y),X)
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y) = Y ) ) ).

% inf_absorb2
tff(fact_2283_inf__absorb2,axiom,
    ! [Y: assn,X: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,Y),X)
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y) = Y ) ) ).

% inf_absorb2
tff(fact_2284_inf__absorb2,axiom,
    ! [Y: nat,X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y) = Y ) ) ).

% inf_absorb2
tff(fact_2285_inf__absorb2,axiom,
    ! [Y: int,X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),X)
     => ( aa_int_int(aa_int_fun_int_int(inf_inf_int,X),Y) = Y ) ) ).

% inf_absorb2
tff(fact_2286_inf__absorb2,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Y),X)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y) = Y ) ) ).

% inf_absorb2
tff(fact_2287_inf__absorb1,axiom,
    ! [X: product_unit,Y: product_unit] :
      ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,X),Y)
     => ( aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),Y) = X ) ) ).

% inf_absorb1
tff(fact_2288_inf__absorb1,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X),Y)
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y) = X ) ) ).

% inf_absorb1
tff(fact_2289_inf__absorb1,axiom,
    ! [X: assn,Y: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),Y)
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y) = X ) ) ).

% inf_absorb1
tff(fact_2290_inf__absorb1,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y) = X ) ) ).

% inf_absorb1
tff(fact_2291_inf__absorb1,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
     => ( aa_int_int(aa_int_fun_int_int(inf_inf_int,X),Y) = X ) ) ).

% inf_absorb1
tff(fact_2292_inf__absorb1,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Y)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y) = X ) ) ).

% inf_absorb1
tff(fact_2293_inf_Oabsorb2,axiom,
    ! [B: product_unit,A: product_unit] :
      ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,B),A)
     => ( aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,A),B) = B ) ) ).

% inf.absorb2
tff(fact_2294_inf_Oabsorb2,axiom,
    ! [B: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,B),A)
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) = B ) ) ).

% inf.absorb2
tff(fact_2295_inf_Oabsorb2,axiom,
    ! [B: assn,A: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),A)
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B) = B ) ) ).

% inf.absorb2
tff(fact_2296_inf_Oabsorb2,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B) = B ) ) ).

% inf.absorb2
tff(fact_2297_inf_Oabsorb2,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B) = B ) ) ).

% inf.absorb2
tff(fact_2298_inf_Oabsorb2,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) = B ) ) ).

% inf.absorb2
tff(fact_2299_inf_Oabsorb1,axiom,
    ! [A: product_unit,B: product_unit] :
      ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,A),B)
     => ( aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,A),B) = A ) ) ).

% inf.absorb1
tff(fact_2300_inf_Oabsorb1,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),B)
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) = A ) ) ).

% inf.absorb1
tff(fact_2301_inf_Oabsorb1,axiom,
    ! [A: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B) = A ) ) ).

% inf.absorb1
tff(fact_2302_inf_Oabsorb1,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B) = A ) ) ).

% inf.absorb1
tff(fact_2303_inf_Oabsorb1,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B) = A ) ) ).

% inf.absorb1
tff(fact_2304_inf_Oabsorb1,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) = A ) ) ).

% inf.absorb1
tff(fact_2305_le__iff__inf,axiom,
    ! [X: product_unit,Y: product_unit] :
      ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,X),Y)
    <=> ( aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),Y) = X ) ) ).

% le_iff_inf
tff(fact_2306_le__iff__inf,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X),Y)
    <=> ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y) = X ) ) ).

% le_iff_inf
tff(fact_2307_le__iff__inf,axiom,
    ! [X: assn,Y: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),Y)
    <=> ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y) = X ) ) ).

% le_iff_inf
tff(fact_2308_le__iff__inf,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y) = X ) ) ).

% le_iff_inf
tff(fact_2309_le__iff__inf,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
    <=> ( aa_int_int(aa_int_fun_int_int(inf_inf_int,X),Y) = X ) ) ).

% le_iff_inf
tff(fact_2310_le__iff__inf,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Y)
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y) = X ) ) ).

% le_iff_inf
tff(fact_2311_inf__unique,axiom,
    ! [F: fun_Pr3553901264588531491t_unit,X: product_unit,Y: product_unit] :
      ( ! [X2: product_unit,Y2: product_unit] : aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(F,X2),Y2)),X2)
     => ( ! [X2: product_unit,Y2: product_unit] : aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(F,X2),Y2)),Y2)
       => ( ! [X2: product_unit,Y2: product_unit,Z3: product_unit] :
              ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,X2),Y2)
             => ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,X2),Z3)
               => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,X2),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(F,Y2),Z3)) ) )
         => ( aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),Y) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(F,X),Y) ) ) ) ) ).

% inf_unique
tff(fact_2312_inf__unique,axiom,
    ! [F: fun_se3560235952666129655at_nat,X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ! [X2: set_Pr1261947904930325089at_nat,Y2: set_Pr1261947904930325089at_nat] : aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(F,X2),Y2)),X2)
     => ( ! [X2: set_Pr1261947904930325089at_nat,Y2: set_Pr1261947904930325089at_nat] : aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(F,X2),Y2)),Y2)
       => ( ! [X2: set_Pr1261947904930325089at_nat,Y2: set_Pr1261947904930325089at_nat,Z3: set_Pr1261947904930325089at_nat] :
              ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X2),Y2)
             => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X2),Z3)
               => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(F,Y2),Z3)) ) )
         => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(F,X),Y) ) ) ) ) ).

% inf_unique
tff(fact_2313_inf__unique,axiom,
    ! [F: fun_as3983628580203722482n_assn,X: assn,Y: assn] :
      ( ! [X2: assn,Y2: assn] : aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(F,X2),Y2)),X2)
     => ( ! [X2: assn,Y2: assn] : aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(F,X2),Y2)),Y2)
       => ( ! [X2: assn,Y2: assn,Z3: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X2),Y2)
             => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X2),Z3)
               => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X2),aa_assn_assn(aa_ass954072293579886259n_assn(F,Y2),Z3)) ) )
         => ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y) = aa_assn_assn(aa_ass954072293579886259n_assn(F,X),Y) ) ) ) ) ).

% inf_unique
tff(fact_2314_inf__unique,axiom,
    ! [F: fun_nat_fun_nat_nat,X: nat,Y: nat] :
      ( ! [X2: nat,Y2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(F,X2),Y2)),X2)
     => ( ! [X2: nat,Y2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(F,X2),Y2)),Y2)
       => ( ! [X2: nat,Y2: nat,Z3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Y2)
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Z3)
               => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),aa_nat_nat(aa_nat_fun_nat_nat(F,Y2),Z3)) ) )
         => ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y) = aa_nat_nat(aa_nat_fun_nat_nat(F,X),Y) ) ) ) ) ).

% inf_unique
tff(fact_2315_inf__unique,axiom,
    ! [F: fun_int_fun_int_int,X: int,Y: int] :
      ( ! [X2: int,Y2: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(F,X2),Y2)),X2)
     => ( ! [X2: int,Y2: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(F,X2),Y2)),Y2)
       => ( ! [X2: int,Y2: int,Z3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X2),Y2)
             => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X2),Z3)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X2),aa_int_int(aa_int_fun_int_int(F,Y2),Z3)) ) )
         => ( aa_int_int(aa_int_fun_int_int(inf_inf_int,X),Y) = aa_int_int(aa_int_fun_int_int(F,X),Y) ) ) ) ) ).

% inf_unique
tff(fact_2316_inf__unique,axiom,
    ! [F: fun_se1909708154429037836et_nat,X: set_nat,Y: set_nat] :
      ( ! [X2: set_nat,Y2: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(F,X2),Y2)),X2)
     => ( ! [X2: set_nat,Y2: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(F,X2),Y2)),Y2)
       => ( ! [X2: set_nat,Y2: set_nat,Z3: set_nat] :
              ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X2),Y2)
             => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X2),Z3)
               => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(F,Y2),Z3)) ) )
         => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(F,X),Y) ) ) ) ) ).

% inf_unique
tff(fact_2317_inf_OorderI,axiom,
    ! [A: product_unit,B: product_unit] :
      ( ( A = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,A),B) )
     => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,A),B) ) ).

% inf.orderI
tff(fact_2318_inf_OorderI,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( A = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) )
     => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),B) ) ).

% inf.orderI
tff(fact_2319_inf_OorderI,axiom,
    ! [A: assn,B: assn] :
      ( ( A = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B) )
     => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B) ) ).

% inf.orderI
tff(fact_2320_inf_OorderI,axiom,
    ! [A: nat,B: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% inf.orderI
tff(fact_2321_inf_OorderI,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% inf.orderI
tff(fact_2322_inf_OorderI,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( A = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) )
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B) ) ).

% inf.orderI
tff(fact_2323_inf_OorderE,axiom,
    ! [A: product_unit,B: product_unit] :
      ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,A),B)
     => ( A = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,A),B) ) ) ).

% inf.orderE
tff(fact_2324_inf_OorderE,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),B)
     => ( A = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) ) ) ).

% inf.orderE
tff(fact_2325_inf_OorderE,axiom,
    ! [A: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
     => ( A = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B) ) ) ).

% inf.orderE
tff(fact_2326_inf_OorderE,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( A = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B) ) ) ).

% inf.orderE
tff(fact_2327_inf_OorderE,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( A = aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B) ) ) ).

% inf.orderE
tff(fact_2328_inf_OorderE,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B)
     => ( A = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) ) ) ).

% inf.orderE
tff(fact_2329_le__infI2,axiom,
    ! [B: product_unit,X: product_unit,A: product_unit] :
      ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,B),X)
     => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,A),B)),X) ) ).

% le_infI2
tff(fact_2330_le__infI2,axiom,
    ! [B: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,B),X)
     => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)),X) ) ).

% le_infI2
tff(fact_2331_le__infI2,axiom,
    ! [B: assn,X: assn,A: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),X)
     => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B)),X) ) ).

% le_infI2
tff(fact_2332_le__infI2,axiom,
    ! [B: nat,X: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),X)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)),X) ) ).

% le_infI2
tff(fact_2333_le__infI2,axiom,
    ! [B: int,X: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),X)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B)),X) ) ).

% le_infI2
tff(fact_2334_le__infI2,axiom,
    ! [B: set_nat,X: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),X)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)),X) ) ).

% le_infI2
tff(fact_2335_le__infI1,axiom,
    ! [A: product_unit,X: product_unit,B: product_unit] :
      ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,A),X)
     => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,A),B)),X) ) ).

% le_infI1
tff(fact_2336_le__infI1,axiom,
    ! [A: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),X)
     => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)),X) ) ).

% le_infI1
tff(fact_2337_le__infI1,axiom,
    ! [A: assn,X: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),X)
     => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B)),X) ) ).

% le_infI1
tff(fact_2338_le__infI1,axiom,
    ! [A: nat,X: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),X)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)),X) ) ).

% le_infI1
tff(fact_2339_le__infI1,axiom,
    ! [A: int,X: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),X)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B)),X) ) ).

% le_infI1
tff(fact_2340_le__infI1,axiom,
    ! [A: set_nat,X: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),X)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)),X) ) ).

% le_infI1
tff(fact_2341_inf__mono,axiom,
    ! [A: product_unit,C2: product_unit,B: product_unit,D2: product_unit] :
      ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,A),C2)
     => ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,B),D2)
       => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,A),B)),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,C2),D2)) ) ) ).

% inf_mono
tff(fact_2342_inf__mono,axiom,
    ! [A: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,D2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),C2)
     => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,B),D2)
       => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,C2),D2)) ) ) ).

% inf_mono
tff(fact_2343_inf__mono,axiom,
    ! [A: assn,C2: assn,B: assn,D2: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),C2)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),D2)
       => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B)),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,C2),D2)) ) ) ).

% inf_mono
tff(fact_2344_inf__mono,axiom,
    ! [A: nat,C2: nat,B: nat,D2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),D2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,C2),D2)) ) ) ).

% inf_mono
tff(fact_2345_inf__mono,axiom,
    ! [A: int,C2: int,B: int,D2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),D2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B)),aa_int_int(aa_int_fun_int_int(inf_inf_int,C2),D2)) ) ) ).

% inf_mono
tff(fact_2346_inf__mono,axiom,
    ! [A: set_nat,C2: set_nat,B: set_nat,D2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),C2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),D2)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,C2),D2)) ) ) ).

% inf_mono
tff(fact_2347_le__infI,axiom,
    ! [X: product_unit,A: product_unit,B: product_unit] :
      ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,X),A)
     => ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,X),B)
       => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,X),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,A),B)) ) ) ).

% le_infI
tff(fact_2348_le__infI,axiom,
    ! [X: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X),A)
     => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X),B)
       => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)) ) ) ).

% le_infI
tff(fact_2349_le__infI,axiom,
    ! [X: assn,A: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),A)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),B)
       => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B)) ) ) ).

% le_infI
tff(fact_2350_le__infI,axiom,
    ! [X: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)) ) ) ).

% le_infI
tff(fact_2351_le__infI,axiom,
    ! [X: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B)) ) ) ).

% le_infI
tff(fact_2352_le__infI,axiom,
    ! [X: set_nat,A: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),A)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),B)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)) ) ) ).

% le_infI
tff(fact_2353_le__infE,axiom,
    ! [X: product_unit,A: product_unit,B: product_unit] :
      ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,X),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,A),B))
     => ~ ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,X),A)
         => ~ aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,X),B) ) ) ).

% le_infE
tff(fact_2354_le__infE,axiom,
    ! [X: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B))
     => ~ ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X),A)
         => ~ aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X),B) ) ) ).

% le_infE
tff(fact_2355_le__infE,axiom,
    ! [X: assn,A: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B))
     => ~ ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),A)
         => ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),B) ) ) ).

% le_infE
tff(fact_2356_le__infE,axiom,
    ! [X: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B))
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),A)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),B) ) ) ).

% le_infE
tff(fact_2357_le__infE,axiom,
    ! [X: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B))
     => ~ ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),A)
         => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),B) ) ) ).

% le_infE
tff(fact_2358_le__infE,axiom,
    ! [X: set_nat,A: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B))
     => ~ ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),A)
         => ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),B) ) ) ).

% le_infE
tff(fact_2359_inf__le2,axiom,
    ! [X: product_unit,Y: product_unit] : aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),Y)),Y) ).

% inf_le2
tff(fact_2360_inf__le2,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y)),Y) ).

% inf_le2
tff(fact_2361_inf__le2,axiom,
    ! [X: assn,Y: assn] : aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)),Y) ).

% inf_le2
tff(fact_2362_inf__le2,axiom,
    ! [X: nat,Y: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y)),Y) ).

% inf_le2
tff(fact_2363_inf__le2,axiom,
    ! [X: int,Y: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,X),Y)),Y) ).

% inf_le2
tff(fact_2364_inf__le2,axiom,
    ! [X: set_nat,Y: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)),Y) ).

% inf_le2
tff(fact_2365_inf__le1,axiom,
    ! [X: product_unit,Y: product_unit] : aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),Y)),X) ).

% inf_le1
tff(fact_2366_inf__le1,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y)),X) ).

% inf_le1
tff(fact_2367_inf__le1,axiom,
    ! [X: assn,Y: assn] : aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)),X) ).

% inf_le1
tff(fact_2368_inf__le1,axiom,
    ! [X: nat,Y: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y)),X) ).

% inf_le1
tff(fact_2369_inf__le1,axiom,
    ! [X: int,Y: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,X),Y)),X) ).

% inf_le1
tff(fact_2370_inf__le1,axiom,
    ! [X: set_nat,Y: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)),X) ).

% inf_le1
tff(fact_2371_inf__sup__ord_I1_J,axiom,
    ! [X: product_unit,Y: product_unit] : aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),Y)),X) ).

% inf_sup_ord(1)
tff(fact_2372_inf__sup__ord_I1_J,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y)),X) ).

% inf_sup_ord(1)
tff(fact_2373_inf__sup__ord_I1_J,axiom,
    ! [X: assn,Y: assn] : aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)),X) ).

% inf_sup_ord(1)
tff(fact_2374_inf__sup__ord_I1_J,axiom,
    ! [X: nat,Y: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y)),X) ).

% inf_sup_ord(1)
tff(fact_2375_inf__sup__ord_I1_J,axiom,
    ! [X: int,Y: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,X),Y)),X) ).

% inf_sup_ord(1)
tff(fact_2376_inf__sup__ord_I1_J,axiom,
    ! [X: set_nat,Y: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)),X) ).

% inf_sup_ord(1)
tff(fact_2377_inf__sup__ord_I2_J,axiom,
    ! [X: product_unit,Y: product_unit] : aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),Y)),Y) ).

% inf_sup_ord(2)
tff(fact_2378_inf__sup__ord_I2_J,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y)),Y) ).

% inf_sup_ord(2)
tff(fact_2379_inf__sup__ord_I2_J,axiom,
    ! [X: assn,Y: assn] : aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)),Y) ).

% inf_sup_ord(2)
tff(fact_2380_inf__sup__ord_I2_J,axiom,
    ! [X: nat,Y: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y)),Y) ).

% inf_sup_ord(2)
tff(fact_2381_inf__sup__ord_I2_J,axiom,
    ! [X: int,Y: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,X),Y)),Y) ).

% inf_sup_ord(2)
tff(fact_2382_inf__sup__ord_I2_J,axiom,
    ! [X: set_nat,Y: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)),Y) ).

% inf_sup_ord(2)
tff(fact_2383_inf__sup__ord_I4_J,axiom,
    ! [Y: assn,X: assn] : aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,Y),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y)) ).

% inf_sup_ord(4)
tff(fact_2384_inf__sup__ord_I4_J,axiom,
    ! [Y: nat,X: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y)) ).

% inf_sup_ord(4)
tff(fact_2385_inf__sup__ord_I4_J,axiom,
    ! [Y: int,X: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y)) ).

% inf_sup_ord(4)
tff(fact_2386_inf__sup__ord_I4_J,axiom,
    ! [Y: set_nat,X: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Y),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y)) ).

% inf_sup_ord(4)
tff(fact_2387_inf__sup__ord_I3_J,axiom,
    ! [X: assn,Y: assn] : aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y)) ).

% inf_sup_ord(3)
tff(fact_2388_inf__sup__ord_I3_J,axiom,
    ! [X: nat,Y: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y)) ).

% inf_sup_ord(3)
tff(fact_2389_inf__sup__ord_I3_J,axiom,
    ! [X: int,Y: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y)) ).

% inf_sup_ord(3)
tff(fact_2390_inf__sup__ord_I3_J,axiom,
    ! [X: set_nat,Y: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y)) ).

% inf_sup_ord(3)
tff(fact_2391_le__supE,axiom,
    ! [A: assn,B: assn,X: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B)),X)
     => ~ ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),X)
         => ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),X) ) ) ).

% le_supE
tff(fact_2392_le__supE,axiom,
    ! [A: nat,B: nat,X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)),X)
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),X)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),X) ) ) ).

% le_supE
tff(fact_2393_le__supE,axiom,
    ! [A: int,B: int,X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)),X)
     => ~ ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),X)
         => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),X) ) ) ).

% le_supE
tff(fact_2394_le__supE,axiom,
    ! [A: set_nat,B: set_nat,X: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)),X)
     => ~ ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),X)
         => ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),X) ) ) ).

% le_supE
tff(fact_2395_le__supI,axiom,
    ! [A: assn,X: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),X)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),X)
       => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B)),X) ) ) ).

% le_supI
tff(fact_2396_le__supI,axiom,
    ! [A: nat,X: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),X)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),X)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)),X) ) ) ).

% le_supI
tff(fact_2397_le__supI,axiom,
    ! [A: int,X: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),X)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),X)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)),X) ) ) ).

% le_supI
tff(fact_2398_le__supI,axiom,
    ! [A: set_nat,X: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),X)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),X)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)),X) ) ) ).

% le_supI
tff(fact_2399_sup__ge1,axiom,
    ! [X: assn,Y: assn] : aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y)) ).

% sup_ge1
tff(fact_2400_sup__ge1,axiom,
    ! [X: nat,Y: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y)) ).

% sup_ge1
tff(fact_2401_sup__ge1,axiom,
    ! [X: int,Y: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y)) ).

% sup_ge1
tff(fact_2402_sup__ge1,axiom,
    ! [X: set_nat,Y: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y)) ).

% sup_ge1
tff(fact_2403_sup__ge2,axiom,
    ! [Y: assn,X: assn] : aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,Y),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y)) ).

% sup_ge2
tff(fact_2404_sup__ge2,axiom,
    ! [Y: nat,X: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y)) ).

% sup_ge2
tff(fact_2405_sup__ge2,axiom,
    ! [Y: int,X: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y)) ).

% sup_ge2
tff(fact_2406_sup__ge2,axiom,
    ! [Y: set_nat,X: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Y),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y)) ).

% sup_ge2
tff(fact_2407_le__supI1,axiom,
    ! [X: assn,A: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),A)
     => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B)) ) ).

% le_supI1
tff(fact_2408_le__supI1,axiom,
    ! [X: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)) ) ).

% le_supI1
tff(fact_2409_le__supI1,axiom,
    ! [X: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)) ) ).

% le_supI1
tff(fact_2410_le__supI1,axiom,
    ! [X: set_nat,A: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),A)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) ) ).

% le_supI1
tff(fact_2411_le__supI2,axiom,
    ! [X: assn,B: assn,A: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),B)
     => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B)) ) ).

% le_supI2
tff(fact_2412_le__supI2,axiom,
    ! [X: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)) ) ).

% le_supI2
tff(fact_2413_le__supI2,axiom,
    ! [X: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)) ) ).

% le_supI2
tff(fact_2414_le__supI2,axiom,
    ! [X: set_nat,B: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),B)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) ) ).

% le_supI2
tff(fact_2415_sup_Omono,axiom,
    ! [C2: assn,A: assn,D2: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,C2),A)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,D2),B)
       => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,C2),D2)),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B)) ) ) ).

% sup.mono
tff(fact_2416_sup_Omono,axiom,
    ! [C2: nat,A: nat,D2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,D2),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,C2),D2)),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)) ) ) ).

% sup.mono
tff(fact_2417_sup_Omono,axiom,
    ! [C2: int,A: int,D2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,D2),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,C2),D2)),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)) ) ) ).

% sup.mono
tff(fact_2418_sup_Omono,axiom,
    ! [C2: set_nat,A: set_nat,D2: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C2),A)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,D2),B)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,C2),D2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) ) ) ).

% sup.mono
tff(fact_2419_sup__mono,axiom,
    ! [A: assn,C2: assn,B: assn,D2: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),C2)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),D2)
       => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B)),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,C2),D2)) ) ) ).

% sup_mono
tff(fact_2420_sup__mono,axiom,
    ! [A: nat,C2: nat,B: nat,D2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),D2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,C2),D2)) ) ) ).

% sup_mono
tff(fact_2421_sup__mono,axiom,
    ! [A: int,C2: int,B: int,D2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),D2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)),aa_int_int(aa_int_fun_int_int(sup_sup_int,C2),D2)) ) ) ).

% sup_mono
tff(fact_2422_sup__mono,axiom,
    ! [A: set_nat,C2: set_nat,B: set_nat,D2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),C2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),D2)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,C2),D2)) ) ) ).

% sup_mono
tff(fact_2423_sup__least,axiom,
    ! [Y: assn,X: assn,Z: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,Y),X)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,Z),X)
       => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),Z)),X) ) ) ).

% sup_least
tff(fact_2424_sup__least,axiom,
    ! [Y: nat,X: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),X)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Y),Z)),X) ) ) ).

% sup_least
tff(fact_2425_sup__least,axiom,
    ! [Y: int,X: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),X)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),X)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,Y),Z)),X) ) ) ).

% sup_least
tff(fact_2426_sup__least,axiom,
    ! [Y: set_nat,X: set_nat,Z: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Y),X)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Z),X)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),Z)),X) ) ) ).

% sup_least
tff(fact_2427_le__iff__sup,axiom,
    ! [X: assn,Y: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),Y)
    <=> ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y) = Y ) ) ).

% le_iff_sup
tff(fact_2428_le__iff__sup,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y) = Y ) ) ).

% le_iff_sup
tff(fact_2429_le__iff__sup,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
    <=> ( aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y) = Y ) ) ).

% le_iff_sup
tff(fact_2430_le__iff__sup,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Y)
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y) = Y ) ) ).

% le_iff_sup
tff(fact_2431_sup_OorderE,axiom,
    ! [B: assn,A: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),A)
     => ( A = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B) ) ) ).

% sup.orderE
tff(fact_2432_sup_OorderE,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( A = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B) ) ) ).

% sup.orderE
tff(fact_2433_sup_OorderE,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( A = aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B) ) ) ).

% sup.orderE
tff(fact_2434_sup_OorderE,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A)
     => ( A = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) ) ) ).

% sup.orderE
tff(fact_2435_sup_OorderI,axiom,
    ! [A: assn,B: assn] :
      ( ( A = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B) )
     => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),A) ) ).

% sup.orderI
tff(fact_2436_sup_OorderI,axiom,
    ! [A: nat,B: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A) ) ).

% sup.orderI
tff(fact_2437_sup_OorderI,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ).

% sup.orderI
tff(fact_2438_sup_OorderI,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( A = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) )
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A) ) ).

% sup.orderI
tff(fact_2439_sup__unique,axiom,
    ! [F: fun_as3983628580203722482n_assn,X: assn,Y: assn] :
      ( ! [X2: assn,Y2: assn] : aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X2),aa_assn_assn(aa_ass954072293579886259n_assn(F,X2),Y2))
     => ( ! [X2: assn,Y2: assn] : aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,Y2),aa_assn_assn(aa_ass954072293579886259n_assn(F,X2),Y2))
       => ( ! [X2: assn,Y2: assn,Z3: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,Y2),X2)
             => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,Z3),X2)
               => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(F,Y2),Z3)),X2) ) )
         => ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y) = aa_assn_assn(aa_ass954072293579886259n_assn(F,X),Y) ) ) ) ) ).

% sup_unique
tff(fact_2440_sup__unique,axiom,
    ! [F: fun_nat_fun_nat_nat,X: nat,Y: nat] :
      ( ! [X2: nat,Y2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),aa_nat_nat(aa_nat_fun_nat_nat(F,X2),Y2))
     => ( ! [X2: nat,Y2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y2),aa_nat_nat(aa_nat_fun_nat_nat(F,X2),Y2))
       => ( ! [X2: nat,Y2: nat,Z3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y2),X2)
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z3),X2)
               => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(F,Y2),Z3)),X2) ) )
         => ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y) = aa_nat_nat(aa_nat_fun_nat_nat(F,X),Y) ) ) ) ) ).

% sup_unique
tff(fact_2441_sup__unique,axiom,
    ! [F: fun_int_fun_int_int,X: int,Y: int] :
      ( ! [X2: int,Y2: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X2),aa_int_int(aa_int_fun_int_int(F,X2),Y2))
     => ( ! [X2: int,Y2: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y2),aa_int_int(aa_int_fun_int_int(F,X2),Y2))
       => ( ! [X2: int,Y2: int,Z3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y2),X2)
             => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z3),X2)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(F,Y2),Z3)),X2) ) )
         => ( aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y) = aa_int_int(aa_int_fun_int_int(F,X),Y) ) ) ) ) ).

% sup_unique
tff(fact_2442_sup__unique,axiom,
    ! [F: fun_se1909708154429037836et_nat,X: set_nat,Y: set_nat] :
      ( ! [X2: set_nat,Y2: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(F,X2),Y2))
     => ( ! [X2: set_nat,Y2: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Y2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(F,X2),Y2))
       => ( ! [X2: set_nat,Y2: set_nat,Z3: set_nat] :
              ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Y2),X2)
             => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Z3),X2)
               => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(F,Y2),Z3)),X2) ) )
         => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(F,X),Y) ) ) ) ) ).

% sup_unique
tff(fact_2443_sup_Oabsorb1,axiom,
    ! [B: assn,A: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),A)
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B) = A ) ) ).

% sup.absorb1
tff(fact_2444_sup_Oabsorb1,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B) = A ) ) ).

% sup.absorb1
tff(fact_2445_sup_Oabsorb1,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B) = A ) ) ).

% sup.absorb1
tff(fact_2446_sup_Oabsorb1,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) = A ) ) ).

% sup.absorb1
tff(fact_2447_sup_Oabsorb2,axiom,
    ! [A: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B) = B ) ) ).

% sup.absorb2
tff(fact_2448_sup_Oabsorb2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B) = B ) ) ).

% sup.absorb2
tff(fact_2449_sup_Oabsorb2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B) = B ) ) ).

% sup.absorb2
tff(fact_2450_sup_Oabsorb2,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) = B ) ) ).

% sup.absorb2
tff(fact_2451_sup__absorb1,axiom,
    ! [Y: assn,X: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,Y),X)
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y) = X ) ) ).

% sup_absorb1
tff(fact_2452_sup__absorb1,axiom,
    ! [Y: nat,X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y) = X ) ) ).

% sup_absorb1
tff(fact_2453_sup__absorb1,axiom,
    ! [Y: int,X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),X)
     => ( aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y) = X ) ) ).

% sup_absorb1
tff(fact_2454_sup__absorb1,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Y),X)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y) = X ) ) ).

% sup_absorb1
tff(fact_2455_sup__absorb2,axiom,
    ! [X: assn,Y: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),Y)
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y) = Y ) ) ).

% sup_absorb2
tff(fact_2456_sup__absorb2,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y) = Y ) ) ).

% sup_absorb2
tff(fact_2457_sup__absorb2,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
     => ( aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y) = Y ) ) ).

% sup_absorb2
tff(fact_2458_sup__absorb2,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Y)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y) = Y ) ) ).

% sup_absorb2
tff(fact_2459_sup_OboundedE,axiom,
    ! [B: assn,C2: assn,A: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,B),C2)),A)
     => ~ ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),A)
         => ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,C2),A) ) ) ).

% sup.boundedE
tff(fact_2460_sup_OboundedE,axiom,
    ! [B: nat,C2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,B),C2)),A)
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),A) ) ) ).

% sup.boundedE
tff(fact_2461_sup_OboundedE,axiom,
    ! [B: int,C2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,B),C2)),A)
     => ~ ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
         => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),A) ) ) ).

% sup.boundedE
tff(fact_2462_sup_OboundedE,axiom,
    ! [B: set_nat,C2: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B),C2)),A)
     => ~ ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A)
         => ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C2),A) ) ) ).

% sup.boundedE
tff(fact_2463_sup_OboundedI,axiom,
    ! [B: assn,A: assn,C2: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),A)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,C2),A)
       => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,B),C2)),A) ) ) ).

% sup.boundedI
tff(fact_2464_sup_OboundedI,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,B),C2)),A) ) ) ).

% sup.boundedI
tff(fact_2465_sup_OboundedI,axiom,
    ! [B: int,A: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,B),C2)),A) ) ) ).

% sup.boundedI
tff(fact_2466_sup_OboundedI,axiom,
    ! [B: set_nat,A: set_nat,C2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C2),A)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B),C2)),A) ) ) ).

% sup.boundedI
tff(fact_2467_sup_Oorder__iff,axiom,
    ! [B: assn,A: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),A)
    <=> ( A = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B) ) ) ).

% sup.order_iff
tff(fact_2468_sup_Oorder__iff,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
    <=> ( A = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B) ) ) ).

% sup.order_iff
tff(fact_2469_sup_Oorder__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
    <=> ( A = aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B) ) ) ).

% sup.order_iff
tff(fact_2470_sup_Oorder__iff,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A)
    <=> ( A = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) ) ) ).

% sup.order_iff
tff(fact_2471_sup_Ocobounded1,axiom,
    ! [A: assn,B: assn] : aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B)) ).

% sup.cobounded1
tff(fact_2472_sup_Ocobounded1,axiom,
    ! [A: nat,B: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)) ).

% sup.cobounded1
tff(fact_2473_sup_Ocobounded1,axiom,
    ! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)) ).

% sup.cobounded1
tff(fact_2474_sup_Ocobounded1,axiom,
    ! [A: set_nat,B: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) ).

% sup.cobounded1
tff(fact_2475_sup_Ocobounded2,axiom,
    ! [B: assn,A: assn] : aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B)) ).

% sup.cobounded2
tff(fact_2476_sup_Ocobounded2,axiom,
    ! [B: nat,A: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)) ).

% sup.cobounded2
tff(fact_2477_sup_Ocobounded2,axiom,
    ! [B: int,A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)) ).

% sup.cobounded2
tff(fact_2478_sup_Ocobounded2,axiom,
    ! [B: set_nat,A: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) ).

% sup.cobounded2
tff(fact_2479_sup_Oabsorb__iff1,axiom,
    ! [B: assn,A: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),A)
    <=> ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B) = A ) ) ).

% sup.absorb_iff1
tff(fact_2480_sup_Oabsorb__iff1,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B) = A ) ) ).

% sup.absorb_iff1
tff(fact_2481_sup_Oabsorb__iff1,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
    <=> ( aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B) = A ) ) ).

% sup.absorb_iff1
tff(fact_2482_sup_Oabsorb__iff1,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A)
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) = A ) ) ).

% sup.absorb_iff1
tff(fact_2483_sup_Oabsorb__iff2,axiom,
    ! [A: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
    <=> ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B) = B ) ) ).

% sup.absorb_iff2
tff(fact_2484_sup_Oabsorb__iff2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B) = B ) ) ).

% sup.absorb_iff2
tff(fact_2485_sup_Oabsorb__iff2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
    <=> ( aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B) = B ) ) ).

% sup.absorb_iff2
tff(fact_2486_sup_Oabsorb__iff2,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B)
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) = B ) ) ).

% sup.absorb_iff2
tff(fact_2487_sup_OcoboundedI1,axiom,
    ! [C2: assn,A: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,C2),A)
     => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,C2),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B)) ) ).

% sup.coboundedI1
tff(fact_2488_sup_OcoboundedI1,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)) ) ).

% sup.coboundedI1
tff(fact_2489_sup_OcoboundedI1,axiom,
    ! [C2: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)) ) ).

% sup.coboundedI1
tff(fact_2490_sup_OcoboundedI1,axiom,
    ! [C2: set_nat,A: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C2),A)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) ) ).

% sup.coboundedI1
tff(fact_2491_sup_OcoboundedI2,axiom,
    ! [C2: assn,B: assn,A: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,C2),B)
     => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,C2),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B)) ) ).

% sup.coboundedI2
tff(fact_2492_sup_OcoboundedI2,axiom,
    ! [C2: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)) ) ).

% sup.coboundedI2
tff(fact_2493_sup_OcoboundedI2,axiom,
    ! [C2: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)) ) ).

% sup.coboundedI2
tff(fact_2494_sup_OcoboundedI2,axiom,
    ! [C2: set_nat,B: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C2),B)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) ) ).

% sup.coboundedI2
tff(fact_2495_Int__Collect__mono,axiom,
    ! [A2: set_fu3555266933746557007_nat_o,B2: set_fu3555266933746557007_nat_o,Pa: fun_fu5135969500543783852at_o_o,Q: fun_fu5135969500543783852at_o_o] :
      ( aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(A2),B2)
     => ( ! [X2: fun_Pr6460944699146269935_nat_o] :
            ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X2),A2)
           => ( aa_fun1675921005893230629at_o_o(Pa,X2)
             => aa_fun1675921005893230629at_o_o(Q,X2) ) )
       => aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(inf_in3376754634429691197_nat_o(A2,collec6407654713642994138_nat_o(Pa))),inf_in3376754634429691197_nat_o(B2,collec6407654713642994138_nat_o(Q))) ) ) ).

% Int_Collect_mono
tff(fact_2496_Int__Collect__mono,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,Pa: fun_Pr4796943543331071514_nat_o,Q: fun_Pr4796943543331071514_nat_o] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A2),B2)
     => ( ! [X2: product_prod_nat_nat] :
            ( member8440522571783428010at_nat(X2,A2)
           => ( aa_Pro6150637604445964947_nat_o(Pa,X2)
             => aa_Pro6150637604445964947_nat_o(Q,X2) ) )
       => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),collec3392354462482085612at_nat(Pa))),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B2),collec3392354462482085612at_nat(Q))) ) ) ).

% Int_Collect_mono
tff(fact_2497_Int__Collect__mono,axiom,
    ! [A2: set_nat,B2: set_nat,Pa: fun_nat_o,Q: fun_nat_o] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B2)
     => ( ! [X2: nat] :
            ( aa_set_nat_o(member_nat2(X2),A2)
           => ( aa_nat_o(Pa,X2)
             => aa_nat_o(Q,X2) ) )
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),collect_nat(Pa))),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),collect_nat(Q))) ) ) ).

% Int_Collect_mono
tff(fact_2498_Int__greatest,axiom,
    ! [C3: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,C3),A2)
     => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,C3),B2)
       => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,C3),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2)) ) ) ).

% Int_greatest
tff(fact_2499_Int__greatest,axiom,
    ! [C3: set_nat,A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C3),A2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C3),B2)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2)) ) ) ).

% Int_greatest
tff(fact_2500_Int__absorb2,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A2),B2)
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2) = A2 ) ) ).

% Int_absorb2
tff(fact_2501_Int__absorb2,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B2)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) = A2 ) ) ).

% Int_absorb2
tff(fact_2502_Int__absorb1,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,B2),A2)
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2) = B2 ) ) ).

% Int_absorb1
tff(fact_2503_Int__absorb1,axiom,
    ! [B2: set_nat,A2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B2),A2)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) = B2 ) ) ).

% Int_absorb1
tff(fact_2504_Int__lower2,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2)),B2) ).

% Int_lower2
tff(fact_2505_Int__lower2,axiom,
    ! [A2: set_nat,B2: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2)),B2) ).

% Int_lower2
tff(fact_2506_Int__lower1,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2)),A2) ).

% Int_lower1
tff(fact_2507_Int__lower1,axiom,
    ! [A2: set_nat,B2: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2)),A2) ).

% Int_lower1
tff(fact_2508_Int__mono,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,C3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,D: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A2),C3)
     => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,B2),D)
       => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,C3),D)) ) ) ).

% Int_mono
tff(fact_2509_Int__mono,axiom,
    ! [A2: set_nat,C3: set_nat,B2: set_nat,D: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),C3)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B2),D)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,C3),D)) ) ) ).

% Int_mono
tff(fact_2510_inter__eq__subsetI,axiom,
    ! [S: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,S),S3)
     => ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),S3) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B2),S3) )
       => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),S) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B2),S) ) ) ) ).

% inter_eq_subsetI
tff(fact_2511_inter__eq__subsetI,axiom,
    ! [S: set_nat,S3: set_nat,A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S),S3)
     => ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),S3) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),S3) )
       => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),S) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),S) ) ) ) ).

% inter_eq_subsetI
tff(fact_2512_Un__mono,axiom,
    ! [A2: set_nat,C3: set_nat,B2: set_nat,D: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),C3)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B2),D)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,C3),D)) ) ) ).

% Un_mono
tff(fact_2513_Un__least,axiom,
    ! [A2: set_nat,C3: set_nat,B2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),C3)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B2),C3)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)),C3) ) ) ).

% Un_least
tff(fact_2514_Un__upper1,axiom,
    ! [A2: set_nat,B2: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)) ).

% Un_upper1
tff(fact_2515_Un__upper2,axiom,
    ! [B2: set_nat,A2: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)) ).

% Un_upper2
tff(fact_2516_Un__absorb1,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B2)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2) = B2 ) ) ).

% Un_absorb1
tff(fact_2517_Un__absorb2,axiom,
    ! [B2: set_nat,A2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B2),A2)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2) = A2 ) ) ).

% Un_absorb2
tff(fact_2518_subset__UnE,axiom,
    ! [C3: set_nat,A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2))
     => ~ ! [A5: set_nat] :
            ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A5),A2)
           => ! [B6: set_nat] :
                ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B6),B2)
               => ( C3 != aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A5),B6) ) ) ) ) ).

% subset_UnE
tff(fact_2519_subset__Un__eq,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B2)
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2) = B2 ) ) ).

% subset_Un_eq
tff(fact_2520_subset__iff__psubset__eq,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B2)
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A2),B2)
        | ( A2 = B2 ) ) ) ).

% subset_iff_psubset_eq
tff(fact_2521_subset__psubset__trans,axiom,
    ! [A2: set_nat,B2: set_nat,C3: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,B2),C3)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A2),C3) ) ) ).

% subset_psubset_trans
tff(fact_2522_subset__not__subset__eq,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A2),B2)
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B2)
        & ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B2),A2) ) ) ).

% subset_not_subset_eq
tff(fact_2523_psubset__subset__trans,axiom,
    ! [A2: set_nat,B2: set_nat,C3: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A2),B2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B2),C3)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A2),C3) ) ) ).

% psubset_subset_trans
tff(fact_2524_psubset__imp__subset,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A2),B2)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B2) ) ).

% psubset_imp_subset
tff(fact_2525_psubset__eq,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A2),B2)
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B2)
        & ( A2 != B2 ) ) ) ).

% psubset_eq
tff(fact_2526_psubsetE,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A2),B2)
     => ~ ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B2)
         => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B2),A2) ) ) ).

% psubsetE
tff(fact_2527_less__eq__assn__def,axiom,
    ! [A: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
    <=> ( A = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B) ) ) ).

% less_eq_assn_def
tff(fact_2528_Max__mono,axiom,
    ! [M: set_o,N4: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(M),N4)
     => ( ( M != bot_bot_set_o )
       => ( aa_set_o_o(finite_finite_o,N4)
         => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,aa_set_o_o(lattic1921953407002678535_Max_o,M)),aa_set_o_o(lattic1921953407002678535_Max_o,N4)) ) ) ) ).

% Max_mono
tff(fact_2529_Max__mono,axiom,
    ! [M: set_int,N4: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(M),N4)
     => ( ( M != bot_bot_set_int )
       => ( aa_set_int_o(finite_finite_int,N4)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,lattic8263393255366662781ax_int(M)),lattic8263393255366662781ax_int(N4)) ) ) ) ).

% Max_mono
tff(fact_2530_Max__mono,axiom,
    ! [M: set_nat,N4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,M),N4)
     => ( ( M != bot_bot_set_nat )
       => ( aa_set_nat_o(finite_finite_nat,N4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,lattic8265883725875713057ax_nat(M)),lattic8265883725875713057ax_nat(N4)) ) ) ) ).

% Max_mono
tff(fact_2531_Max_Osubset__imp,axiom,
    ! [A2: set_o,B2: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(A2),B2)
     => ( ( A2 != bot_bot_set_o )
       => ( aa_set_o_o(finite_finite_o,B2)
         => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,aa_set_o_o(lattic1921953407002678535_Max_o,A2)),aa_set_o_o(lattic1921953407002678535_Max_o,B2)) ) ) ) ).

% Max.subset_imp
tff(fact_2532_Max_Osubset__imp,axiom,
    ! [A2: set_int,B2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),B2)
     => ( ( A2 != bot_bot_set_int )
       => ( aa_set_int_o(finite_finite_int,B2)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,lattic8263393255366662781ax_int(A2)),lattic8263393255366662781ax_int(B2)) ) ) ) ).

% Max.subset_imp
tff(fact_2533_Max_Osubset__imp,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B2)
     => ( ( A2 != bot_bot_set_nat )
       => ( aa_set_nat_o(finite_finite_nat,B2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,lattic8265883725875713057ax_nat(A2)),lattic8265883725875713057ax_nat(B2)) ) ) ) ).

% Max.subset_imp
tff(fact_2534_Min__antimono,axiom,
    ! [M: set_o,N4: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(M),N4)
     => ( ( M != bot_bot_set_o )
       => ( aa_set_o_o(finite_finite_o,N4)
         => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,aa_set_o_o(lattic1973801136483472281_Min_o,N4)),aa_set_o_o(lattic1973801136483472281_Min_o,M)) ) ) ) ).

% Min_antimono
tff(fact_2535_Min__antimono,axiom,
    ! [M: set_int,N4: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(M),N4)
     => ( ( M != bot_bot_set_int )
       => ( aa_set_int_o(finite_finite_int,N4)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,lattic8718645017227715691in_int(N4)),lattic8718645017227715691in_int(M)) ) ) ) ).

% Min_antimono
tff(fact_2536_Min__antimono,axiom,
    ! [M: set_nat,N4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,M),N4)
     => ( ( M != bot_bot_set_nat )
       => ( aa_set_nat_o(finite_finite_nat,N4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,lattic8721135487736765967in_nat(N4)),lattic8721135487736765967in_nat(M)) ) ) ) ).

% Min_antimono
tff(fact_2537_Min_Osubset__imp,axiom,
    ! [A2: set_o,B2: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(A2),B2)
     => ( ( A2 != bot_bot_set_o )
       => ( aa_set_o_o(finite_finite_o,B2)
         => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,aa_set_o_o(lattic1973801136483472281_Min_o,B2)),aa_set_o_o(lattic1973801136483472281_Min_o,A2)) ) ) ) ).

% Min.subset_imp
tff(fact_2538_Min_Osubset__imp,axiom,
    ! [A2: set_int,B2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),B2)
     => ( ( A2 != bot_bot_set_int )
       => ( aa_set_int_o(finite_finite_int,B2)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,lattic8718645017227715691in_int(B2)),lattic8718645017227715691in_int(A2)) ) ) ) ).

% Min.subset_imp
tff(fact_2539_Min_Osubset__imp,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B2)
     => ( ( A2 != bot_bot_set_nat )
       => ( aa_set_nat_o(finite_finite_nat,B2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,lattic8721135487736765967in_nat(B2)),lattic8721135487736765967in_nat(A2)) ) ) ) ).

% Min.subset_imp
tff(fact_2540_Inf__fin_Osubset__imp,axiom,
    ! [A2: set_o,B2: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(A2),B2)
     => ( ( A2 != bot_bot_set_o )
       => ( aa_set_o_o(finite_finite_o,B2)
         => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,aa_set_o_o(lattic4107685809792843317_fin_o,B2)),aa_set_o_o(lattic4107685809792843317_fin_o,A2)) ) ) ) ).

% Inf_fin.subset_imp
tff(fact_2541_Inf__fin_Osubset__imp,axiom,
    ! [A2: set_assn,B2: set_assn] :
      ( ord_less_eq_set_assn(A2,B2)
     => ( ( A2 != bot_bot_set_assn )
       => ( finite_finite_assn(B2)
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_set_assn_assn(lattic47131356835913163n_assn,B2)),aa_set_assn_assn(lattic47131356835913163n_assn,A2)) ) ) ) ).

% Inf_fin.subset_imp
tff(fact_2542_Inf__fin_Osubset__imp,axiom,
    ! [A2: set_int,B2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),B2)
     => ( ( A2 != bot_bot_set_int )
       => ( aa_set_int_o(finite_finite_int,B2)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,lattic5235898064620869839in_int(B2)),lattic5235898064620869839in_int(A2)) ) ) ) ).

% Inf_fin.subset_imp
tff(fact_2543_Inf__fin_Osubset__imp,axiom,
    ! [A2: set_set_nat,B2: set_set_nat] :
      ( ord_le6893508408891458716et_nat(A2,B2)
     => ( ( A2 != bot_bot_set_set_nat )
       => ( finite1152437895449049373et_nat(B2)
         => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set6364576160234933248et_nat(lattic3014633134055518761et_nat,B2)),aa_set6364576160234933248et_nat(lattic3014633134055518761et_nat,A2)) ) ) ) ).

% Inf_fin.subset_imp
tff(fact_2544_Inf__fin_Osubset__imp,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B2)
     => ( ( A2 != bot_bot_set_nat )
       => ( aa_set_nat_o(finite_finite_nat,B2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(lattic5238388535129920115in_nat,B2)),aa_set_nat_nat(lattic5238388535129920115in_nat,A2)) ) ) ) ).

% Inf_fin.subset_imp
tff(fact_2545_Sup__fin_Osubset__imp,axiom,
    ! [A2: set_o,B2: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(A2),B2)
     => ( ( A2 != bot_bot_set_o )
       => ( aa_set_o_o(finite_finite_o,B2)
         => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,aa_set_o_o(lattic1508158080041050831_fin_o,A2)),aa_set_o_o(lattic1508158080041050831_fin_o,B2)) ) ) ) ).

% Sup_fin.subset_imp
tff(fact_2546_Sup__fin_Osubset__imp,axiom,
    ! [A2: set_assn,B2: set_assn] :
      ( ord_less_eq_set_assn(A2,B2)
     => ( ( A2 != bot_bot_set_assn )
       => ( finite_finite_assn(B2)
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,lattic2150320897289308081n_assn(A2)),lattic2150320897289308081n_assn(B2)) ) ) ) ).

% Sup_fin.subset_imp
tff(fact_2547_Sup__fin_Osubset__imp,axiom,
    ! [A2: set_int,B2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),B2)
     => ( ( A2 != bot_bot_set_int )
       => ( aa_set_int_o(finite_finite_int,B2)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,lattic1091506334969745077in_int(A2)),lattic1091506334969745077in_int(B2)) ) ) ) ).

% Sup_fin.subset_imp
tff(fact_2548_Sup__fin_Osubset__imp,axiom,
    ! [A2: set_set_nat,B2: set_set_nat] :
      ( ord_le6893508408891458716et_nat(A2,B2)
     => ( ( A2 != bot_bot_set_set_nat )
       => ( finite1152437895449049373et_nat(B2)
         => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,lattic3835124923745554447et_nat(A2)),lattic3835124923745554447et_nat(B2)) ) ) ) ).

% Sup_fin.subset_imp
tff(fact_2549_Sup__fin_Osubset__imp,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B2)
     => ( ( A2 != bot_bot_set_nat )
       => ( aa_set_nat_o(finite_finite_nat,B2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,lattic1093996805478795353in_nat(A2)),lattic1093996805478795353in_nat(B2)) ) ) ) ).

% Sup_fin.subset_imp
tff(fact_2550_infinite__finite__induct,axiom,
    ! [Pa: fun_se7459991829846211660at_o_o,A2: set_fu3555266933746557007_nat_o] :
      ( ! [A6: set_fu3555266933746557007_nat_o] :
          ( ~ aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,A6)
         => aa_set5091322549519198405at_o_o(Pa,A6) )
     => ( aa_set5091322549519198405at_o_o(Pa,bot_bo8632255151191677115_nat_o)
       => ( ! [X2: fun_Pr6460944699146269935_nat_o,F3: set_fu3555266933746557007_nat_o] :
              ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,F3)
             => ( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X2),F3)
               => ( aa_set5091322549519198405at_o_o(Pa,F3)
                 => aa_set5091322549519198405at_o_o(Pa,insert4923026906511862463_nat_o(X2,F3)) ) ) )
         => aa_set5091322549519198405at_o_o(Pa,A2) ) ) ) ).

% infinite_finite_induct
tff(fact_2551_infinite__finite__induct,axiom,
    ! [Pa: fun_set_o_o,A2: set_o] :
      ( ! [A6: set_o] :
          ( ~ aa_set_o_o(finite_finite_o,A6)
         => aa_set_o_o(Pa,A6) )
     => ( aa_set_o_o(Pa,bot_bot_set_o)
       => ( ! [X2: $o,F3: set_o] :
              ( aa_set_o_o(finite_finite_o,F3)
             => ( ~ aa_set_o_o(member_o((X2)),F3)
               => ( aa_set_o_o(Pa,F3)
                 => aa_set_o_o(Pa,insert_o((X2),F3)) ) ) )
         => aa_set_o_o(Pa,A2) ) ) ) ).

% infinite_finite_induct
tff(fact_2552_infinite__finite__induct,axiom,
    ! [Pa: fun_set_nat_o,A2: set_nat] :
      ( ! [A6: set_nat] :
          ( ~ aa_set_nat_o(finite_finite_nat,A6)
         => aa_set_nat_o(Pa,A6) )
     => ( aa_set_nat_o(Pa,bot_bot_set_nat)
       => ( ! [X2: nat,F3: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,F3)
             => ( ~ aa_set_nat_o(member_nat2(X2),F3)
               => ( aa_set_nat_o(Pa,F3)
                 => aa_set_nat_o(Pa,insert_nat2(X2,F3)) ) ) )
         => aa_set_nat_o(Pa,A2) ) ) ) ).

% infinite_finite_induct
tff(fact_2553_finite__ne__induct,axiom,
    ! [F2: set_fu3555266933746557007_nat_o,Pa: fun_se7459991829846211660at_o_o] :
      ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,F2)
     => ( ( F2 != bot_bo8632255151191677115_nat_o )
       => ( ! [X2: fun_Pr6460944699146269935_nat_o] : aa_set5091322549519198405at_o_o(Pa,insert4923026906511862463_nat_o(X2,bot_bo8632255151191677115_nat_o))
         => ( ! [X2: fun_Pr6460944699146269935_nat_o,F3: set_fu3555266933746557007_nat_o] :
                ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,F3)
               => ( ( F3 != bot_bo8632255151191677115_nat_o )
                 => ( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X2),F3)
                   => ( aa_set5091322549519198405at_o_o(Pa,F3)
                     => aa_set5091322549519198405at_o_o(Pa,insert4923026906511862463_nat_o(X2,F3)) ) ) ) )
           => aa_set5091322549519198405at_o_o(Pa,F2) ) ) ) ) ).

% finite_ne_induct
tff(fact_2554_finite__ne__induct,axiom,
    ! [F2: set_o,Pa: fun_set_o_o] :
      ( aa_set_o_o(finite_finite_o,F2)
     => ( ( F2 != bot_bot_set_o )
       => ( ! [X2: $o] : aa_set_o_o(Pa,insert_o((X2),bot_bot_set_o))
         => ( ! [X2: $o,F3: set_o] :
                ( aa_set_o_o(finite_finite_o,F3)
               => ( ( F3 != bot_bot_set_o )
                 => ( ~ aa_set_o_o(member_o((X2)),F3)
                   => ( aa_set_o_o(Pa,F3)
                     => aa_set_o_o(Pa,insert_o((X2),F3)) ) ) ) )
           => aa_set_o_o(Pa,F2) ) ) ) ) ).

% finite_ne_induct
tff(fact_2555_finite__ne__induct,axiom,
    ! [F2: set_nat,Pa: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,F2)
     => ( ( F2 != bot_bot_set_nat )
       => ( ! [X2: nat] : aa_set_nat_o(Pa,insert_nat2(X2,bot_bot_set_nat))
         => ( ! [X2: nat,F3: set_nat] :
                ( aa_set_nat_o(finite_finite_nat,F3)
               => ( ( F3 != bot_bot_set_nat )
                 => ( ~ aa_set_nat_o(member_nat2(X2),F3)
                   => ( aa_set_nat_o(Pa,F3)
                     => aa_set_nat_o(Pa,insert_nat2(X2,F3)) ) ) ) )
           => aa_set_nat_o(Pa,F2) ) ) ) ) ).

% finite_ne_induct
tff(fact_2556_finite__induct,axiom,
    ! [F2: set_fu3555266933746557007_nat_o,Pa: fun_se7459991829846211660at_o_o] :
      ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,F2)
     => ( aa_set5091322549519198405at_o_o(Pa,bot_bo8632255151191677115_nat_o)
       => ( ! [X2: fun_Pr6460944699146269935_nat_o,F3: set_fu3555266933746557007_nat_o] :
              ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,F3)
             => ( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X2),F3)
               => ( aa_set5091322549519198405at_o_o(Pa,F3)
                 => aa_set5091322549519198405at_o_o(Pa,insert4923026906511862463_nat_o(X2,F3)) ) ) )
         => aa_set5091322549519198405at_o_o(Pa,F2) ) ) ) ).

% finite_induct
tff(fact_2557_finite__induct,axiom,
    ! [F2: set_o,Pa: fun_set_o_o] :
      ( aa_set_o_o(finite_finite_o,F2)
     => ( aa_set_o_o(Pa,bot_bot_set_o)
       => ( ! [X2: $o,F3: set_o] :
              ( aa_set_o_o(finite_finite_o,F3)
             => ( ~ aa_set_o_o(member_o((X2)),F3)
               => ( aa_set_o_o(Pa,F3)
                 => aa_set_o_o(Pa,insert_o((X2),F3)) ) ) )
         => aa_set_o_o(Pa,F2) ) ) ) ).

% finite_induct
tff(fact_2558_finite__induct,axiom,
    ! [F2: set_nat,Pa: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,F2)
     => ( aa_set_nat_o(Pa,bot_bot_set_nat)
       => ( ! [X2: nat,F3: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,F3)
             => ( ~ aa_set_nat_o(member_nat2(X2),F3)
               => ( aa_set_nat_o(Pa,F3)
                 => aa_set_nat_o(Pa,insert_nat2(X2,F3)) ) ) )
         => aa_set_nat_o(Pa,F2) ) ) ) ).

% finite_induct
tff(fact_2559_finite_Osimps,axiom,
    ! [A: set_o] :
      ( aa_set_o_o(finite_finite_o,A)
    <=> ( ( A = bot_bot_set_o )
        | ? [A7: set_o,A4: $o] :
            ( ( A = insert_o((A4),A7) )
            & aa_set_o_o(finite_finite_o,A7) ) ) ) ).

% finite.simps
tff(fact_2560_finite_Osimps,axiom,
    ! [A: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
    <=> ( ( A = bot_bot_set_nat )
        | ? [A7: set_nat,A4: nat] :
            ( ( A = insert_nat2(A4,A7) )
            & aa_set_nat_o(finite_finite_nat,A7) ) ) ) ).

% finite.simps
tff(fact_2561_finite_Ocases,axiom,
    ! [A: set_o] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( ( A != bot_bot_set_o )
       => ~ ! [A6: set_o] :
              ( ? [A3: $o] : A = insert_o((A3),A6)
             => ~ aa_set_o_o(finite_finite_o,A6) ) ) ) ).

% finite.cases
tff(fact_2562_finite_Ocases,axiom,
    ! [A: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( ( A != bot_bot_set_nat )
       => ~ ! [A6: set_nat] :
              ( ? [A3: nat] : A = insert_nat2(A3,A6)
             => ~ aa_set_nat_o(finite_finite_nat,A6) ) ) ) ).

% finite.cases
tff(fact_2563_insert__is__Un,axiom,
    ! [A: $o,A2: set_o] : insert_o((A),A2) = sup_sup_set_o(insert_o((A),bot_bot_set_o),A2) ).

% insert_is_Un
tff(fact_2564_insert__is__Un,axiom,
    ! [A: nat,A2: set_nat] : insert_nat2(A,A2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,insert_nat2(A,bot_bot_set_nat)),A2) ).

% insert_is_Un
tff(fact_2565_Un__singleton__iff,axiom,
    ! [A2: set_o,B2: set_o,X: $o] :
      ( ( sup_sup_set_o(A2,B2) = insert_o((X),bot_bot_set_o) )
    <=> ( ( ( A2 = bot_bot_set_o )
          & ( B2 = insert_o((X),bot_bot_set_o) ) )
        | ( ( A2 = insert_o((X),bot_bot_set_o) )
          & ( B2 = bot_bot_set_o ) )
        | ( ( A2 = insert_o((X),bot_bot_set_o) )
          & ( B2 = insert_o((X),bot_bot_set_o) ) ) ) ) ).

% Un_singleton_iff
tff(fact_2566_Un__singleton__iff,axiom,
    ! [A2: set_nat,B2: set_nat,X: nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2) = insert_nat2(X,bot_bot_set_nat) )
    <=> ( ( ( A2 = bot_bot_set_nat )
          & ( B2 = insert_nat2(X,bot_bot_set_nat) ) )
        | ( ( A2 = insert_nat2(X,bot_bot_set_nat) )
          & ( B2 = bot_bot_set_nat ) )
        | ( ( A2 = insert_nat2(X,bot_bot_set_nat) )
          & ( B2 = insert_nat2(X,bot_bot_set_nat) ) ) ) ) ).

% Un_singleton_iff
tff(fact_2567_singleton__Un__iff,axiom,
    ! [X: $o,A2: set_o,B2: set_o] :
      ( ( insert_o((X),bot_bot_set_o) = sup_sup_set_o(A2,B2) )
    <=> ( ( ( A2 = bot_bot_set_o )
          & ( B2 = insert_o((X),bot_bot_set_o) ) )
        | ( ( A2 = insert_o((X),bot_bot_set_o) )
          & ( B2 = bot_bot_set_o ) )
        | ( ( A2 = insert_o((X),bot_bot_set_o) )
          & ( B2 = insert_o((X),bot_bot_set_o) ) ) ) ) ).

% singleton_Un_iff
tff(fact_2568_singleton__Un__iff,axiom,
    ! [X: nat,A2: set_nat,B2: set_nat] :
      ( ( insert_nat2(X,bot_bot_set_nat) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2) )
    <=> ( ( ( A2 = bot_bot_set_nat )
          & ( B2 = insert_nat2(X,bot_bot_set_nat) ) )
        | ( ( A2 = insert_nat2(X,bot_bot_set_nat) )
          & ( B2 = bot_bot_set_nat ) )
        | ( ( A2 = insert_nat2(X,bot_bot_set_nat) )
          & ( B2 = insert_nat2(X,bot_bot_set_nat) ) ) ) ) ).

% singleton_Un_iff
tff(fact_2569_finite__has__minimal,axiom,
    ! [A2: set_o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ( A2 != bot_bot_set_o )
       => ? [X2: $o] :
            ( aa_set_o_o(member_o((X2)),A2)
            & ! [Xa3: $o] :
                ( aa_set_o_o(member_o((Xa3)),A2)
               => ( aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(Xa3)),(X2))
                 => ( (X2)
                  <=> (Xa3) ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_2570_finite__has__minimal,axiom,
    ! [A2: set_assn] :
      ( finite_finite_assn(A2)
     => ( ( A2 != bot_bot_set_assn )
       => ? [X2: assn] :
            ( member_assn(X2,A2)
            & ! [Xa3: assn] :
                ( member_assn(Xa3,A2)
               => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,Xa3),X2)
                 => ( X2 = Xa3 ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_2571_finite__has__minimal,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( A2 != bot_bot_set_nat )
       => ? [X2: nat] :
            ( aa_set_nat_o(member_nat2(X2),A2)
            & ! [Xa3: nat] :
                ( aa_set_nat_o(member_nat2(Xa3),A2)
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa3),X2)
                 => ( X2 = Xa3 ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_2572_finite__has__minimal,axiom,
    ! [A2: set_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( A2 != bot_bot_set_int )
       => ? [X2: int] :
            ( aa_set_int_o(member_int2(X2),A2)
            & ! [Xa3: int] :
                ( aa_set_int_o(member_int2(Xa3),A2)
               => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa3),X2)
                 => ( X2 = Xa3 ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_2573_finite__has__minimal,axiom,
    ! [A2: set_set_nat] :
      ( finite1152437895449049373et_nat(A2)
     => ( ( A2 != bot_bot_set_set_nat )
       => ? [X2: set_nat] :
            ( member_set_nat(X2,A2)
            & ! [Xa3: set_nat] :
                ( member_set_nat(Xa3,A2)
               => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Xa3),X2)
                 => ( X2 = Xa3 ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_2574_finite__has__maximal,axiom,
    ! [A2: set_o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ( A2 != bot_bot_set_o )
       => ? [X2: $o] :
            ( aa_set_o_o(member_o((X2)),A2)
            & ! [Xa3: $o] :
                ( aa_set_o_o(member_o((Xa3)),A2)
               => ( aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(X2)),(Xa3))
                 => ( (X2)
                  <=> (Xa3) ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_2575_finite__has__maximal,axiom,
    ! [A2: set_assn] :
      ( finite_finite_assn(A2)
     => ( ( A2 != bot_bot_set_assn )
       => ? [X2: assn] :
            ( member_assn(X2,A2)
            & ! [Xa3: assn] :
                ( member_assn(Xa3,A2)
               => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X2),Xa3)
                 => ( X2 = Xa3 ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_2576_finite__has__maximal,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( A2 != bot_bot_set_nat )
       => ? [X2: nat] :
            ( aa_set_nat_o(member_nat2(X2),A2)
            & ! [Xa3: nat] :
                ( aa_set_nat_o(member_nat2(Xa3),A2)
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Xa3)
                 => ( X2 = Xa3 ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_2577_finite__has__maximal,axiom,
    ! [A2: set_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( A2 != bot_bot_set_int )
       => ? [X2: int] :
            ( aa_set_int_o(member_int2(X2),A2)
            & ! [Xa3: int] :
                ( aa_set_int_o(member_int2(Xa3),A2)
               => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X2),Xa3)
                 => ( X2 = Xa3 ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_2578_finite__has__maximal,axiom,
    ! [A2: set_set_nat] :
      ( finite1152437895449049373et_nat(A2)
     => ( ( A2 != bot_bot_set_set_nat )
       => ? [X2: set_nat] :
            ( member_set_nat(X2,A2)
            & ! [Xa3: set_nat] :
                ( member_set_nat(Xa3,A2)
               => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X2),Xa3)
                 => ( X2 = Xa3 ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_2579_distrib__inf__le,axiom,
    ! [X: product_unit,Y: product_unit,Z: product_unit] : aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),Y)),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),Z))),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,Y),Z))) ).

% distrib_inf_le
tff(fact_2580_distrib__inf__le,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] : aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Z))),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Y),Z))) ).

% distrib_inf_le
tff(fact_2581_distrib__inf__le,axiom,
    ! [X: assn,Y: assn,Z: assn] : aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Z))),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),Z))) ).

% distrib_inf_le
tff(fact_2582_distrib__inf__le,axiom,
    ! [X: nat,Y: nat,Z: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y)),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Z))),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Y),Z))) ).

% distrib_inf_le
tff(fact_2583_distrib__inf__le,axiom,
    ! [X: int,Y: int,Z: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,X),Y)),aa_int_int(aa_int_fun_int_int(inf_inf_int,X),Z))),aa_int_int(aa_int_fun_int_int(inf_inf_int,X),aa_int_int(aa_int_fun_int_int(sup_sup_int,Y),Z))) ).

% distrib_inf_le
tff(fact_2584_distrib__inf__le,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Z))),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),Z))) ).

% distrib_inf_le
tff(fact_2585_distrib__sup__le,axiom,
    ! [X: product_unit,Y: product_unit,Z: product_unit] : aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,X),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,Y),Z))),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,X),Y)),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,X),Z))) ).

% distrib_sup_le
tff(fact_2586_distrib__sup__le,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] : aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y),Z))),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X),Y)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X),Z))) ).

% distrib_sup_le
tff(fact_2587_distrib__sup__le,axiom,
    ! [X: assn,Y: assn,Z: assn] : aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),Z))),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y)),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Z))) ).

% distrib_sup_le
tff(fact_2588_distrib__sup__le,axiom,
    ! [X: nat,Y: nat,Z: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Y),Z))),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y)),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Z))) ).

% distrib_sup_le
tff(fact_2589_distrib__sup__le,axiom,
    ! [X: int,Y: int,Z: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,X),aa_int_int(aa_int_fun_int_int(inf_inf_int,Y),Z))),aa_int_int(aa_int_fun_int_int(inf_inf_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y)),aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Z))) ).

% distrib_sup_le
tff(fact_2590_distrib__sup__le,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),Z))),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Z))) ).

% distrib_sup_le
tff(fact_2591_disjoint__mono,axiom,
    ! [A: set_Pr1261947904930325089at_nat,A8: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,B7: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),A8)
     => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,B),B7)
       => ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A8),B7) = bot_bo2099793752762293965at_nat )
         => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) = bot_bo2099793752762293965at_nat ) ) ) ) ).

% disjoint_mono
tff(fact_2592_disjoint__mono,axiom,
    ! [A: set_o,A8: set_o,B: set_o,B7: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(A),A8)
     => ( aa_set_o_o(ord_less_eq_set_o(B),B7)
       => ( ( inf_inf_set_o(A8,B7) = bot_bot_set_o )
         => ( inf_inf_set_o(A,B) = bot_bot_set_o ) ) ) ) ).

% disjoint_mono
tff(fact_2593_disjoint__mono,axiom,
    ! [A: set_nat,A8: set_nat,B: set_nat,B7: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),A8)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),B7)
       => ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A8),B7) = bot_bot_set_nat )
         => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) = bot_bot_set_nat ) ) ) ) ).

% disjoint_mono
tff(fact_2594_Un__Int__assoc__eq,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C3: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2)),C3) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,B2),C3)) )
    <=> aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,C3),A2) ) ).

% Un_Int_assoc_eq
tff(fact_2595_Un__Int__assoc__eq,axiom,
    ! [A2: set_nat,B2: set_nat,C3: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2)),C3) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B2),C3)) )
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C3),A2) ) ).

% Un_Int_assoc_eq
tff(fact_2596_finite__linorder__max__induct,axiom,
    ! [A2: set_o,Pa: fun_set_o_o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( aa_set_o_o(Pa,bot_bot_set_o)
       => ( ! [B3: $o,A6: set_o] :
              ( aa_set_o_o(finite_finite_o,A6)
             => ( ! [X4: $o] :
                    ( aa_set_o_o(member_o((X4)),A6)
                   => aa_o_o(aa_o_fun_o_o(ord_less_o,(X4)),(B3)) )
               => ( aa_set_o_o(Pa,A6)
                 => aa_set_o_o(Pa,insert_o((B3),A6)) ) ) )
         => aa_set_o_o(Pa,A2) ) ) ) ).

% finite_linorder_max_induct
tff(fact_2597_finite__linorder__max__induct,axiom,
    ! [A2: set_nat,Pa: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(Pa,bot_bot_set_nat)
       => ( ! [B3: nat,A6: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,A6)
             => ( ! [X4: nat] :
                    ( aa_set_nat_o(member_nat2(X4),A6)
                   => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),B3) )
               => ( aa_set_nat_o(Pa,A6)
                 => aa_set_nat_o(Pa,insert_nat2(B3,A6)) ) ) )
         => aa_set_nat_o(Pa,A2) ) ) ) ).

% finite_linorder_max_induct
tff(fact_2598_finite__linorder__max__induct,axiom,
    ! [A2: set_int,Pa: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(Pa,bot_bot_set_int)
       => ( ! [B3: int,A6: set_int] :
              ( aa_set_int_o(finite_finite_int,A6)
             => ( ! [X4: int] :
                    ( aa_set_int_o(member_int2(X4),A6)
                   => aa_int_o(aa_int_fun_int_o(ord_less_int,X4),B3) )
               => ( aa_set_int_o(Pa,A6)
                 => aa_set_int_o(Pa,insert_int2(B3,A6)) ) ) )
         => aa_set_int_o(Pa,A2) ) ) ) ).

% finite_linorder_max_induct
tff(fact_2599_finite__linorder__min__induct,axiom,
    ! [A2: set_o,Pa: fun_set_o_o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( aa_set_o_o(Pa,bot_bot_set_o)
       => ( ! [B3: $o,A6: set_o] :
              ( aa_set_o_o(finite_finite_o,A6)
             => ( ! [X4: $o] :
                    ( aa_set_o_o(member_o((X4)),A6)
                   => aa_o_o(aa_o_fun_o_o(ord_less_o,(B3)),(X4)) )
               => ( aa_set_o_o(Pa,A6)
                 => aa_set_o_o(Pa,insert_o((B3),A6)) ) ) )
         => aa_set_o_o(Pa,A2) ) ) ) ).

% finite_linorder_min_induct
tff(fact_2600_finite__linorder__min__induct,axiom,
    ! [A2: set_nat,Pa: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(Pa,bot_bot_set_nat)
       => ( ! [B3: nat,A6: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,A6)
             => ( ! [X4: nat] :
                    ( aa_set_nat_o(member_nat2(X4),A6)
                   => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B3),X4) )
               => ( aa_set_nat_o(Pa,A6)
                 => aa_set_nat_o(Pa,insert_nat2(B3,A6)) ) ) )
         => aa_set_nat_o(Pa,A2) ) ) ) ).

% finite_linorder_min_induct
tff(fact_2601_finite__linorder__min__induct,axiom,
    ! [A2: set_int,Pa: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(Pa,bot_bot_set_int)
       => ( ! [B3: int,A6: set_int] :
              ( aa_set_int_o(finite_finite_int,A6)
             => ( ! [X4: int] :
                    ( aa_set_int_o(member_int2(X4),A6)
                   => aa_int_o(aa_int_fun_int_o(ord_less_int,B3),X4) )
               => ( aa_set_int_o(Pa,A6)
                 => aa_set_int_o(Pa,insert_int2(B3,A6)) ) ) )
         => aa_set_int_o(Pa,A2) ) ) ) ).

% finite_linorder_min_induct
tff(fact_2602_ShiftD,axiom,
    ! [Kl: list_nat,Kl2: set_list_nat,K: nat] :
      ( member_list_nat(Kl,bNF_Gr1872714664788909425ft_nat(Kl2,K))
     => member_list_nat(aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,K),Kl),Kl2) ) ).

% ShiftD
tff(fact_2603_ShiftD,axiom,
    ! [Kl: list_int,Kl2: set_list_int,K: int] :
      ( member_list_int(Kl,bNF_Gr1870224194279859149ft_int(Kl2,K))
     => member_list_int(aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,K),Kl),Kl2) ) ).

% ShiftD
tff(fact_2604_ShiftD,axiom,
    ! [Kl: list_P8527749157015355191n_assn,Kl2: set_li5131720305576846103n_assn,K: produc6575502325842934193n_assn] :
      ( member852475432509897056n_assn(Kl,bNF_Gr4113829767105464016n_assn(Kl2,K))
     => member852475432509897056n_assn(aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,K),Kl),Kl2) ) ).

% ShiftD
tff(fact_2605_Max_OboundedI,axiom,
    ! [A2: set_o,X: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ( A2 != bot_bot_set_o )
       => ( ! [A3: $o] :
              ( aa_set_o_o(member_o((A3)),A2)
             => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(A3)),(X)) )
         => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,aa_set_o_o(lattic1921953407002678535_Max_o,A2)),(X)) ) ) ) ).

% Max.boundedI
tff(fact_2606_Max_OboundedI,axiom,
    ! [A2: set_nat,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( A2 != bot_bot_set_nat )
       => ( ! [A3: nat] :
              ( aa_set_nat_o(member_nat2(A3),A2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A3),X) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,lattic8265883725875713057ax_nat(A2)),X) ) ) ) ).

% Max.boundedI
tff(fact_2607_Max_OboundedI,axiom,
    ! [A2: set_int,X: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( A2 != bot_bot_set_int )
       => ( ! [A3: int] :
              ( aa_set_int_o(member_int2(A3),A2)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A3),X) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,lattic8263393255366662781ax_int(A2)),X) ) ) ) ).

% Max.boundedI
tff(fact_2608_Max_OboundedE,axiom,
    ! [A2: set_o,X: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ( A2 != bot_bot_set_o )
       => ( aa_o_o(aa_o_fun_o_o(ord_less_eq_o,aa_set_o_o(lattic1921953407002678535_Max_o,A2)),(X))
         => ! [A9: $o] :
              ( aa_set_o_o(member_o((A9)),A2)
             => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(A9)),(X)) ) ) ) ) ).

% Max.boundedE
tff(fact_2609_Max_OboundedE,axiom,
    ! [A2: set_nat,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( A2 != bot_bot_set_nat )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,lattic8265883725875713057ax_nat(A2)),X)
         => ! [A9: nat] :
              ( aa_set_nat_o(member_nat2(A9),A2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A9),X) ) ) ) ) ).

% Max.boundedE
tff(fact_2610_Max_OboundedE,axiom,
    ! [A2: set_int,X: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( A2 != bot_bot_set_int )
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,lattic8263393255366662781ax_int(A2)),X)
         => ! [A9: int] :
              ( aa_set_int_o(member_int2(A9),A2)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A9),X) ) ) ) ) ).

% Max.boundedE
tff(fact_2611_eq__Max__iff,axiom,
    ! [A2: set_o,M2: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ( A2 != bot_bot_set_o )
       => ( ( (M2)
          <=> aa_set_o_o(lattic1921953407002678535_Max_o,A2) )
        <=> ( aa_set_o_o(member_o((M2)),A2)
            & ! [X3: $o] :
                ( aa_set_o_o(member_o((X3)),A2)
               => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(X3)),(M2)) ) ) ) ) ) ).

% eq_Max_iff
tff(fact_2612_eq__Max__iff,axiom,
    ! [A2: set_nat,M2: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( A2 != bot_bot_set_nat )
       => ( ( M2 = lattic8265883725875713057ax_nat(A2) )
        <=> ( aa_set_nat_o(member_nat2(M2),A2)
            & ! [X3: nat] :
                ( aa_set_nat_o(member_nat2(X3),A2)
               => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),M2) ) ) ) ) ) ).

% eq_Max_iff
tff(fact_2613_eq__Max__iff,axiom,
    ! [A2: set_int,M2: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( A2 != bot_bot_set_int )
       => ( ( M2 = lattic8263393255366662781ax_int(A2) )
        <=> ( aa_set_int_o(member_int2(M2),A2)
            & ! [X3: int] :
                ( aa_set_int_o(member_int2(X3),A2)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),M2) ) ) ) ) ) ).

% eq_Max_iff
tff(fact_2614_Max__ge__iff,axiom,
    ! [A2: set_o,X: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ( A2 != bot_bot_set_o )
       => ( aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(X)),aa_set_o_o(lattic1921953407002678535_Max_o,A2))
        <=> ? [X3: $o] :
              ( aa_set_o_o(member_o((X3)),A2)
              & aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(X)),(X3)) ) ) ) ) ).

% Max_ge_iff
tff(fact_2615_Max__ge__iff,axiom,
    ! [A2: set_nat,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( A2 != bot_bot_set_nat )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),lattic8265883725875713057ax_nat(A2))
        <=> ? [X3: nat] :
              ( aa_set_nat_o(member_nat2(X3),A2)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),X3) ) ) ) ) ).

% Max_ge_iff
tff(fact_2616_Max__ge__iff,axiom,
    ! [A2: set_int,X: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( A2 != bot_bot_set_int )
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),lattic8263393255366662781ax_int(A2))
        <=> ? [X3: int] :
              ( aa_set_int_o(member_int2(X3),A2)
              & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),X3) ) ) ) ) ).

% Max_ge_iff
tff(fact_2617_Max__eq__iff,axiom,
    ! [A2: set_o,M2: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ( A2 != bot_bot_set_o )
       => ( ( aa_set_o_o(lattic1921953407002678535_Max_o,A2)
          <=> (M2) )
        <=> ( aa_set_o_o(member_o((M2)),A2)
            & ! [X3: $o] :
                ( aa_set_o_o(member_o((X3)),A2)
               => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(X3)),(M2)) ) ) ) ) ) ).

% Max_eq_iff
tff(fact_2618_Max__eq__iff,axiom,
    ! [A2: set_nat,M2: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( A2 != bot_bot_set_nat )
       => ( ( lattic8265883725875713057ax_nat(A2) = M2 )
        <=> ( aa_set_nat_o(member_nat2(M2),A2)
            & ! [X3: nat] :
                ( aa_set_nat_o(member_nat2(X3),A2)
               => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),M2) ) ) ) ) ) ).

% Max_eq_iff
tff(fact_2619_Max__eq__iff,axiom,
    ! [A2: set_int,M2: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( A2 != bot_bot_set_int )
       => ( ( lattic8263393255366662781ax_int(A2) = M2 )
        <=> ( aa_set_int_o(member_int2(M2),A2)
            & ! [X3: int] :
                ( aa_set_int_o(member_int2(X3),A2)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),M2) ) ) ) ) ) ).

% Max_eq_iff
tff(fact_2620_Min_OboundedI,axiom,
    ! [A2: set_o,X: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ( A2 != bot_bot_set_o )
       => ( ! [A3: $o] :
              ( aa_set_o_o(member_o((A3)),A2)
             => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(X)),(A3)) )
         => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(X)),aa_set_o_o(lattic1973801136483472281_Min_o,A2)) ) ) ) ).

% Min.boundedI
tff(fact_2621_Min_OboundedI,axiom,
    ! [A2: set_nat,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( A2 != bot_bot_set_nat )
       => ( ! [A3: nat] :
              ( aa_set_nat_o(member_nat2(A3),A2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),A3) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),lattic8721135487736765967in_nat(A2)) ) ) ) ).

% Min.boundedI
tff(fact_2622_Min_OboundedI,axiom,
    ! [A2: set_int,X: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( A2 != bot_bot_set_int )
       => ( ! [A3: int] :
              ( aa_set_int_o(member_int2(A3),A2)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),A3) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),lattic8718645017227715691in_int(A2)) ) ) ) ).

% Min.boundedI
tff(fact_2623_Min_OboundedE,axiom,
    ! [A2: set_o,X: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ( A2 != bot_bot_set_o )
       => ( aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(X)),aa_set_o_o(lattic1973801136483472281_Min_o,A2))
         => ! [A9: $o] :
              ( aa_set_o_o(member_o((A9)),A2)
             => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(X)),(A9)) ) ) ) ) ).

% Min.boundedE
tff(fact_2624_Min_OboundedE,axiom,
    ! [A2: set_nat,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( A2 != bot_bot_set_nat )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),lattic8721135487736765967in_nat(A2))
         => ! [A9: nat] :
              ( aa_set_nat_o(member_nat2(A9),A2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),A9) ) ) ) ) ).

% Min.boundedE
tff(fact_2625_Min_OboundedE,axiom,
    ! [A2: set_int,X: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( A2 != bot_bot_set_int )
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),lattic8718645017227715691in_int(A2))
         => ! [A9: int] :
              ( aa_set_int_o(member_int2(A9),A2)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),A9) ) ) ) ) ).

% Min.boundedE
tff(fact_2626_eq__Min__iff,axiom,
    ! [A2: set_o,M2: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ( A2 != bot_bot_set_o )
       => ( ( (M2)
          <=> aa_set_o_o(lattic1973801136483472281_Min_o,A2) )
        <=> ( aa_set_o_o(member_o((M2)),A2)
            & ! [X3: $o] :
                ( aa_set_o_o(member_o((X3)),A2)
               => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(M2)),(X3)) ) ) ) ) ) ).

% eq_Min_iff
tff(fact_2627_eq__Min__iff,axiom,
    ! [A2: set_nat,M2: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( A2 != bot_bot_set_nat )
       => ( ( M2 = lattic8721135487736765967in_nat(A2) )
        <=> ( aa_set_nat_o(member_nat2(M2),A2)
            & ! [X3: nat] :
                ( aa_set_nat_o(member_nat2(X3),A2)
               => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),X3) ) ) ) ) ) ).

% eq_Min_iff
tff(fact_2628_eq__Min__iff,axiom,
    ! [A2: set_int,M2: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( A2 != bot_bot_set_int )
       => ( ( M2 = lattic8718645017227715691in_int(A2) )
        <=> ( aa_set_int_o(member_int2(M2),A2)
            & ! [X3: int] :
                ( aa_set_int_o(member_int2(X3),A2)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,M2),X3) ) ) ) ) ) ).

% eq_Min_iff
tff(fact_2629_Min__le__iff,axiom,
    ! [A2: set_o,X: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ( A2 != bot_bot_set_o )
       => ( aa_o_o(aa_o_fun_o_o(ord_less_eq_o,aa_set_o_o(lattic1973801136483472281_Min_o,A2)),(X))
        <=> ? [X3: $o] :
              ( aa_set_o_o(member_o((X3)),A2)
              & aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(X3)),(X)) ) ) ) ) ).

% Min_le_iff
tff(fact_2630_Min__le__iff,axiom,
    ! [A2: set_nat,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( A2 != bot_bot_set_nat )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,lattic8721135487736765967in_nat(A2)),X)
        <=> ? [X3: nat] :
              ( aa_set_nat_o(member_nat2(X3),A2)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),X) ) ) ) ) ).

% Min_le_iff
tff(fact_2631_Min__le__iff,axiom,
    ! [A2: set_int,X: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( A2 != bot_bot_set_int )
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,lattic8718645017227715691in_int(A2)),X)
        <=> ? [X3: int] :
              ( aa_set_int_o(member_int2(X3),A2)
              & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),X) ) ) ) ) ).

% Min_le_iff
tff(fact_2632_Min__eq__iff,axiom,
    ! [A2: set_o,M2: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ( A2 != bot_bot_set_o )
       => ( ( aa_set_o_o(lattic1973801136483472281_Min_o,A2)
          <=> (M2) )
        <=> ( aa_set_o_o(member_o((M2)),A2)
            & ! [X3: $o] :
                ( aa_set_o_o(member_o((X3)),A2)
               => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(M2)),(X3)) ) ) ) ) ) ).

% Min_eq_iff
tff(fact_2633_Min__eq__iff,axiom,
    ! [A2: set_nat,M2: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( A2 != bot_bot_set_nat )
       => ( ( lattic8721135487736765967in_nat(A2) = M2 )
        <=> ( aa_set_nat_o(member_nat2(M2),A2)
            & ! [X3: nat] :
                ( aa_set_nat_o(member_nat2(X3),A2)
               => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),X3) ) ) ) ) ) ).

% Min_eq_iff
tff(fact_2634_Min__eq__iff,axiom,
    ! [A2: set_int,M2: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( A2 != bot_bot_set_int )
       => ( ( lattic8718645017227715691in_int(A2) = M2 )
        <=> ( aa_set_int_o(member_int2(M2),A2)
            & ! [X3: int] :
                ( aa_set_int_o(member_int2(X3),A2)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,M2),X3) ) ) ) ) ) ).

% Min_eq_iff
tff(fact_2635_Sup__fin_Obounded__iff,axiom,
    ! [A2: set_o,X: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ( A2 != bot_bot_set_o )
       => ( aa_o_o(aa_o_fun_o_o(ord_less_eq_o,aa_set_o_o(lattic1508158080041050831_fin_o,A2)),(X))
        <=> ! [X3: $o] :
              ( aa_set_o_o(member_o((X3)),A2)
             => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(X3)),(X)) ) ) ) ) ).

% Sup_fin.bounded_iff
tff(fact_2636_Sup__fin_Obounded__iff,axiom,
    ! [A2: set_assn,X: assn] :
      ( finite_finite_assn(A2)
     => ( ( A2 != bot_bot_set_assn )
       => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,lattic2150320897289308081n_assn(A2)),X)
        <=> ! [X3: assn] :
              ( member_assn(X3,A2)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X3),X) ) ) ) ) ).

% Sup_fin.bounded_iff
tff(fact_2637_Sup__fin_Obounded__iff,axiom,
    ! [A2: set_nat,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( A2 != bot_bot_set_nat )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,lattic1093996805478795353in_nat(A2)),X)
        <=> ! [X3: nat] :
              ( aa_set_nat_o(member_nat2(X3),A2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),X) ) ) ) ) ).

% Sup_fin.bounded_iff
tff(fact_2638_Sup__fin_Obounded__iff,axiom,
    ! [A2: set_int,X: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( A2 != bot_bot_set_int )
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,lattic1091506334969745077in_int(A2)),X)
        <=> ! [X3: int] :
              ( aa_set_int_o(member_int2(X3),A2)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),X) ) ) ) ) ).

% Sup_fin.bounded_iff
tff(fact_2639_Sup__fin_Obounded__iff,axiom,
    ! [A2: set_set_nat,X: set_nat] :
      ( finite1152437895449049373et_nat(A2)
     => ( ( A2 != bot_bot_set_set_nat )
       => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,lattic3835124923745554447et_nat(A2)),X)
        <=> ! [X3: set_nat] :
              ( member_set_nat(X3,A2)
             => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X3),X) ) ) ) ) ).

% Sup_fin.bounded_iff
tff(fact_2640_Inf__fin_Obounded__iff,axiom,
    ! [A2: set_o,X: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ( A2 != bot_bot_set_o )
       => ( aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(X)),aa_set_o_o(lattic4107685809792843317_fin_o,A2))
        <=> ! [X3: $o] :
              ( aa_set_o_o(member_o((X3)),A2)
             => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(X)),(X3)) ) ) ) ) ).

% Inf_fin.bounded_iff
tff(fact_2641_Inf__fin_Obounded__iff,axiom,
    ! [A2: set_assn,X: assn] :
      ( finite_finite_assn(A2)
     => ( ( A2 != bot_bot_set_assn )
       => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),aa_set_assn_assn(lattic47131356835913163n_assn,A2))
        <=> ! [X3: assn] :
              ( member_assn(X3,A2)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),X3) ) ) ) ) ).

% Inf_fin.bounded_iff
tff(fact_2642_Inf__fin_Obounded__iff,axiom,
    ! [A2: set_nat,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( A2 != bot_bot_set_nat )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),aa_set_nat_nat(lattic5238388535129920115in_nat,A2))
        <=> ! [X3: nat] :
              ( aa_set_nat_o(member_nat2(X3),A2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),X3) ) ) ) ) ).

% Inf_fin.bounded_iff
tff(fact_2643_Inf__fin_Obounded__iff,axiom,
    ! [A2: set_int,X: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( A2 != bot_bot_set_int )
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),lattic5235898064620869839in_int(A2))
        <=> ! [X3: int] :
              ( aa_set_int_o(member_int2(X3),A2)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),X3) ) ) ) ) ).

% Inf_fin.bounded_iff
tff(fact_2644_Inf__fin_Obounded__iff,axiom,
    ! [A2: set_set_nat,X: set_nat] :
      ( finite1152437895449049373et_nat(A2)
     => ( ( A2 != bot_bot_set_set_nat )
       => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),aa_set6364576160234933248et_nat(lattic3014633134055518761et_nat,A2))
        <=> ! [X3: set_nat] :
              ( member_set_nat(X3,A2)
             => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),X3) ) ) ) ) ).

% Inf_fin.bounded_iff
tff(fact_2645_Sup__fin_OboundedI,axiom,
    ! [A2: set_fu3555266933746557007_nat_o,X: fun_Pr6460944699146269935_nat_o] :
      ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,A2)
     => ( ( A2 != bot_bo8632255151191677115_nat_o )
       => ( ! [A3: fun_Pr6460944699146269935_nat_o] :
              ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A3),A2)
             => aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(ord_le5018287362838994063_nat_o,A3),X) )
         => aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(ord_le5018287362838994063_nat_o,lattic7388253182610234406_nat_o(A2)),X) ) ) ) ).

% Sup_fin.boundedI
tff(fact_2646_Sup__fin_OboundedI,axiom,
    ! [A2: set_o,X: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ( A2 != bot_bot_set_o )
       => ( ! [A3: $o] :
              ( aa_set_o_o(member_o((A3)),A2)
             => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(A3)),(X)) )
         => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,aa_set_o_o(lattic1508158080041050831_fin_o,A2)),(X)) ) ) ) ).

% Sup_fin.boundedI
tff(fact_2647_Sup__fin_OboundedI,axiom,
    ! [A2: set_assn,X: assn] :
      ( finite_finite_assn(A2)
     => ( ( A2 != bot_bot_set_assn )
       => ( ! [A3: assn] :
              ( member_assn(A3,A2)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A3),X) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,lattic2150320897289308081n_assn(A2)),X) ) ) ) ).

% Sup_fin.boundedI
tff(fact_2648_Sup__fin_OboundedI,axiom,
    ! [A2: set_nat,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( A2 != bot_bot_set_nat )
       => ( ! [A3: nat] :
              ( aa_set_nat_o(member_nat2(A3),A2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A3),X) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,lattic1093996805478795353in_nat(A2)),X) ) ) ) ).

% Sup_fin.boundedI
tff(fact_2649_Sup__fin_OboundedI,axiom,
    ! [A2: set_int,X: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( A2 != bot_bot_set_int )
       => ( ! [A3: int] :
              ( aa_set_int_o(member_int2(A3),A2)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A3),X) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,lattic1091506334969745077in_int(A2)),X) ) ) ) ).

% Sup_fin.boundedI
tff(fact_2650_Sup__fin_OboundedI,axiom,
    ! [A2: set_set_nat,X: set_nat] :
      ( finite1152437895449049373et_nat(A2)
     => ( ( A2 != bot_bot_set_set_nat )
       => ( ! [A3: set_nat] :
              ( member_set_nat(A3,A2)
             => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A3),X) )
         => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,lattic3835124923745554447et_nat(A2)),X) ) ) ) ).

% Sup_fin.boundedI
tff(fact_2651_Sup__fin_OboundedE,axiom,
    ! [A2: set_fu3555266933746557007_nat_o,X: fun_Pr6460944699146269935_nat_o] :
      ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,A2)
     => ( ( A2 != bot_bo8632255151191677115_nat_o )
       => ( aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(ord_le5018287362838994063_nat_o,lattic7388253182610234406_nat_o(A2)),X)
         => ! [A9: fun_Pr6460944699146269935_nat_o] :
              ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A9),A2)
             => aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(ord_le5018287362838994063_nat_o,A9),X) ) ) ) ) ).

% Sup_fin.boundedE
tff(fact_2652_Sup__fin_OboundedE,axiom,
    ! [A2: set_o,X: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ( A2 != bot_bot_set_o )
       => ( aa_o_o(aa_o_fun_o_o(ord_less_eq_o,aa_set_o_o(lattic1508158080041050831_fin_o,A2)),(X))
         => ! [A9: $o] :
              ( aa_set_o_o(member_o((A9)),A2)
             => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(A9)),(X)) ) ) ) ) ).

% Sup_fin.boundedE
tff(fact_2653_Sup__fin_OboundedE,axiom,
    ! [A2: set_assn,X: assn] :
      ( finite_finite_assn(A2)
     => ( ( A2 != bot_bot_set_assn )
       => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,lattic2150320897289308081n_assn(A2)),X)
         => ! [A9: assn] :
              ( member_assn(A9,A2)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A9),X) ) ) ) ) ).

% Sup_fin.boundedE
tff(fact_2654_Sup__fin_OboundedE,axiom,
    ! [A2: set_nat,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( A2 != bot_bot_set_nat )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,lattic1093996805478795353in_nat(A2)),X)
         => ! [A9: nat] :
              ( aa_set_nat_o(member_nat2(A9),A2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A9),X) ) ) ) ) ).

% Sup_fin.boundedE
tff(fact_2655_Sup__fin_OboundedE,axiom,
    ! [A2: set_int,X: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( A2 != bot_bot_set_int )
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,lattic1091506334969745077in_int(A2)),X)
         => ! [A9: int] :
              ( aa_set_int_o(member_int2(A9),A2)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A9),X) ) ) ) ) ).

% Sup_fin.boundedE
tff(fact_2656_Sup__fin_OboundedE,axiom,
    ! [A2: set_set_nat,X: set_nat] :
      ( finite1152437895449049373et_nat(A2)
     => ( ( A2 != bot_bot_set_set_nat )
       => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,lattic3835124923745554447et_nat(A2)),X)
         => ! [A9: set_nat] :
              ( member_set_nat(A9,A2)
             => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A9),X) ) ) ) ) ).

% Sup_fin.boundedE
tff(fact_2657_Inf__fin_OboundedI,axiom,
    ! [A2: set_fu3555266933746557007_nat_o,X: fun_Pr6460944699146269935_nat_o] :
      ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,A2)
     => ( ( A2 != bot_bo8632255151191677115_nat_o )
       => ( ! [A3: fun_Pr6460944699146269935_nat_o] :
              ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A3),A2)
             => aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(ord_le5018287362838994063_nat_o,X),A3) )
         => aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(ord_le5018287362838994063_nat_o,X),lattic6089491107740071820_nat_o(A2)) ) ) ) ).

% Inf_fin.boundedI
tff(fact_2658_Inf__fin_OboundedI,axiom,
    ! [A2: set_o,X: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ( A2 != bot_bot_set_o )
       => ( ! [A3: $o] :
              ( aa_set_o_o(member_o((A3)),A2)
             => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(X)),(A3)) )
         => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(X)),aa_set_o_o(lattic4107685809792843317_fin_o,A2)) ) ) ) ).

% Inf_fin.boundedI
tff(fact_2659_Inf__fin_OboundedI,axiom,
    ! [A2: set_assn,X: assn] :
      ( finite_finite_assn(A2)
     => ( ( A2 != bot_bot_set_assn )
       => ( ! [A3: assn] :
              ( member_assn(A3,A2)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),A3) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),aa_set_assn_assn(lattic47131356835913163n_assn,A2)) ) ) ) ).

% Inf_fin.boundedI
tff(fact_2660_Inf__fin_OboundedI,axiom,
    ! [A2: set_nat,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( A2 != bot_bot_set_nat )
       => ( ! [A3: nat] :
              ( aa_set_nat_o(member_nat2(A3),A2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),A3) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),aa_set_nat_nat(lattic5238388535129920115in_nat,A2)) ) ) ) ).

% Inf_fin.boundedI
tff(fact_2661_Inf__fin_OboundedI,axiom,
    ! [A2: set_int,X: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( A2 != bot_bot_set_int )
       => ( ! [A3: int] :
              ( aa_set_int_o(member_int2(A3),A2)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),A3) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),lattic5235898064620869839in_int(A2)) ) ) ) ).

% Inf_fin.boundedI
tff(fact_2662_Inf__fin_OboundedI,axiom,
    ! [A2: set_set_nat,X: set_nat] :
      ( finite1152437895449049373et_nat(A2)
     => ( ( A2 != bot_bot_set_set_nat )
       => ( ! [A3: set_nat] :
              ( member_set_nat(A3,A2)
             => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),A3) )
         => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),aa_set6364576160234933248et_nat(lattic3014633134055518761et_nat,A2)) ) ) ) ).

% Inf_fin.boundedI
tff(fact_2663_Inf__fin_OboundedE,axiom,
    ! [A2: set_fu3555266933746557007_nat_o,X: fun_Pr6460944699146269935_nat_o] :
      ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,A2)
     => ( ( A2 != bot_bo8632255151191677115_nat_o )
       => ( aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(ord_le5018287362838994063_nat_o,X),lattic6089491107740071820_nat_o(A2))
         => ! [A9: fun_Pr6460944699146269935_nat_o] :
              ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A9),A2)
             => aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(ord_le5018287362838994063_nat_o,X),A9) ) ) ) ) ).

% Inf_fin.boundedE
tff(fact_2664_Inf__fin_OboundedE,axiom,
    ! [A2: set_o,X: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ( A2 != bot_bot_set_o )
       => ( aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(X)),aa_set_o_o(lattic4107685809792843317_fin_o,A2))
         => ! [A9: $o] :
              ( aa_set_o_o(member_o((A9)),A2)
             => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(X)),(A9)) ) ) ) ) ).

% Inf_fin.boundedE
tff(fact_2665_Inf__fin_OboundedE,axiom,
    ! [A2: set_assn,X: assn] :
      ( finite_finite_assn(A2)
     => ( ( A2 != bot_bot_set_assn )
       => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),aa_set_assn_assn(lattic47131356835913163n_assn,A2))
         => ! [A9: assn] :
              ( member_assn(A9,A2)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),A9) ) ) ) ) ).

% Inf_fin.boundedE
tff(fact_2666_Inf__fin_OboundedE,axiom,
    ! [A2: set_nat,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( A2 != bot_bot_set_nat )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),aa_set_nat_nat(lattic5238388535129920115in_nat,A2))
         => ! [A9: nat] :
              ( aa_set_nat_o(member_nat2(A9),A2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),A9) ) ) ) ) ).

% Inf_fin.boundedE
tff(fact_2667_Inf__fin_OboundedE,axiom,
    ! [A2: set_int,X: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( A2 != bot_bot_set_int )
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),lattic5235898064620869839in_int(A2))
         => ! [A9: int] :
              ( aa_set_int_o(member_int2(A9),A2)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),A9) ) ) ) ) ).

% Inf_fin.boundedE
tff(fact_2668_Inf__fin_OboundedE,axiom,
    ! [A2: set_set_nat,X: set_nat] :
      ( finite1152437895449049373et_nat(A2)
     => ( ( A2 != bot_bot_set_set_nat )
       => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),aa_set6364576160234933248et_nat(lattic3014633134055518761et_nat,A2))
         => ! [A9: set_nat] :
              ( member_set_nat(A9,A2)
             => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),A9) ) ) ) ) ).

% Inf_fin.boundedE
tff(fact_2669_arg__min__least,axiom,
    ! [S: set_fu3555266933746557007_nat_o,Y: fun_Pr6460944699146269935_nat_o,F: fun_fu3504661237974713170_o_nat] :
      ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,S)
     => ( ( S != bot_bo8632255151191677115_nat_o )
       => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Y),S)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_fun767828674014743683_o_nat(F,lattic7301374186475077990_o_nat(F,S))),aa_fun767828674014743683_o_nat(F,Y)) ) ) ) ).

% arg_min_least
tff(fact_2670_arg__min__least,axiom,
    ! [S: set_o,Y: $o,F: fun_o_nat] :
      ( aa_set_o_o(finite_finite_o,S)
     => ( ( S != bot_bot_set_o )
       => ( aa_set_o_o(member_o((Y)),S)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_o_nat(F,aa_set_o_o(lattic2775856028456453135_o_nat(F),S))),aa_o_nat(F,(Y))) ) ) ) ).

% arg_min_least
tff(fact_2671_arg__min__least,axiom,
    ! [S: set_nat,Y: nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( ( S != bot_bot_set_nat )
       => ( aa_set_nat_o(member_nat2(Y),S)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,lattic7446932960582359483at_nat(F,S))),aa_nat_nat(F,Y)) ) ) ) ).

% arg_min_least
tff(fact_2672_arg__min__least,axiom,
    ! [S: set_fu3555266933746557007_nat_o,Y: fun_Pr6460944699146269935_nat_o,F: fun_fu8550182255320292270_o_int] :
      ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,S)
     => ( ( S != bot_bo8632255151191677115_nat_o )
       => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Y),S)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_fun765338203505693407_o_int(F,lattic7298883715966027714_o_int(F,S))),aa_fun765338203505693407_o_int(F,Y)) ) ) ) ).

% arg_min_least
tff(fact_2673_arg__min__least,axiom,
    ! [S: set_o,Y: $o,F: fun_o_int] :
      ( aa_set_o_o(finite_finite_o,S)
     => ( ( S != bot_bot_set_o )
       => ( aa_set_o_o(member_o((Y)),S)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_o_int(F,aa_set_o_o(lattic2773365557947402859_o_int(F),S))),aa_o_int(F,(Y))) ) ) ) ).

% arg_min_least
tff(fact_2674_arg__min__least,axiom,
    ! [S: set_nat,Y: nat,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( ( S != bot_bot_set_nat )
       => ( aa_set_nat_o(member_nat2(Y),S)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,lattic7444442490073309207at_int(F,S))),aa_nat_int(F,Y)) ) ) ) ).

% arg_min_least
tff(fact_2675_Inf__fin_Oinsert__not__elem,axiom,
    ! [A2: set_fu3555266933746557007_nat_o,X: fun_Pr6460944699146269935_nat_o] :
      ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,A2)
     => ( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),A2)
       => ( ( A2 != bot_bo8632255151191677115_nat_o )
         => ( lattic6089491107740071820_nat_o(insert4923026906511862463_nat_o(X,A2)) = aa_fun2638059832795814268_nat_o(aa_fun6000889628616002998_nat_o(inf_in4424744116413743581_nat_o,X),lattic6089491107740071820_nat_o(A2)) ) ) ) ) ).

% Inf_fin.insert_not_elem
tff(fact_2676_Inf__fin_Oinsert__not__elem,axiom,
    ! [A2: set_assn,X: assn] :
      ( finite_finite_assn(A2)
     => ( ~ member_assn(X,A2)
       => ( ( A2 != bot_bot_set_assn )
         => ( aa_set_assn_assn(lattic47131356835913163n_assn,insert_assn(X,A2)) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),aa_set_assn_assn(lattic47131356835913163n_assn,A2)) ) ) ) ) ).

% Inf_fin.insert_not_elem
tff(fact_2677_Inf__fin_Oinsert__not__elem,axiom,
    ! [A2: set_set_nat,X: set_nat] :
      ( finite1152437895449049373et_nat(A2)
     => ( ~ member_set_nat(X,A2)
       => ( ( A2 != bot_bot_set_set_nat )
         => ( aa_set6364576160234933248et_nat(lattic3014633134055518761et_nat,insert_set_nat(X,A2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),aa_set6364576160234933248et_nat(lattic3014633134055518761et_nat,A2)) ) ) ) ) ).

% Inf_fin.insert_not_elem
tff(fact_2678_Inf__fin_Oinsert__not__elem,axiom,
    ! [A2: set_Product_unit,X: product_unit] :
      ( finite4290736615968046902t_unit(A2)
     => ( ~ member_Product_unit(X,A2)
       => ( ( A2 != bot_bo3957492148770167129t_unit )
         => ( aa_set6350468789499455260t_unit(lattic1263872656861969706t_unit,insert_Product_unit(X,A2)) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),aa_set6350468789499455260t_unit(lattic1263872656861969706t_unit,A2)) ) ) ) ) ).

% Inf_fin.insert_not_elem
tff(fact_2679_Inf__fin_Oinsert__not__elem,axiom,
    ! [A2: set_se7855581050983116737at_nat,X: set_Pr1261947904930325089at_nat] :
      ( finite9047747110432174090at_nat(A2)
     => ( ~ member2643936169264416010at_nat(X,A2)
       => ( ( A2 != bot_bo3083307316010499117at_nat )
         => ( aa_set3914448398961233820at_nat(lattic30941717366863870at_nat,insert9200635055090092081at_nat(X,A2)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),aa_set3914448398961233820at_nat(lattic30941717366863870at_nat,A2)) ) ) ) ) ).

% Inf_fin.insert_not_elem
tff(fact_2680_Inf__fin_Oinsert__not__elem,axiom,
    ! [A2: set_o,X: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ~ aa_set_o_o(member_o((X)),A2)
       => ( ( A2 != bot_bot_set_o )
         => ( aa_set_o_o(lattic4107685809792843317_fin_o,insert_o((X),A2))
          <=> aa_o_o(aa_o_fun_o_o(inf_inf_o,(X)),aa_set_o_o(lattic4107685809792843317_fin_o,A2)) ) ) ) ) ).

% Inf_fin.insert_not_elem
tff(fact_2681_Inf__fin_Oinsert__not__elem,axiom,
    ! [A2: set_nat,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ~ aa_set_nat_o(member_nat2(X),A2)
       => ( ( A2 != bot_bot_set_nat )
         => ( aa_set_nat_nat(lattic5238388535129920115in_nat,insert_nat2(X,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),aa_set_nat_nat(lattic5238388535129920115in_nat,A2)) ) ) ) ) ).

% Inf_fin.insert_not_elem
tff(fact_2682_Inf__fin_Oclosed,axiom,
    ! [A2: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,A2)
     => ( ( A2 != bot_bo8632255151191677115_nat_o )
       => ( ! [X2: fun_Pr6460944699146269935_nat_o,Y2: fun_Pr6460944699146269935_nat_o] : aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(aa_fun2638059832795814268_nat_o(aa_fun6000889628616002998_nat_o(inf_in4424744116413743581_nat_o,X2),Y2)),insert4923026906511862463_nat_o(X2,insert4923026906511862463_nat_o(Y2,bot_bo8632255151191677115_nat_o)))
         => aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(lattic6089491107740071820_nat_o(A2)),A2) ) ) ) ).

% Inf_fin.closed
tff(fact_2683_Inf__fin_Oclosed,axiom,
    ! [A2: set_assn] :
      ( finite_finite_assn(A2)
     => ( ( A2 != bot_bot_set_assn )
       => ( ! [X2: assn,Y2: assn] : member_assn(aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X2),Y2),insert_assn(X2,insert_assn(Y2,bot_bot_set_assn)))
         => member_assn(aa_set_assn_assn(lattic47131356835913163n_assn,A2),A2) ) ) ) ).

% Inf_fin.closed
tff(fact_2684_Inf__fin_Oclosed,axiom,
    ! [A2: set_set_nat] :
      ( finite1152437895449049373et_nat(A2)
     => ( ( A2 != bot_bot_set_set_nat )
       => ( ! [X2: set_nat,Y2: set_nat] : member_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X2),Y2),insert_set_nat(X2,insert_set_nat(Y2,bot_bot_set_set_nat)))
         => member_set_nat(aa_set6364576160234933248et_nat(lattic3014633134055518761et_nat,A2),A2) ) ) ) ).

% Inf_fin.closed
tff(fact_2685_Inf__fin_Oclosed,axiom,
    ! [A2: set_Product_unit] :
      ( finite4290736615968046902t_unit(A2)
     => ( ( A2 != bot_bo3957492148770167129t_unit )
       => ( ! [X2: product_unit,Y2: product_unit] : member_Product_unit(aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X2),Y2),insert_Product_unit(X2,insert_Product_unit(Y2,bot_bo3957492148770167129t_unit)))
         => member_Product_unit(aa_set6350468789499455260t_unit(lattic1263872656861969706t_unit,A2),A2) ) ) ) ).

% Inf_fin.closed
tff(fact_2686_Inf__fin_Oclosed,axiom,
    ! [A2: set_se7855581050983116737at_nat] :
      ( finite9047747110432174090at_nat(A2)
     => ( ( A2 != bot_bo3083307316010499117at_nat )
       => ( ! [X2: set_Pr1261947904930325089at_nat,Y2: set_Pr1261947904930325089at_nat] : member2643936169264416010at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X2),Y2),insert9200635055090092081at_nat(X2,insert9200635055090092081at_nat(Y2,bot_bo3083307316010499117at_nat)))
         => member2643936169264416010at_nat(aa_set3914448398961233820at_nat(lattic30941717366863870at_nat,A2),A2) ) ) ) ).

% Inf_fin.closed
tff(fact_2687_Inf__fin_Oclosed,axiom,
    ! [A2: set_o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ( A2 != bot_bot_set_o )
       => ( ! [X2: $o,Y2: $o] : aa_set_o_o(member_o(aa_o_o(aa_o_fun_o_o(inf_inf_o,(X2)),(Y2))),insert_o((X2),insert_o((Y2),bot_bot_set_o)))
         => aa_set_o_o(member_o(aa_set_o_o(lattic4107685809792843317_fin_o,A2)),A2) ) ) ) ).

% Inf_fin.closed
tff(fact_2688_Inf__fin_Oclosed,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( A2 != bot_bot_set_nat )
       => ( ! [X2: nat,Y2: nat] : aa_set_nat_o(member_nat2(aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X2),Y2)),insert_nat2(X2,insert_nat2(Y2,bot_bot_set_nat)))
         => aa_set_nat_o(member_nat2(aa_set_nat_nat(lattic5238388535129920115in_nat,A2)),A2) ) ) ) ).

% Inf_fin.closed
tff(fact_2689_mergesort__def,axiom,
    mergesort_assn = merges2400687674486622701l_assn(ord_less_eq_assn) ).

% mergesort_def
tff(fact_2690_mergesort__def,axiom,
    mergesort_nat = mergesort_by_rel_nat(ord_less_eq_nat) ).

% mergesort_def
tff(fact_2691_mergesort__def,axiom,
    mergesort_int = mergesort_by_rel_int(ord_less_eq_int) ).

% mergesort_def
tff(fact_2692_mergesort__def,axiom,
    mergesort_set_nat = merges7492048612793653835et_nat(ord_less_eq_set_nat) ).

% mergesort_def
tff(fact_2693_the__elem__eq,axiom,
    ! [X: $o] :
      ( aa_set_o_o(the_elem_o,insert_o((X),bot_bot_set_o))
    <=> (X) ) ).

% the_elem_eq
tff(fact_2694_the__elem__eq,axiom,
    ! [X: nat] : the_elem_nat(insert_nat2(X,bot_bot_set_nat)) = X ).

% the_elem_eq
tff(fact_2695_is__singletonI,axiom,
    ! [X: $o] : aa_set_o_o(is_singleton_o,insert_o((X),bot_bot_set_o)) ).

% is_singletonI
tff(fact_2696_is__singletonI,axiom,
    ! [X: nat] : aa_set_nat_o(is_singleton_nat,insert_nat2(X,bot_bot_set_nat)) ).

% is_singletonI
tff(fact_2697_subset__emptyI,axiom,
    ! [A2: set_fu3555266933746557007_nat_o] :
      ( ! [X2: fun_Pr6460944699146269935_nat_o] : ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X2),A2)
     => aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(A2),bot_bo8632255151191677115_nat_o) ) ).

% subset_emptyI
tff(fact_2698_subset__emptyI,axiom,
    ! [A2: set_o] :
      ( ! [X2: $o] : ~ aa_set_o_o(member_o((X2)),A2)
     => aa_set_o_o(ord_less_eq_set_o(A2),bot_bot_set_o) ) ).

% subset_emptyI
tff(fact_2699_subset__emptyI,axiom,
    ! [A2: set_nat] :
      ( ! [X2: nat] : ~ aa_set_nat_o(member_nat2(X2),A2)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),bot_bot_set_nat) ) ).

% subset_emptyI
tff(fact_2700_sorted__list__of__set__nonempty,axiom,
    ! [A2: set_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( A2 != bot_bot_set_int )
       => ( linord2612477271533052124et_int(A2) = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,lattic8718645017227715691in_int(A2)),linord2612477271533052124et_int(minus_minus_set_int(A2,insert_int2(lattic8718645017227715691in_int(A2),bot_bot_set_int)))) ) ) ) ).

% sorted_list_of_set_nonempty
tff(fact_2701_sorted__list__of__set__nonempty,axiom,
    ! [A2: set_o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ( A2 != bot_bot_set_o )
       => ( linord3142498349692569832_set_o(A2) = cons_o(aa_set_o_o(lattic1973801136483472281_Min_o,A2),linord3142498349692569832_set_o(minus_minus_set_o(A2,insert_o(aa_set_o_o(lattic1973801136483472281_Min_o,A2),bot_bot_set_o)))) ) ) ) ).

% sorted_list_of_set_nonempty
tff(fact_2702_sorted__list__of__set__nonempty,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( A2 != bot_bot_set_nat )
       => ( linord2614967742042102400et_nat(A2) = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,lattic8721135487736765967in_nat(A2)),linord2614967742042102400et_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),insert_nat2(lattic8721135487736765967in_nat(A2),bot_bot_set_nat)))) ) ) ) ).

% sorted_list_of_set_nonempty
tff(fact_2703_Sup__fin_Oinsert__remove,axiom,
    ! [A2: set_assn,X: assn] :
      ( finite_finite_assn(A2)
     => ( lattic2150320897289308081n_assn(insert_assn(X,A2)) = $ite(minus_minus_set_assn(A2,insert_assn(X,bot_bot_set_assn)) = bot_bot_set_assn,X,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),lattic2150320897289308081n_assn(minus_minus_set_assn(A2,insert_assn(X,bot_bot_set_assn))))) ) ) ).

% Sup_fin.insert_remove
tff(fact_2704_Sup__fin_Oinsert__remove,axiom,
    ! [A2: set_set_nat,X: set_nat] :
      ( finite1152437895449049373et_nat(A2)
     => ( lattic3835124923745554447et_nat(insert_set_nat(X,A2)) = $ite(minus_2163939370556025621et_nat(A2,insert_set_nat(X,bot_bot_set_set_nat)) = bot_bot_set_set_nat,X,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),lattic3835124923745554447et_nat(minus_2163939370556025621et_nat(A2,insert_set_nat(X,bot_bot_set_set_nat))))) ) ) ).

% Sup_fin.insert_remove
tff(fact_2705_Sup__fin_Oinsert__remove,axiom,
    ! [A2: set_o,X: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( aa_set_o_o(lattic1508158080041050831_fin_o,insert_o((X),A2))
      <=> $ite(minus_minus_set_o(A2,insert_o((X),bot_bot_set_o)) = bot_bot_set_o,(X),aa_o_o(aa_o_fun_o_o(sup_sup_o,(X)),aa_set_o_o(lattic1508158080041050831_fin_o,minus_minus_set_o(A2,insert_o((X),bot_bot_set_o))))) ) ) ).

% Sup_fin.insert_remove
tff(fact_2706_Sup__fin_Oinsert__remove,axiom,
    ! [A2: set_nat,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( lattic1093996805478795353in_nat(insert_nat2(X,A2)) = $ite(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),insert_nat2(X,bot_bot_set_nat)) = bot_bot_set_nat,X,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),lattic1093996805478795353in_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),insert_nat2(X,bot_bot_set_nat))))) ) ) ).

% Sup_fin.insert_remove
tff(fact_2707_Sup__fin_Oremove,axiom,
    ! [A2: set_fu3555266933746557007_nat_o,X: fun_Pr6460944699146269935_nat_o] :
      ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,A2)
     => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),A2)
       => ( lattic7388253182610234406_nat_o(A2) = $ite(minus_6456595905716665654_nat_o(A2,insert4923026906511862463_nat_o(X,bot_bo8632255151191677115_nat_o)) = bot_bo8632255151191677115_nat_o,X,aa_fun2638059832795814268_nat_o(aa_fun6000889628616002998_nat_o(sup_su6210301778384944323_nat_o,X),lattic7388253182610234406_nat_o(minus_6456595905716665654_nat_o(A2,insert4923026906511862463_nat_o(X,bot_bo8632255151191677115_nat_o))))) ) ) ) ).

% Sup_fin.remove
tff(fact_2708_Sup__fin_Oremove,axiom,
    ! [A2: set_assn,X: assn] :
      ( finite_finite_assn(A2)
     => ( member_assn(X,A2)
       => ( lattic2150320897289308081n_assn(A2) = $ite(minus_minus_set_assn(A2,insert_assn(X,bot_bot_set_assn)) = bot_bot_set_assn,X,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),lattic2150320897289308081n_assn(minus_minus_set_assn(A2,insert_assn(X,bot_bot_set_assn))))) ) ) ) ).

% Sup_fin.remove
tff(fact_2709_Sup__fin_Oremove,axiom,
    ! [A2: set_set_nat,X: set_nat] :
      ( finite1152437895449049373et_nat(A2)
     => ( member_set_nat(X,A2)
       => ( lattic3835124923745554447et_nat(A2) = $ite(minus_2163939370556025621et_nat(A2,insert_set_nat(X,bot_bot_set_set_nat)) = bot_bot_set_set_nat,X,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),lattic3835124923745554447et_nat(minus_2163939370556025621et_nat(A2,insert_set_nat(X,bot_bot_set_set_nat))))) ) ) ) ).

% Sup_fin.remove
tff(fact_2710_Sup__fin_Oremove,axiom,
    ! [A2: set_o,X: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( aa_set_o_o(member_o((X)),A2)
       => ( aa_set_o_o(lattic1508158080041050831_fin_o,A2)
        <=> $ite(minus_minus_set_o(A2,insert_o((X),bot_bot_set_o)) = bot_bot_set_o,(X),aa_o_o(aa_o_fun_o_o(sup_sup_o,(X)),aa_set_o_o(lattic1508158080041050831_fin_o,minus_minus_set_o(A2,insert_o((X),bot_bot_set_o))))) ) ) ) ).

% Sup_fin.remove
tff(fact_2711_Sup__fin_Oremove,axiom,
    ! [A2: set_nat,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(member_nat2(X),A2)
       => ( lattic1093996805478795353in_nat(A2) = $ite(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),insert_nat2(X,bot_bot_set_nat)) = bot_bot_set_nat,X,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),lattic1093996805478795353in_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),insert_nat2(X,bot_bot_set_nat))))) ) ) ) ).

% Sup_fin.remove
tff(fact_2712_Inf__fin_Oremove,axiom,
    ! [A2: set_fu3555266933746557007_nat_o,X: fun_Pr6460944699146269935_nat_o] :
      ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,A2)
     => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),A2)
       => ( lattic6089491107740071820_nat_o(A2) = $ite(minus_6456595905716665654_nat_o(A2,insert4923026906511862463_nat_o(X,bot_bo8632255151191677115_nat_o)) = bot_bo8632255151191677115_nat_o,X,aa_fun2638059832795814268_nat_o(aa_fun6000889628616002998_nat_o(inf_in4424744116413743581_nat_o,X),lattic6089491107740071820_nat_o(minus_6456595905716665654_nat_o(A2,insert4923026906511862463_nat_o(X,bot_bo8632255151191677115_nat_o))))) ) ) ) ).

% Inf_fin.remove
tff(fact_2713_Inf__fin_Oremove,axiom,
    ! [A2: set_assn,X: assn] :
      ( finite_finite_assn(A2)
     => ( member_assn(X,A2)
       => ( aa_set_assn_assn(lattic47131356835913163n_assn,A2) = $ite(minus_minus_set_assn(A2,insert_assn(X,bot_bot_set_assn)) = bot_bot_set_assn,X,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),aa_set_assn_assn(lattic47131356835913163n_assn,minus_minus_set_assn(A2,insert_assn(X,bot_bot_set_assn))))) ) ) ) ).

% Inf_fin.remove
tff(fact_2714_Inf__fin_Oremove,axiom,
    ! [A2: set_set_nat,X: set_nat] :
      ( finite1152437895449049373et_nat(A2)
     => ( member_set_nat(X,A2)
       => ( aa_set6364576160234933248et_nat(lattic3014633134055518761et_nat,A2) = $ite(minus_2163939370556025621et_nat(A2,insert_set_nat(X,bot_bot_set_set_nat)) = bot_bot_set_set_nat,X,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),aa_set6364576160234933248et_nat(lattic3014633134055518761et_nat,minus_2163939370556025621et_nat(A2,insert_set_nat(X,bot_bot_set_set_nat))))) ) ) ) ).

% Inf_fin.remove
tff(fact_2715_Inf__fin_Oremove,axiom,
    ! [A2: set_Product_unit,X: product_unit] :
      ( finite4290736615968046902t_unit(A2)
     => ( member_Product_unit(X,A2)
       => ( aa_set6350468789499455260t_unit(lattic1263872656861969706t_unit,A2) = $ite(minus_6452836326544984404t_unit(A2,insert_Product_unit(X,bot_bo3957492148770167129t_unit)) = bot_bo3957492148770167129t_unit,X,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),aa_set6350468789499455260t_unit(lattic1263872656861969706t_unit,minus_6452836326544984404t_unit(A2,insert_Product_unit(X,bot_bo3957492148770167129t_unit))))) ) ) ) ).

% Inf_fin.remove
tff(fact_2716_Inf__fin_Oremove,axiom,
    ! [A2: set_se7855581050983116737at_nat,X: set_Pr1261947904930325089at_nat] :
      ( finite9047747110432174090at_nat(A2)
     => ( member2643936169264416010at_nat(X,A2)
       => ( aa_set3914448398961233820at_nat(lattic30941717366863870at_nat,A2) = $ite(minus_4207664762107033000at_nat(A2,insert9200635055090092081at_nat(X,bot_bo3083307316010499117at_nat)) = bot_bo3083307316010499117at_nat,X,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),aa_set3914448398961233820at_nat(lattic30941717366863870at_nat,minus_4207664762107033000at_nat(A2,insert9200635055090092081at_nat(X,bot_bo3083307316010499117at_nat))))) ) ) ) ).

% Inf_fin.remove
tff(fact_2717_Inf__fin_Oremove,axiom,
    ! [A2: set_o,X: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( aa_set_o_o(member_o((X)),A2)
       => ( aa_set_o_o(lattic4107685809792843317_fin_o,A2)
        <=> $ite(minus_minus_set_o(A2,insert_o((X),bot_bot_set_o)) = bot_bot_set_o,(X),aa_o_o(aa_o_fun_o_o(inf_inf_o,(X)),aa_set_o_o(lattic4107685809792843317_fin_o,minus_minus_set_o(A2,insert_o((X),bot_bot_set_o))))) ) ) ) ).

% Inf_fin.remove
tff(fact_2718_Inf__fin_Oremove,axiom,
    ! [A2: set_nat,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(member_nat2(X),A2)
       => ( aa_set_nat_nat(lattic5238388535129920115in_nat,A2) = $ite(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),insert_nat2(X,bot_bot_set_nat)) = bot_bot_set_nat,X,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),aa_set_nat_nat(lattic5238388535129920115in_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),insert_nat2(X,bot_bot_set_nat))))) ) ) ) ).

% Inf_fin.remove
tff(fact_2719_Inf__fin_Oinsert__remove,axiom,
    ! [A2: set_assn,X: assn] :
      ( finite_finite_assn(A2)
     => ( aa_set_assn_assn(lattic47131356835913163n_assn,insert_assn(X,A2)) = $ite(minus_minus_set_assn(A2,insert_assn(X,bot_bot_set_assn)) = bot_bot_set_assn,X,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),aa_set_assn_assn(lattic47131356835913163n_assn,minus_minus_set_assn(A2,insert_assn(X,bot_bot_set_assn))))) ) ) ).

% Inf_fin.insert_remove
tff(fact_2720_Inf__fin_Oinsert__remove,axiom,
    ! [A2: set_set_nat,X: set_nat] :
      ( finite1152437895449049373et_nat(A2)
     => ( aa_set6364576160234933248et_nat(lattic3014633134055518761et_nat,insert_set_nat(X,A2)) = $ite(minus_2163939370556025621et_nat(A2,insert_set_nat(X,bot_bot_set_set_nat)) = bot_bot_set_set_nat,X,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),aa_set6364576160234933248et_nat(lattic3014633134055518761et_nat,minus_2163939370556025621et_nat(A2,insert_set_nat(X,bot_bot_set_set_nat))))) ) ) ).

% Inf_fin.insert_remove
tff(fact_2721_Inf__fin_Oinsert__remove,axiom,
    ! [A2: set_Product_unit,X: product_unit] :
      ( finite4290736615968046902t_unit(A2)
     => ( aa_set6350468789499455260t_unit(lattic1263872656861969706t_unit,insert_Product_unit(X,A2)) = $ite(minus_6452836326544984404t_unit(A2,insert_Product_unit(X,bot_bo3957492148770167129t_unit)) = bot_bo3957492148770167129t_unit,X,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),aa_set6350468789499455260t_unit(lattic1263872656861969706t_unit,minus_6452836326544984404t_unit(A2,insert_Product_unit(X,bot_bo3957492148770167129t_unit))))) ) ) ).

% Inf_fin.insert_remove
tff(fact_2722_Inf__fin_Oinsert__remove,axiom,
    ! [A2: set_se7855581050983116737at_nat,X: set_Pr1261947904930325089at_nat] :
      ( finite9047747110432174090at_nat(A2)
     => ( aa_set3914448398961233820at_nat(lattic30941717366863870at_nat,insert9200635055090092081at_nat(X,A2)) = $ite(minus_4207664762107033000at_nat(A2,insert9200635055090092081at_nat(X,bot_bo3083307316010499117at_nat)) = bot_bo3083307316010499117at_nat,X,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),aa_set3914448398961233820at_nat(lattic30941717366863870at_nat,minus_4207664762107033000at_nat(A2,insert9200635055090092081at_nat(X,bot_bo3083307316010499117at_nat))))) ) ) ).

% Inf_fin.insert_remove
tff(fact_2723_Inf__fin_Oinsert__remove,axiom,
    ! [A2: set_o,X: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( aa_set_o_o(lattic4107685809792843317_fin_o,insert_o((X),A2))
      <=> $ite(minus_minus_set_o(A2,insert_o((X),bot_bot_set_o)) = bot_bot_set_o,(X),aa_o_o(aa_o_fun_o_o(inf_inf_o,(X)),aa_set_o_o(lattic4107685809792843317_fin_o,minus_minus_set_o(A2,insert_o((X),bot_bot_set_o))))) ) ) ).

% Inf_fin.insert_remove
tff(fact_2724_Inf__fin_Oinsert__remove,axiom,
    ! [A2: set_nat,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_nat(lattic5238388535129920115in_nat,insert_nat2(X,A2)) = $ite(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),insert_nat2(X,bot_bot_set_nat)) = bot_bot_set_nat,X,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),aa_set_nat_nat(lattic5238388535129920115in_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),insert_nat2(X,bot_bot_set_nat))))) ) ) ).

% Inf_fin.insert_remove
tff(fact_2725_is__singletonE,axiom,
    ! [A2: set_o] :
      ( aa_set_o_o(is_singleton_o,A2)
     => ~ ! [X2: $o] : A2 != insert_o((X2),bot_bot_set_o) ) ).

% is_singletonE
tff(fact_2726_is__singletonE,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(is_singleton_nat,A2)
     => ~ ! [X2: nat] : A2 != insert_nat2(X2,bot_bot_set_nat) ) ).

% is_singletonE
tff(fact_2727_Diff__iff,axiom,
    ! [C2: fun_Pr6460944699146269935_nat_o,A2: set_fu3555266933746557007_nat_o,B2: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C2),minus_6456595905716665654_nat_o(A2,B2))
    <=> ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C2),A2)
        & ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C2),B2) ) ) ).

% Diff_iff
tff(fact_2728_DiffI,axiom,
    ! [C2: fun_Pr6460944699146269935_nat_o,A2: set_fu3555266933746557007_nat_o,B2: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C2),A2)
     => ( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C2),B2)
       => aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C2),minus_6456595905716665654_nat_o(A2,B2)) ) ) ).

% DiffI
tff(fact_2729_Diff__empty,axiom,
    ! [A2: set_o] : minus_minus_set_o(A2,bot_bot_set_o) = A2 ).

% Diff_empty
tff(fact_2730_Diff__empty,axiom,
    ! [A2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),bot_bot_set_nat) = A2 ).

% Diff_empty
tff(fact_2731_empty__Diff,axiom,
    ! [A2: set_o] : minus_minus_set_o(bot_bot_set_o,A2) = bot_bot_set_o ).

% empty_Diff
tff(fact_2732_empty__Diff,axiom,
    ! [A2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,bot_bot_set_nat),A2) = bot_bot_set_nat ).

% empty_Diff
tff(fact_2733_Diff__cancel,axiom,
    ! [A2: set_o] : minus_minus_set_o(A2,A2) = bot_bot_set_o ).

% Diff_cancel
tff(fact_2734_Diff__cancel,axiom,
    ! [A2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),A2) = bot_bot_set_nat ).

% Diff_cancel
tff(fact_2735_insert__Diff1,axiom,
    ! [X: $o,B2: set_o,A2: set_o] :
      ( aa_set_o_o(member_o((X)),B2)
     => ( minus_minus_set_o(insert_o((X),A2),B2) = minus_minus_set_o(A2,B2) ) ) ).

% insert_Diff1
tff(fact_2736_insert__Diff1,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,B2: set_fu3555266933746557007_nat_o,A2: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),B2)
     => ( minus_6456595905716665654_nat_o(insert4923026906511862463_nat_o(X,A2),B2) = minus_6456595905716665654_nat_o(A2,B2) ) ) ).

% insert_Diff1
tff(fact_2737_Diff__insert0,axiom,
    ! [X: $o,A2: set_o,B2: set_o] :
      ( ~ aa_set_o_o(member_o((X)),A2)
     => ( minus_minus_set_o(A2,insert_o((X),B2)) = minus_minus_set_o(A2,B2) ) ) ).

% Diff_insert0
tff(fact_2738_Diff__insert0,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,A2: set_fu3555266933746557007_nat_o,B2: set_fu3555266933746557007_nat_o] :
      ( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),A2)
     => ( minus_6456595905716665654_nat_o(A2,insert4923026906511862463_nat_o(X,B2)) = minus_6456595905716665654_nat_o(A2,B2) ) ) ).

% Diff_insert0
tff(fact_2739_Un__Diff__cancel2,axiom,
    ! [B2: set_nat,A2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B2),A2)),A2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B2),A2) ).

% Un_Diff_cancel2
tff(fact_2740_Un__Diff__cancel,axiom,
    ! [A2: set_nat,B2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B2),A2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2) ).

% Un_Diff_cancel
tff(fact_2741_Diff__eq__empty__iff,axiom,
    ! [A2: set_o,B2: set_o] :
      ( ( minus_minus_set_o(A2,B2) = bot_bot_set_o )
    <=> aa_set_o_o(ord_less_eq_set_o(A2),B2) ) ).

% Diff_eq_empty_iff
tff(fact_2742_Diff__eq__empty__iff,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2) = bot_bot_set_nat )
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B2) ) ).

% Diff_eq_empty_iff
tff(fact_2743_insert__Diff__single,axiom,
    ! [A: $o,A2: set_o] : insert_o((A),minus_minus_set_o(A2,insert_o((A),bot_bot_set_o))) = insert_o((A),A2) ).

% insert_Diff_single
tff(fact_2744_insert__Diff__single,axiom,
    ! [A: nat,A2: set_nat] : insert_nat2(A,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),insert_nat2(A,bot_bot_set_nat))) = insert_nat2(A,A2) ).

% insert_Diff_single
tff(fact_2745_Diff__disjoint,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,B2),A2)) = bot_bo2099793752762293965at_nat ).

% Diff_disjoint
tff(fact_2746_Diff__disjoint,axiom,
    ! [A2: set_o,B2: set_o] : inf_inf_set_o(A2,minus_minus_set_o(B2,A2)) = bot_bot_set_o ).

% Diff_disjoint
tff(fact_2747_Diff__disjoint,axiom,
    ! [A2: set_nat,B2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B2),A2)) = bot_bot_set_nat ).

% Diff_disjoint
tff(fact_2748_less__assn__def,axiom,
    ! [A: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
    <=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
        & ( A != B ) ) ) ).

% less_assn_def
tff(fact_2749_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A: nat,C2: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),C2)),B) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)),C2) ).

% cancel_ab_semigroup_add_class.diff_right_commute
tff(fact_2750_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A: int,C2: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C2)),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C2) ).

% cancel_ab_semigroup_add_class.diff_right_commute
tff(fact_2751_diff__eq__diff__eq,axiom,
    ! [A: int,B: int,C2: int,D2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,C2),D2) )
     => ( ( A = B )
      <=> ( C2 = D2 ) ) ) ).

% diff_eq_diff_eq
tff(fact_2752_DiffD2,axiom,
    ! [C2: fun_Pr6460944699146269935_nat_o,A2: set_fu3555266933746557007_nat_o,B2: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C2),minus_6456595905716665654_nat_o(A2,B2))
     => ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C2),B2) ) ).

% DiffD2
tff(fact_2753_DiffD1,axiom,
    ! [C2: fun_Pr6460944699146269935_nat_o,A2: set_fu3555266933746557007_nat_o,B2: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C2),minus_6456595905716665654_nat_o(A2,B2))
     => aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C2),A2) ) ).

% DiffD1
tff(fact_2754_DiffE,axiom,
    ! [C2: fun_Pr6460944699146269935_nat_o,A2: set_fu3555266933746557007_nat_o,B2: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C2),minus_6456595905716665654_nat_o(A2,B2))
     => ~ ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C2),A2)
         => aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C2),B2) ) ) ).

% DiffE
tff(fact_2755_diff__eq__diff__less__eq,axiom,
    ! [A: int,B: int,C2: int,D2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,C2),D2) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),D2) ) ) ).

% diff_eq_diff_less_eq
tff(fact_2756_diff__right__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C2)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C2)) ) ).

% diff_right_mono
tff(fact_2757_diff__left__mono,axiom,
    ! [B: int,A: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,C2),A)),aa_int_int(aa_int_fun_int_int(minus_minus_int,C2),B)) ) ).

% diff_left_mono
tff(fact_2758_diff__mono,axiom,
    ! [A: int,B: int,D2: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,D2),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C2)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),D2)) ) ) ).

% diff_mono
tff(fact_2759_diff__strict__right__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C2)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C2)) ) ).

% diff_strict_right_mono
tff(fact_2760_diff__strict__left__mono,axiom,
    ! [B: int,A: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,C2),A)),aa_int_int(aa_int_fun_int_int(minus_minus_int,C2),B)) ) ).

% diff_strict_left_mono
tff(fact_2761_diff__eq__diff__less,axiom,
    ! [A: int,B: int,C2: int,D2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,C2),D2) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,C2),D2) ) ) ).

% diff_eq_diff_less
tff(fact_2762_diff__strict__mono,axiom,
    ! [A: int,B: int,D2: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,D2),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C2)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),D2)) ) ) ).

% diff_strict_mono
tff(fact_2763_left__diff__distrib,axiom,
    ! [A: int,B: int,C2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C2) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)) ).

% left_diff_distrib
tff(fact_2764_right__diff__distrib,axiom,
    ! [A: int,B: int,C2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)) ).

% right_diff_distrib
tff(fact_2765_left__diff__distrib_H,axiom,
    ! [B: int,C2: int,A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C2)),A) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)) ).

% left_diff_distrib'
tff(fact_2766_left__diff__distrib_H,axiom,
    ! [B: nat,C2: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),C2)),A) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),A)) ).

% left_diff_distrib'
tff(fact_2767_right__diff__distrib_H,axiom,
    ! [A: int,B: int,C2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)) ).

% right_diff_distrib'
tff(fact_2768_right__diff__distrib_H,axiom,
    ! [A: nat,B: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)) ).

% right_diff_distrib'
tff(fact_2769_double__diff,axiom,
    ! [A2: set_nat,B2: set_nat,C3: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B2),C3)
       => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,C3),A2)) = A2 ) ) ) ).

% double_diff
tff(fact_2770_Diff__subset,axiom,
    ! [A2: set_nat,B2: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2)),A2) ).

% Diff_subset
tff(fact_2771_Diff__mono,axiom,
    ! [A2: set_nat,C3: set_nat,D: set_nat,B2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),C3)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,D),B2)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,C3),D)) ) ) ).

% Diff_mono
tff(fact_2772_insert__Diff__if,axiom,
    ! [X: $o,A2: set_o,B2: set_o] :
      minus_minus_set_o(insert_o((X),A2),B2) = $ite(aa_set_o_o(member_o((X)),B2),minus_minus_set_o(A2,B2),insert_o((X),minus_minus_set_o(A2,B2))) ).

% insert_Diff_if
tff(fact_2773_insert__Diff__if,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,A2: set_fu3555266933746557007_nat_o,B2: set_fu3555266933746557007_nat_o] :
      minus_6456595905716665654_nat_o(insert4923026906511862463_nat_o(X,A2),B2) = $ite(aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),B2),minus_6456595905716665654_nat_o(A2,B2),insert4923026906511862463_nat_o(X,minus_6456595905716665654_nat_o(A2,B2))) ).

% insert_Diff_if
tff(fact_2774_Int__Diff,axiom,
    ! [A2: set_nat,B2: set_nat,C3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2)),C3) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B2),C3)) ).

% Int_Diff
tff(fact_2775_Int__Diff,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C3: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2)),C3) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,B2),C3)) ).

% Int_Diff
tff(fact_2776_Diff__Int2,axiom,
    ! [A2: set_nat,C3: set_nat,B2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),C3)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),C3)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),C3)),B2) ).

% Diff_Int2
tff(fact_2777_Diff__Int2,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,C3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),C3)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B2),C3)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),C3)),B2) ).

% Diff_Int2
tff(fact_2778_Diff__Diff__Int,axiom,
    ! [A2: set_nat,B2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) ).

% Diff_Diff_Int
tff(fact_2779_Diff__Diff__Int,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),B2)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2) ).

% Diff_Diff_Int
tff(fact_2780_Diff__Int__distrib,axiom,
    ! [C3: set_nat,A2: set_nat,B2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,C3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,C3),A2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,C3),B2)) ).

% Diff_Int_distrib
tff(fact_2781_Diff__Int__distrib,axiom,
    ! [C3: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,C3),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),B2)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,C3),A2)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,C3),B2)) ).

% Diff_Int_distrib
tff(fact_2782_Diff__Int__distrib2,axiom,
    ! [A2: set_nat,B2: set_nat,C3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2)),C3) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),C3)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),C3)) ).

% Diff_Int_distrib2
tff(fact_2783_Diff__Int__distrib2,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C3: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),B2)),C3) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),C3)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B2),C3)) ).

% Diff_Int_distrib2
tff(fact_2784_set__diff__diff__left,axiom,
    ! [A2: set_nat,B2: set_nat,C3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2)),C3) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B2),C3)) ).

% set_diff_diff_left
tff(fact_2785_Un__Diff,axiom,
    ! [A2: set_nat,B2: set_nat,C3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)),C3) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),C3)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B2),C3)) ).

% Un_Diff
tff(fact_2786_psubset__imp__ex__mem,axiom,
    ! [A2: set_fu3555266933746557007_nat_o,B2: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(ord_le1274849205035319547_nat_o(A2),B2)
     => ? [B3: fun_Pr6460944699146269935_nat_o] : aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(B3),minus_6456595905716665654_nat_o(B2,A2)) ) ).

% psubset_imp_ex_mem
tff(fact_2787_diff__shunt__var,axiom,
    ! [X: set_o,Y: set_o] :
      ( ( minus_minus_set_o(X,Y) = bot_bot_set_o )
    <=> aa_set_o_o(ord_less_eq_set_o(X),Y) ) ).

% diff_shunt_var
tff(fact_2788_diff__shunt__var,axiom,
    ! [X: assn,Y: assn] :
      ( ( aa_assn_assn(aa_ass954072293579886259n_assn(minus_minus_assn,X),Y) = bot_bot_assn )
    <=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),Y) ) ).

% diff_shunt_var
tff(fact_2789_diff__shunt__var,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,X),Y) = bot_bot_set_nat )
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Y) ) ).

% diff_shunt_var
tff(fact_2790_subset__minus__empty,axiom,
    ! [A2: set_o,B2: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(A2),B2)
     => ( minus_minus_set_o(A2,B2) = bot_bot_set_o ) ) ).

% subset_minus_empty
tff(fact_2791_subset__minus__empty,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B2)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2) = bot_bot_set_nat ) ) ).

% subset_minus_empty
tff(fact_2792_Diff__insert__absorb,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,A2: set_fu3555266933746557007_nat_o] :
      ( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),A2)
     => ( minus_6456595905716665654_nat_o(insert4923026906511862463_nat_o(X,A2),insert4923026906511862463_nat_o(X,bot_bo8632255151191677115_nat_o)) = A2 ) ) ).

% Diff_insert_absorb
tff(fact_2793_Diff__insert__absorb,axiom,
    ! [X: $o,A2: set_o] :
      ( ~ aa_set_o_o(member_o((X)),A2)
     => ( minus_minus_set_o(insert_o((X),A2),insert_o((X),bot_bot_set_o)) = A2 ) ) ).

% Diff_insert_absorb
tff(fact_2794_Diff__insert__absorb,axiom,
    ! [X: nat,A2: set_nat] :
      ( ~ aa_set_nat_o(member_nat2(X),A2)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,insert_nat2(X,A2)),insert_nat2(X,bot_bot_set_nat)) = A2 ) ) ).

% Diff_insert_absorb
tff(fact_2795_Diff__insert2,axiom,
    ! [A2: set_o,A: $o,B2: set_o] : minus_minus_set_o(A2,insert_o((A),B2)) = minus_minus_set_o(minus_minus_set_o(A2,insert_o((A),bot_bot_set_o)),B2) ).

% Diff_insert2
tff(fact_2796_Diff__insert2,axiom,
    ! [A2: set_nat,A: nat,B2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),insert_nat2(A,B2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),insert_nat2(A,bot_bot_set_nat))),B2) ).

% Diff_insert2
tff(fact_2797_insert__Diff,axiom,
    ! [A: fun_Pr6460944699146269935_nat_o,A2: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A),A2)
     => ( insert4923026906511862463_nat_o(A,minus_6456595905716665654_nat_o(A2,insert4923026906511862463_nat_o(A,bot_bo8632255151191677115_nat_o))) = A2 ) ) ).

% insert_Diff
tff(fact_2798_insert__Diff,axiom,
    ! [A: $o,A2: set_o] :
      ( aa_set_o_o(member_o((A)),A2)
     => ( insert_o((A),minus_minus_set_o(A2,insert_o((A),bot_bot_set_o))) = A2 ) ) ).

% insert_Diff
tff(fact_2799_insert__Diff,axiom,
    ! [A: nat,A2: set_nat] :
      ( aa_set_nat_o(member_nat2(A),A2)
     => ( insert_nat2(A,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),insert_nat2(A,bot_bot_set_nat))) = A2 ) ) ).

% insert_Diff
tff(fact_2800_Diff__insert,axiom,
    ! [A2: set_o,A: $o,B2: set_o] : minus_minus_set_o(A2,insert_o((A),B2)) = minus_minus_set_o(minus_minus_set_o(A2,B2),insert_o((A),bot_bot_set_o)) ).

% Diff_insert
tff(fact_2801_Diff__insert,axiom,
    ! [A2: set_nat,A: nat,B2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),insert_nat2(A,B2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2)),insert_nat2(A,bot_bot_set_nat)) ).

% Diff_insert
tff(fact_2802_insert__minus__eq,axiom,
    ! [X: $o,Y: $o,A2: set_o] :
      ( ~ ( (X)
        <=> (Y) )
     => ( minus_minus_set_o(insert_o((X),A2),insert_o((Y),bot_bot_set_o)) = insert_o((X),minus_minus_set_o(A2,insert_o((Y),bot_bot_set_o))) ) ) ).

% insert_minus_eq
tff(fact_2803_insert__minus__eq,axiom,
    ! [X: nat,Y: nat,A2: set_nat] :
      ( ( X != Y )
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,insert_nat2(X,A2)),insert_nat2(Y,bot_bot_set_nat)) = insert_nat2(X,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),insert_nat2(Y,bot_bot_set_nat))) ) ) ).

% insert_minus_eq
tff(fact_2804_set__minus__singleton__eq,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,X5: set_fu3555266933746557007_nat_o] :
      ( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),X5)
     => ( minus_6456595905716665654_nat_o(X5,insert4923026906511862463_nat_o(X,bot_bo8632255151191677115_nat_o)) = X5 ) ) ).

% set_minus_singleton_eq
tff(fact_2805_set__minus__singleton__eq,axiom,
    ! [X: $o,X5: set_o] :
      ( ~ aa_set_o_o(member_o((X)),X5)
     => ( minus_minus_set_o(X5,insert_o((X),bot_bot_set_o)) = X5 ) ) ).

% set_minus_singleton_eq
tff(fact_2806_set__minus__singleton__eq,axiom,
    ! [X: nat,X5: set_nat] :
      ( ~ aa_set_nat_o(member_nat2(X),X5)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,X5),insert_nat2(X,bot_bot_set_nat)) = X5 ) ) ).

% set_minus_singleton_eq
tff(fact_2807_subset__Diff__insert,axiom,
    ! [A2: set_o,B2: set_o,X: $o,C3: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(A2),minus_minus_set_o(B2,insert_o((X),C3)))
    <=> ( aa_set_o_o(ord_less_eq_set_o(A2),minus_minus_set_o(B2,C3))
        & ~ aa_set_o_o(member_o((X)),A2) ) ) ).

% subset_Diff_insert
tff(fact_2808_subset__Diff__insert,axiom,
    ! [A2: set_fu3555266933746557007_nat_o,B2: set_fu3555266933746557007_nat_o,X: fun_Pr6460944699146269935_nat_o,C3: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(A2),minus_6456595905716665654_nat_o(B2,insert4923026906511862463_nat_o(X,C3)))
    <=> ( aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(A2),minus_6456595905716665654_nat_o(B2,C3))
        & ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),A2) ) ) ).

% subset_Diff_insert
tff(fact_2809_subset__Diff__insert,axiom,
    ! [A2: set_nat,B2: set_nat,X: nat,C3: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B2),insert_nat2(X,C3)))
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B2),C3))
        & ~ aa_set_nat_o(member_nat2(X),A2) ) ) ).

% subset_Diff_insert
tff(fact_2810_Int__Diff__disjoint,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),B2)) = bot_bo2099793752762293965at_nat ).

% Int_Diff_disjoint
tff(fact_2811_Int__Diff__disjoint,axiom,
    ! [A2: set_o,B2: set_o] : inf_inf_set_o(inf_inf_set_o(A2,B2),minus_minus_set_o(A2,B2)) = bot_bot_set_o ).

% Int_Diff_disjoint
tff(fact_2812_Int__Diff__disjoint,axiom,
    ! [A2: set_nat,B2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2)) = bot_bot_set_nat ).

% Int_Diff_disjoint
tff(fact_2813_Diff__triv,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2) = bot_bo2099793752762293965at_nat )
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),B2) = A2 ) ) ).

% Diff_triv
tff(fact_2814_Diff__triv,axiom,
    ! [A2: set_o,B2: set_o] :
      ( ( inf_inf_set_o(A2,B2) = bot_bot_set_o )
     => ( minus_minus_set_o(A2,B2) = A2 ) ) ).

% Diff_triv
tff(fact_2815_Diff__triv,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) = bot_bot_set_nat )
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2) = A2 ) ) ).

% Diff_triv
tff(fact_2816_disjoint__alt__simp1,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),B2) = A2 )
    <=> ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2) = bot_bo2099793752762293965at_nat ) ) ).

% disjoint_alt_simp1
tff(fact_2817_disjoint__alt__simp1,axiom,
    ! [A2: set_o,B2: set_o] :
      ( ( minus_minus_set_o(A2,B2) = A2 )
    <=> ( inf_inf_set_o(A2,B2) = bot_bot_set_o ) ) ).

% disjoint_alt_simp1
tff(fact_2818_disjoint__alt__simp1,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2) = A2 )
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) = bot_bot_set_nat ) ) ).

% disjoint_alt_simp1
tff(fact_2819_disjoint__alt__simp2,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),B2) != A2 )
    <=> ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2) != bot_bo2099793752762293965at_nat ) ) ).

% disjoint_alt_simp2
tff(fact_2820_disjoint__alt__simp2,axiom,
    ! [A2: set_o,B2: set_o] :
      ( ( minus_minus_set_o(A2,B2) != A2 )
    <=> ( inf_inf_set_o(A2,B2) != bot_bot_set_o ) ) ).

% disjoint_alt_simp2
tff(fact_2821_disjoint__alt__simp2,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2) != A2 )
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) != bot_bot_set_nat ) ) ).

% disjoint_alt_simp2
tff(fact_2822_Diff__subset__conv,axiom,
    ! [A2: set_nat,B2: set_nat,C3: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2)),C3)
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B2),C3)) ) ).

% Diff_subset_conv
tff(fact_2823_Diff__partition,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B2)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B2),A2)) = B2 ) ) ).

% Diff_partition
tff(fact_2824_Diff__Un,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C3: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,B2),C3)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),B2)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),C3)) ).

% Diff_Un
tff(fact_2825_Diff__Un,axiom,
    ! [A2: set_nat,B2: set_nat,C3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B2),C3)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),C3)) ).

% Diff_Un
tff(fact_2826_Diff__Int,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C3: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B2),C3)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),B2)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),C3)) ).

% Diff_Int
tff(fact_2827_Diff__Int,axiom,
    ! [A2: set_nat,B2: set_nat,C3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),C3)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),C3)) ).

% Diff_Int
tff(fact_2828_Int__Diff__Un,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),B2)) = A2 ).

% Int_Diff_Un
tff(fact_2829_Int__Diff__Un,axiom,
    ! [A2: set_nat,B2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2)) = A2 ).

% Int_Diff_Un
tff(fact_2830_Un__Diff__Int,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),B2)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2)) = A2 ).

% Un_Diff_Int
tff(fact_2831_Un__Diff__Int,axiom,
    ! [A2: set_nat,B2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2)) = A2 ).

% Un_Diff_Int
tff(fact_2832_is__singleton__the__elem,axiom,
    ! [A2: set_o] :
      ( aa_set_o_o(is_singleton_o,A2)
    <=> ( A2 = insert_o(aa_set_o_o(the_elem_o,A2),bot_bot_set_o) ) ) ).

% is_singleton_the_elem
tff(fact_2833_is__singleton__the__elem,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(is_singleton_nat,A2)
    <=> ( A2 = insert_nat2(the_elem_nat(A2),bot_bot_set_nat) ) ) ).

% is_singleton_the_elem
tff(fact_2834_is__singletonI_H,axiom,
    ! [A2: set_fu3555266933746557007_nat_o] :
      ( ( A2 != bot_bo8632255151191677115_nat_o )
     => ( ! [X2: fun_Pr6460944699146269935_nat_o,Y2: fun_Pr6460944699146269935_nat_o] :
            ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X2),A2)
           => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Y2),A2)
             => ( X2 = Y2 ) ) )
       => aa_set5091322549519198405at_o_o(is_sin624665226112233955_nat_o,A2) ) ) ).

% is_singletonI'
tff(fact_2835_is__singletonI_H,axiom,
    ! [A2: set_o] :
      ( ( A2 != bot_bot_set_o )
     => ( ! [X2: $o,Y2: $o] :
            ( aa_set_o_o(member_o((X2)),A2)
           => ( aa_set_o_o(member_o((Y2)),A2)
             => ( (X2)
              <=> (Y2) ) ) )
       => aa_set_o_o(is_singleton_o,A2) ) ) ).

% is_singletonI'
tff(fact_2836_is__singletonI_H,axiom,
    ! [A2: set_nat] :
      ( ( A2 != bot_bot_set_nat )
     => ( ! [X2: nat,Y2: nat] :
            ( aa_set_nat_o(member_nat2(X2),A2)
           => ( aa_set_nat_o(member_nat2(Y2),A2)
             => ( X2 = Y2 ) ) )
       => aa_set_nat_o(is_singleton_nat,A2) ) ) ).

% is_singletonI'
tff(fact_2837_infinite__remove,axiom,
    ! [S: set_o,A: $o] :
      ( ~ aa_set_o_o(finite_finite_o,S)
     => ~ aa_set_o_o(finite_finite_o,minus_minus_set_o(S,insert_o((A),bot_bot_set_o))) ) ).

% infinite_remove
tff(fact_2838_infinite__remove,axiom,
    ! [S: set_nat,A: nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,S)
     => ~ aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,S),insert_nat2(A,bot_bot_set_nat))) ) ).

% infinite_remove
tff(fact_2839_infinite__coinduct,axiom,
    ! [X5: fun_set_o_o,A2: set_o] :
      ( aa_set_o_o(X5,A2)
     => ( ! [A6: set_o] :
            ( aa_set_o_o(X5,A6)
           => ? [X4: $o] :
                ( aa_set_o_o(member_o((X4)),A6)
                & ( aa_set_o_o(X5,minus_minus_set_o(A6,insert_o((X4),bot_bot_set_o)))
                  | ~ aa_set_o_o(finite_finite_o,minus_minus_set_o(A6,insert_o((X4),bot_bot_set_o))) ) ) )
       => ~ aa_set_o_o(finite_finite_o,A2) ) ) ).

% infinite_coinduct
tff(fact_2840_infinite__coinduct,axiom,
    ! [X5: fun_set_nat_o,A2: set_nat] :
      ( aa_set_nat_o(X5,A2)
     => ( ! [A6: set_nat] :
            ( aa_set_nat_o(X5,A6)
           => ? [X4: nat] :
                ( aa_set_nat_o(member_nat2(X4),A6)
                & ( aa_set_nat_o(X5,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A6),insert_nat2(X4,bot_bot_set_nat)))
                  | ~ aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A6),insert_nat2(X4,bot_bot_set_nat))) ) ) )
       => ~ aa_set_nat_o(finite_finite_nat,A2) ) ) ).

% infinite_coinduct
tff(fact_2841_finite__empty__induct,axiom,
    ! [A2: set_fu3555266933746557007_nat_o,Pa: fun_se7459991829846211660at_o_o] :
      ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,A2)
     => ( aa_set5091322549519198405at_o_o(Pa,A2)
       => ( ! [A3: fun_Pr6460944699146269935_nat_o,A6: set_fu3555266933746557007_nat_o] :
              ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,A6)
             => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A3),A6)
               => ( aa_set5091322549519198405at_o_o(Pa,A6)
                 => aa_set5091322549519198405at_o_o(Pa,minus_6456595905716665654_nat_o(A6,insert4923026906511862463_nat_o(A3,bot_bo8632255151191677115_nat_o))) ) ) )
         => aa_set5091322549519198405at_o_o(Pa,bot_bo8632255151191677115_nat_o) ) ) ) ).

% finite_empty_induct
tff(fact_2842_finite__empty__induct,axiom,
    ! [A2: set_o,Pa: fun_set_o_o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( aa_set_o_o(Pa,A2)
       => ( ! [A3: $o,A6: set_o] :
              ( aa_set_o_o(finite_finite_o,A6)
             => ( aa_set_o_o(member_o((A3)),A6)
               => ( aa_set_o_o(Pa,A6)
                 => aa_set_o_o(Pa,minus_minus_set_o(A6,insert_o((A3),bot_bot_set_o))) ) ) )
         => aa_set_o_o(Pa,bot_bot_set_o) ) ) ) ).

% finite_empty_induct
tff(fact_2843_finite__empty__induct,axiom,
    ! [A2: set_nat,Pa: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(Pa,A2)
       => ( ! [A3: nat,A6: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,A6)
             => ( aa_set_nat_o(member_nat2(A3),A6)
               => ( aa_set_nat_o(Pa,A6)
                 => aa_set_nat_o(Pa,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A6),insert_nat2(A3,bot_bot_set_nat))) ) ) )
         => aa_set_nat_o(Pa,bot_bot_set_nat) ) ) ) ).

% finite_empty_induct
tff(fact_2844_Diff__single__insert,axiom,
    ! [A2: set_o,X: $o,B2: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(minus_minus_set_o(A2,insert_o((X),bot_bot_set_o))),B2)
     => aa_set_o_o(ord_less_eq_set_o(A2),insert_o((X),B2)) ) ).

% Diff_single_insert
tff(fact_2845_Diff__single__insert,axiom,
    ! [A2: set_nat,X: nat,B2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),insert_nat2(X,bot_bot_set_nat))),B2)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),insert_nat2(X,B2)) ) ).

% Diff_single_insert
tff(fact_2846_subset__insert__iff,axiom,
    ! [A2: set_fu3555266933746557007_nat_o,X: fun_Pr6460944699146269935_nat_o,B2: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(A2),insert4923026906511862463_nat_o(X,B2))
    <=> $ite(aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),A2),aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(minus_6456595905716665654_nat_o(A2,insert4923026906511862463_nat_o(X,bot_bo8632255151191677115_nat_o))),B2),aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(A2),B2)) ) ).

% subset_insert_iff
tff(fact_2847_subset__insert__iff,axiom,
    ! [A2: set_o,X: $o,B2: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(A2),insert_o((X),B2))
    <=> $ite(aa_set_o_o(member_o((X)),A2),aa_set_o_o(ord_less_eq_set_o(minus_minus_set_o(A2,insert_o((X),bot_bot_set_o))),B2),aa_set_o_o(ord_less_eq_set_o(A2),B2)) ) ).

% subset_insert_iff
tff(fact_2848_subset__insert__iff,axiom,
    ! [A2: set_nat,X: nat,B2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),insert_nat2(X,B2))
    <=> $ite(aa_set_nat_o(member_nat2(X),A2),aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),insert_nat2(X,bot_bot_set_nat))),B2),aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B2)) ) ).

% subset_insert_iff
tff(fact_2849_remove__subset,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,S: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),S)
     => aa_set5091322549519198405at_o_o(ord_le1274849205035319547_nat_o(minus_6456595905716665654_nat_o(S,insert4923026906511862463_nat_o(X,bot_bo8632255151191677115_nat_o))),S) ) ).

% remove_subset
tff(fact_2850_remove__subset,axiom,
    ! [X: $o,S: set_o] :
      ( aa_set_o_o(member_o((X)),S)
     => aa_set_o_o(ord_less_set_o(minus_minus_set_o(S,insert_o((X),bot_bot_set_o))),S) ) ).

% remove_subset
tff(fact_2851_remove__subset,axiom,
    ! [X: nat,S: set_nat] :
      ( aa_set_nat_o(member_nat2(X),S)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,S),insert_nat2(X,bot_bot_set_nat))),S) ) ).

% remove_subset
tff(fact_2852_disjoint__alt__simp3,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),B2)),A2)
    <=> ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2) != bot_bo2099793752762293965at_nat ) ) ).

% disjoint_alt_simp3
tff(fact_2853_disjoint__alt__simp3,axiom,
    ! [A2: set_o,B2: set_o] :
      ( aa_set_o_o(ord_less_set_o(minus_minus_set_o(A2,B2)),A2)
    <=> ( inf_inf_set_o(A2,B2) != bot_bot_set_o ) ) ).

% disjoint_alt_simp3
tff(fact_2854_disjoint__alt__simp3,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2)),A2)
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) != bot_bot_set_nat ) ) ).

% disjoint_alt_simp3
tff(fact_2855_finite__remove__induct,axiom,
    ! [B2: set_fu3555266933746557007_nat_o,Pa: fun_se7459991829846211660at_o_o] :
      ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,B2)
     => ( aa_set5091322549519198405at_o_o(Pa,bot_bo8632255151191677115_nat_o)
       => ( ! [A6: set_fu3555266933746557007_nat_o] :
              ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,A6)
             => ( ( A6 != bot_bo8632255151191677115_nat_o )
               => ( aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(A6),B2)
                 => ( ! [X4: fun_Pr6460944699146269935_nat_o] :
                        ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X4),A6)
                       => aa_set5091322549519198405at_o_o(Pa,minus_6456595905716665654_nat_o(A6,insert4923026906511862463_nat_o(X4,bot_bo8632255151191677115_nat_o))) )
                   => aa_set5091322549519198405at_o_o(Pa,A6) ) ) ) )
         => aa_set5091322549519198405at_o_o(Pa,B2) ) ) ) ).

% finite_remove_induct
tff(fact_2856_finite__remove__induct,axiom,
    ! [B2: set_o,Pa: fun_set_o_o] :
      ( aa_set_o_o(finite_finite_o,B2)
     => ( aa_set_o_o(Pa,bot_bot_set_o)
       => ( ! [A6: set_o] :
              ( aa_set_o_o(finite_finite_o,A6)
             => ( ( A6 != bot_bot_set_o )
               => ( aa_set_o_o(ord_less_eq_set_o(A6),B2)
                 => ( ! [X4: $o] :
                        ( aa_set_o_o(member_o((X4)),A6)
                       => aa_set_o_o(Pa,minus_minus_set_o(A6,insert_o((X4),bot_bot_set_o))) )
                   => aa_set_o_o(Pa,A6) ) ) ) )
         => aa_set_o_o(Pa,B2) ) ) ) ).

% finite_remove_induct
tff(fact_2857_finite__remove__induct,axiom,
    ! [B2: set_nat,Pa: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,B2)
     => ( aa_set_nat_o(Pa,bot_bot_set_nat)
       => ( ! [A6: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,A6)
             => ( ( A6 != bot_bot_set_nat )
               => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A6),B2)
                 => ( ! [X4: nat] :
                        ( aa_set_nat_o(member_nat2(X4),A6)
                       => aa_set_nat_o(Pa,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A6),insert_nat2(X4,bot_bot_set_nat))) )
                   => aa_set_nat_o(Pa,A6) ) ) ) )
         => aa_set_nat_o(Pa,B2) ) ) ) ).

% finite_remove_induct
tff(fact_2858_remove__induct,axiom,
    ! [Pa: fun_se7459991829846211660at_o_o,B2: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(Pa,bot_bo8632255151191677115_nat_o)
     => ( ( ~ aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,B2)
         => aa_set5091322549519198405at_o_o(Pa,B2) )
       => ( ! [A6: set_fu3555266933746557007_nat_o] :
              ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,A6)
             => ( ( A6 != bot_bo8632255151191677115_nat_o )
               => ( aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(A6),B2)
                 => ( ! [X4: fun_Pr6460944699146269935_nat_o] :
                        ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X4),A6)
                       => aa_set5091322549519198405at_o_o(Pa,minus_6456595905716665654_nat_o(A6,insert4923026906511862463_nat_o(X4,bot_bo8632255151191677115_nat_o))) )
                   => aa_set5091322549519198405at_o_o(Pa,A6) ) ) ) )
         => aa_set5091322549519198405at_o_o(Pa,B2) ) ) ) ).

% remove_induct
tff(fact_2859_remove__induct,axiom,
    ! [Pa: fun_set_o_o,B2: set_o] :
      ( aa_set_o_o(Pa,bot_bot_set_o)
     => ( ( ~ aa_set_o_o(finite_finite_o,B2)
         => aa_set_o_o(Pa,B2) )
       => ( ! [A6: set_o] :
              ( aa_set_o_o(finite_finite_o,A6)
             => ( ( A6 != bot_bot_set_o )
               => ( aa_set_o_o(ord_less_eq_set_o(A6),B2)
                 => ( ! [X4: $o] :
                        ( aa_set_o_o(member_o((X4)),A6)
                       => aa_set_o_o(Pa,minus_minus_set_o(A6,insert_o((X4),bot_bot_set_o))) )
                   => aa_set_o_o(Pa,A6) ) ) ) )
         => aa_set_o_o(Pa,B2) ) ) ) ).

% remove_induct
tff(fact_2860_remove__induct,axiom,
    ! [Pa: fun_set_nat_o,B2: set_nat] :
      ( aa_set_nat_o(Pa,bot_bot_set_nat)
     => ( ( ~ aa_set_nat_o(finite_finite_nat,B2)
         => aa_set_nat_o(Pa,B2) )
       => ( ! [A6: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,A6)
             => ( ( A6 != bot_bot_set_nat )
               => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A6),B2)
                 => ( ! [X4: nat] :
                        ( aa_set_nat_o(member_nat2(X4),A6)
                       => aa_set_nat_o(Pa,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A6),insert_nat2(X4,bot_bot_set_nat))) )
                   => aa_set_nat_o(Pa,A6) ) ) ) )
         => aa_set_nat_o(Pa,B2) ) ) ) ).

% remove_induct
tff(fact_2861_finite__induct__select,axiom,
    ! [S: set_o,Pa: fun_set_o_o] :
      ( aa_set_o_o(finite_finite_o,S)
     => ( aa_set_o_o(Pa,bot_bot_set_o)
       => ( ! [T3: set_o] :
              ( aa_set_o_o(ord_less_set_o(T3),S)
             => ( aa_set_o_o(Pa,T3)
               => ? [X4: $o] :
                    ( aa_set_o_o(member_o((X4)),minus_minus_set_o(S,T3))
                    & aa_set_o_o(Pa,insert_o((X4),T3)) ) ) )
         => aa_set_o_o(Pa,S) ) ) ) ).

% finite_induct_select
tff(fact_2862_finite__induct__select,axiom,
    ! [S: set_nat,Pa: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( aa_set_nat_o(Pa,bot_bot_set_nat)
       => ( ! [T3: set_nat] :
              ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,T3),S)
             => ( aa_set_nat_o(Pa,T3)
               => ? [X4: nat] :
                    ( aa_set_nat_o(member_nat2(X4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,S),T3))
                    & aa_set_nat_o(Pa,insert_nat2(X4,T3)) ) ) )
         => aa_set_nat_o(Pa,S) ) ) ) ).

% finite_induct_select
tff(fact_2863_psubset__insert__iff,axiom,
    ! [A2: set_fu3555266933746557007_nat_o,X: fun_Pr6460944699146269935_nat_o,B2: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(ord_le1274849205035319547_nat_o(A2),insert4923026906511862463_nat_o(X,B2))
    <=> $ite(
          aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),B2),
          aa_set5091322549519198405at_o_o(ord_le1274849205035319547_nat_o(A2),B2),
          $ite(aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),A2),aa_set5091322549519198405at_o_o(ord_le1274849205035319547_nat_o(minus_6456595905716665654_nat_o(A2,insert4923026906511862463_nat_o(X,bot_bo8632255151191677115_nat_o))),B2),aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(A2),B2)) ) ) ).

% psubset_insert_iff
tff(fact_2864_psubset__insert__iff,axiom,
    ! [A2: set_o,X: $o,B2: set_o] :
      ( aa_set_o_o(ord_less_set_o(A2),insert_o((X),B2))
    <=> $ite(
          aa_set_o_o(member_o((X)),B2),
          aa_set_o_o(ord_less_set_o(A2),B2),
          $ite(aa_set_o_o(member_o((X)),A2),aa_set_o_o(ord_less_set_o(minus_minus_set_o(A2,insert_o((X),bot_bot_set_o))),B2),aa_set_o_o(ord_less_eq_set_o(A2),B2)) ) ) ).

% psubset_insert_iff
tff(fact_2865_psubset__insert__iff,axiom,
    ! [A2: set_nat,X: nat,B2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A2),insert_nat2(X,B2))
    <=> $ite(
          aa_set_nat_o(member_nat2(X),B2),
          aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A2),B2),
          $ite(aa_set_nat_o(member_nat2(X),A2),aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),insert_nat2(X,bot_bot_set_nat))),B2),aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B2)) ) ) ).

% psubset_insert_iff
tff(fact_2866_is__singleton__def,axiom,
    ! [A2: set_o] :
      ( aa_set_o_o(is_singleton_o,A2)
    <=> ? [X3: $o] : A2 = insert_o((X3),bot_bot_set_o) ) ).

% is_singleton_def
tff(fact_2867_is__singleton__def,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(is_singleton_nat,A2)
    <=> ? [X3: nat] : A2 = insert_nat2(X3,bot_bot_set_nat) ) ).

% is_singleton_def
tff(fact_2868_inf__period_I2_J,axiom,
    ! [Pa: fun_int_o,D: int,Q: fun_int_o] :
      ( ! [X2: int,K2: int] :
          ( aa_int_o(Pa,X2)
        <=> aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D))) )
     => ( ! [X2: int,K2: int] :
            ( aa_int_o(Q,X2)
          <=> aa_int_o(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D))) )
       => ! [X4: int,K3: int] :
            ( ( aa_int_o(Pa,X4)
              | aa_int_o(Q,X4) )
          <=> ( aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K3),D)))
              | aa_int_o(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K3),D))) ) ) ) ) ).

% inf_period(2)
tff(fact_2869_inf__period_I1_J,axiom,
    ! [Pa: fun_int_o,D: int,Q: fun_int_o] :
      ( ! [X2: int,K2: int] :
          ( aa_int_o(Pa,X2)
        <=> aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D))) )
     => ( ! [X2: int,K2: int] :
            ( aa_int_o(Q,X2)
          <=> aa_int_o(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D))) )
       => ! [X4: int,K3: int] :
            ( ( aa_int_o(Pa,X4)
              & aa_int_o(Q,X4) )
          <=> ( aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K3),D)))
              & aa_int_o(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K3),D))) ) ) ) ) ).

% inf_period(1)
tff(fact_2870_remove__def,axiom,
    ! [X: $o,A2: set_o] : remove_o((X),A2) = minus_minus_set_o(A2,insert_o((X),bot_bot_set_o)) ).

% remove_def
tff(fact_2871_remove__def,axiom,
    ! [X: nat,A2: set_nat] : remove_nat(X,A2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),insert_nat2(X,bot_bot_set_nat)) ).

% remove_def
tff(fact_2872_sorted__list__of__set_Osorted__key__list__of__set__remove,axiom,
    ! [A2: set_o,X: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( linord3142498349692569832_set_o(minus_minus_set_o(A2,insert_o((X),bot_bot_set_o))) = remove1_o((X),linord3142498349692569832_set_o(A2)) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_remove
tff(fact_2873_sorted__list__of__set_Osorted__key__list__of__set__remove,axiom,
    ! [A2: set_nat,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( linord2614967742042102400et_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),insert_nat2(X,bot_bot_set_nat))) = remove1_nat(X,linord2614967742042102400et_nat(A2)) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_remove
tff(fact_2874_Min_Oremove,axiom,
    ! [A2: set_o,X: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( aa_set_o_o(member_o((X)),A2)
       => ( aa_set_o_o(lattic1973801136483472281_Min_o,A2)
        <=> $ite(minus_minus_set_o(A2,insert_o((X),bot_bot_set_o)) = bot_bot_set_o,(X),aa_o_o(aa_o_fun_o_o(ord_min_o,(X)),aa_set_o_o(lattic1973801136483472281_Min_o,minus_minus_set_o(A2,insert_o((X),bot_bot_set_o))))) ) ) ) ).

% Min.remove
tff(fact_2875_Min_Oremove,axiom,
    ! [A2: set_nat,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(member_nat2(X),A2)
       => ( lattic8721135487736765967in_nat(A2) = $ite(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),insert_nat2(X,bot_bot_set_nat)) = bot_bot_set_nat,X,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,X),lattic8721135487736765967in_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),insert_nat2(X,bot_bot_set_nat))))) ) ) ) ).

% Min.remove
tff(fact_2876_Min_Oinsert__remove,axiom,
    ! [A2: set_o,X: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( aa_set_o_o(lattic1973801136483472281_Min_o,insert_o((X),A2))
      <=> $ite(minus_minus_set_o(A2,insert_o((X),bot_bot_set_o)) = bot_bot_set_o,(X),aa_o_o(aa_o_fun_o_o(ord_min_o,(X)),aa_set_o_o(lattic1973801136483472281_Min_o,minus_minus_set_o(A2,insert_o((X),bot_bot_set_o))))) ) ) ).

% Min.insert_remove
tff(fact_2877_Min_Oinsert__remove,axiom,
    ! [A2: set_nat,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( lattic8721135487736765967in_nat(insert_nat2(X,A2)) = $ite(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),insert_nat2(X,bot_bot_set_nat)) = bot_bot_set_nat,X,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,X),lattic8721135487736765967in_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),insert_nat2(X,bot_bot_set_nat))))) ) ) ).

% Min.insert_remove
tff(fact_2878_Max_Oremove,axiom,
    ! [A2: set_o,X: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( aa_set_o_o(member_o((X)),A2)
       => ( aa_set_o_o(lattic1921953407002678535_Max_o,A2)
        <=> $ite(minus_minus_set_o(A2,insert_o((X),bot_bot_set_o)) = bot_bot_set_o,(X),aa_o_o(aa_o_fun_o_o(ord_max_o,(X)),aa_set_o_o(lattic1921953407002678535_Max_o,minus_minus_set_o(A2,insert_o((X),bot_bot_set_o))))) ) ) ) ).

% Max.remove
tff(fact_2879_Max_Oremove,axiom,
    ! [A2: set_nat,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(member_nat2(X),A2)
       => ( lattic8265883725875713057ax_nat(A2) = $ite(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),insert_nat2(X,bot_bot_set_nat)) = bot_bot_set_nat,X,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),lattic8265883725875713057ax_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),insert_nat2(X,bot_bot_set_nat))))) ) ) ) ).

% Max.remove
tff(fact_2880_Max_Oinsert__remove,axiom,
    ! [A2: set_o,X: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( aa_set_o_o(lattic1921953407002678535_Max_o,insert_o((X),A2))
      <=> $ite(minus_minus_set_o(A2,insert_o((X),bot_bot_set_o)) = bot_bot_set_o,(X),aa_o_o(aa_o_fun_o_o(ord_max_o,(X)),aa_set_o_o(lattic1921953407002678535_Max_o,minus_minus_set_o(A2,insert_o((X),bot_bot_set_o))))) ) ) ).

% Max.insert_remove
tff(fact_2881_Max_Oinsert__remove,axiom,
    ! [A2: set_nat,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( lattic8265883725875713057ax_nat(insert_nat2(X,A2)) = $ite(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),insert_nat2(X,bot_bot_set_nat)) = bot_bot_set_nat,X,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),lattic8265883725875713057ax_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),insert_nat2(X,bot_bot_set_nat))))) ) ) ).

% Max.insert_remove
tff(fact_2882_Inf__fin_Osemilattice__order__set__axioms,axiom,
    lattic5111740090977526247t_unit(inf_inf_Product_unit,ord_le3221252021190050221t_unit,ord_le361264281704409273t_unit) ).

% Inf_fin.semilattice_order_set_axioms
tff(fact_2883_Inf__fin_Osemilattice__order__set__axioms,axiom,
    lattic6529551498584149819at_nat(inf_in2572325071724192079at_nat,ord_le3146513528884898305at_nat,ord_le7866589430770878221at_nat) ).

% Inf_fin.semilattice_order_set_axioms
tff(fact_2884_Inf__fin_Osemilattice__order__set__axioms,axiom,
    lattic5623431474481994958t_assn(inf_inf_assn,ord_less_eq_assn,ord_less_assn) ).

% Inf_fin.semilattice_order_set_axioms
tff(fact_2885_Inf__fin_Osemilattice__order__set__axioms,axiom,
    lattic6009151579333465974et_nat(inf_inf_nat,ord_less_eq_nat,ord_less_nat) ).

% Inf_fin.semilattice_order_set_axioms
tff(fact_2886_Inf__fin_Osemilattice__order__set__axioms,axiom,
    lattic6006661108824415698et_int(inf_inf_int,ord_less_eq_int,ord_less_int) ).

% Inf_fin.semilattice_order_set_axioms
tff(fact_2887_Inf__fin_Osemilattice__order__set__axioms,axiom,
    lattic3109210760196336428et_nat(inf_inf_set_nat,ord_less_eq_set_nat,ord_less_set_nat) ).

% Inf_fin.semilattice_order_set_axioms
tff(fact_2888_Min__insert,axiom,
    ! [A2: set_o,X: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ( A2 != bot_bot_set_o )
       => ( aa_set_o_o(lattic1973801136483472281_Min_o,insert_o((X),A2))
        <=> aa_o_o(aa_o_fun_o_o(ord_min_o,(X)),aa_set_o_o(lattic1973801136483472281_Min_o,A2)) ) ) ) ).

% Min_insert
tff(fact_2889_Min__insert,axiom,
    ! [A2: set_nat,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( A2 != bot_bot_set_nat )
       => ( lattic8721135487736765967in_nat(insert_nat2(X,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,X),lattic8721135487736765967in_nat(A2)) ) ) ) ).

% Min_insert
tff(fact_2890_min_Oidem,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A),A) = A ).

% min.idem
tff(fact_2891_min_Oleft__idem,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A),B) ).

% min.left_idem
tff(fact_2892_min_Oright__idem,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A),B)),B) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A),B) ).

% min.right_idem
tff(fact_2893_member__remove,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,Y: fun_Pr6460944699146269935_nat_o,A2: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),remove8193825956935957076_nat_o(Y,A2))
    <=> ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),A2)
        & ( X != Y ) ) ) ).

% member_remove
tff(fact_2894_min_Oabsorb1,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A),B) = A ) ) ).

% min.absorb1
tff(fact_2895_min_Oabsorb1,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_int(aa_int_fun_int_int(ord_min_int,A),B) = A ) ) ).

% min.absorb1
tff(fact_2896_min_Oabsorb2,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A),B) = B ) ) ).

% min.absorb2
tff(fact_2897_min_Oabsorb2,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_int(aa_int_fun_int_int(ord_min_int,A),B) = B ) ) ).

% min.absorb2
tff(fact_2898_min_Obounded__iff,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,B),C2))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C2) ) ) ).

% min.bounded_iff
tff(fact_2899_min_Obounded__iff,axiom,
    ! [A: int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(ord_min_int,B),C2))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C2) ) ) ).

% min.bounded_iff
tff(fact_2900_min__arg__le_I2_J,axiom,
    ! [M2: assn,N: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,M2),aa_assn_assn(aa_ass954072293579886259n_assn(ord_min_assn,M2),N))
    <=> ( aa_assn_assn(aa_ass954072293579886259n_assn(ord_min_assn,M2),N) = M2 ) ) ).

% min_arg_le(2)
tff(fact_2901_min__arg__le_I2_J,axiom,
    ! [M2: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M2),N))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M2),N) = M2 ) ) ).

% min_arg_le(2)
tff(fact_2902_min__arg__le_I2_J,axiom,
    ! [M2: int,N: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,M2),aa_int_int(aa_int_fun_int_int(ord_min_int,M2),N))
    <=> ( aa_int_int(aa_int_fun_int_int(ord_min_int,M2),N) = M2 ) ) ).

% min_arg_le(2)
tff(fact_2903_min__arg__le_I2_J,axiom,
    ! [M2: set_nat,N: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,M2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(ord_min_set_nat,M2),N))
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(ord_min_set_nat,M2),N) = M2 ) ) ).

% min_arg_le(2)
tff(fact_2904_min__arg__le_I1_J,axiom,
    ! [N: assn,M2: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,N),aa_assn_assn(aa_ass954072293579886259n_assn(ord_min_assn,M2),N))
    <=> ( aa_assn_assn(aa_ass954072293579886259n_assn(ord_min_assn,M2),N) = N ) ) ).

% min_arg_le(1)
tff(fact_2905_min__arg__le_I1_J,axiom,
    ! [N: nat,M2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M2),N))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M2),N) = N ) ) ).

% min_arg_le(1)
tff(fact_2906_min__arg__le_I1_J,axiom,
    ! [N: int,M2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,N),aa_int_int(aa_int_fun_int_int(ord_min_int,M2),N))
    <=> ( aa_int_int(aa_int_fun_int_int(ord_min_int,M2),N) = N ) ) ).

% min_arg_le(1)
tff(fact_2907_min__arg__le_I1_J,axiom,
    ! [N: set_nat,M2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,N),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(ord_min_set_nat,M2),N))
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(ord_min_set_nat,M2),N) = N ) ) ).

% min_arg_le(1)
tff(fact_2908_min__eq__arg_I2_J,axiom,
    ! [M2: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M2),N) = N )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M2) ) ).

% min_eq_arg(2)
tff(fact_2909_min__eq__arg_I2_J,axiom,
    ! [M2: int,N: int] :
      ( ( aa_int_int(aa_int_fun_int_int(ord_min_int,M2),N) = N )
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,N),M2) ) ).

% min_eq_arg(2)
tff(fact_2910_min__eq__arg_I1_J,axiom,
    ! [M2: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M2),N) = M2 )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N) ) ).

% min_eq_arg(1)
tff(fact_2911_min__eq__arg_I1_J,axiom,
    ! [M2: int,N: int] :
      ( ( aa_int_int(aa_int_fun_int_int(ord_min_int,M2),N) = M2 )
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,M2),N) ) ).

% min_eq_arg(1)
tff(fact_2912_max_Oabsorb1,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = A ) ) ).

% max.absorb1
tff(fact_2913_max_Oabsorb1,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) = A ) ) ).

% max.absorb1
tff(fact_2914_max_Oabsorb2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = B ) ) ).

% max.absorb2
tff(fact_2915_max_Oabsorb2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) = B ) ) ).

% max.absorb2
tff(fact_2916_max_Obounded__iff,axiom,
    ! [B: nat,C2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B),C2)),A)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),A) ) ) ).

% max.bounded_iff
tff(fact_2917_max_Obounded__iff,axiom,
    ! [B: int,C2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(ord_max_int,B),C2)),A)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),A) ) ) ).

% max.bounded_iff
tff(fact_2918_min_Oabsorb3,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A),B) = A ) ) ).

% min.absorb3
tff(fact_2919_min_Oabsorb3,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_int(aa_int_fun_int_int(ord_min_int,A),B) = A ) ) ).

% min.absorb3
tff(fact_2920_min_Oabsorb4,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A),B) = B ) ) ).

% min.absorb4
tff(fact_2921_min_Oabsorb4,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ( aa_int_int(aa_int_fun_int_int(ord_min_int,A),B) = B ) ) ).

% min.absorb4
tff(fact_2922_min__less__iff__conj,axiom,
    ! [Z: nat,X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z),aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,X),Y))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z),X)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z),Y) ) ) ).

% min_less_iff_conj
tff(fact_2923_min__less__iff__conj,axiom,
    ! [Z: int,X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_int_int(aa_int_fun_int_int(ord_min_int,X),Y))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z),X)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,Z),Y) ) ) ).

% min_less_iff_conj
tff(fact_2924_min__simps_I2_J,axiom,
    ! [B: assn,A: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),A)
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(ord_min_assn,A),B) = B ) ) ).

% min_simps(2)
tff(fact_2925_min__simps_I2_J,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A),B) = B ) ) ).

% min_simps(2)
tff(fact_2926_min__simps_I2_J,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ( aa_int_int(aa_int_fun_int_int(ord_min_int,A),B) = B ) ) ).

% min_simps(2)
tff(fact_2927_min__simps_I1_J,axiom,
    ! [A: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(ord_min_assn,A),B) = A ) ) ).

% min_simps(1)
tff(fact_2928_min__simps_I1_J,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A),B) = A ) ) ).

% min_simps(1)
tff(fact_2929_min__simps_I1_J,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_int(aa_int_fun_int_int(ord_min_int,A),B) = A ) ) ).

% min_simps(1)
tff(fact_2930_min__less__self__conv_I2_J,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A),B)),B)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% min_less_self_conv(2)
tff(fact_2931_min__less__self__conv_I2_J,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(ord_min_int,A),B)),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% min_less_self_conv(2)
tff(fact_2932_min__less__self__conv_I1_J,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A),B)),A)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A) ) ).

% min_less_self_conv(1)
tff(fact_2933_min__less__self__conv_I1_J,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(ord_min_int,A),B)),A)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ).

% min_less_self_conv(1)
tff(fact_2934_min__arg__not__ge_I2_J,axiom,
    ! [M2: nat,N: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M2),N)),N)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M2),N) = N ) ) ).

% min_arg_not_ge(2)
tff(fact_2935_min__arg__not__ge_I2_J,axiom,
    ! [M2: int,N: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(ord_min_int,M2),N)),N)
    <=> ( aa_int_int(aa_int_fun_int_int(ord_min_int,M2),N) = N ) ) ).

% min_arg_not_ge(2)
tff(fact_2936_min__arg__not__ge_I1_J,axiom,
    ! [M2: nat,N: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M2),N)),M2)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M2),N) = M2 ) ) ).

% min_arg_not_ge(1)
tff(fact_2937_min__arg__not__ge_I1_J,axiom,
    ! [M2: int,N: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(ord_min_int,M2),N)),M2)
    <=> ( aa_int_int(aa_int_fun_int_int(ord_min_int,M2),N) = M2 ) ) ).

% min_arg_not_ge(1)
tff(fact_2938_max_Oabsorb3,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = A ) ) ).

% max.absorb3
tff(fact_2939_max_Oabsorb3,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) = A ) ) ).

% max.absorb3
tff(fact_2940_max_Oabsorb4,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = B ) ) ).

% max.absorb4
tff(fact_2941_max_Oabsorb4,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) = B ) ) ).

% max.absorb4
tff(fact_2942_max__less__iff__conj,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Y)),Z)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Z) ) ) ).

% max_less_iff_conj
tff(fact_2943_max__less__iff__conj,axiom,
    ! [X: int,Y: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(ord_max_int,X),Y)),Z)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Z) ) ) ).

% max_less_iff_conj
tff(fact_2944_min__bot2,axiom,
    ! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(ord_min_assn,X),bot_bot_assn) = bot_bot_assn ).

% min_bot2
tff(fact_2945_min__bot2,axiom,
    ! [X: set_o] : ord_min_set_o(X,bot_bot_set_o) = bot_bot_set_o ).

% min_bot2
tff(fact_2946_min__bot2,axiom,
    ! [X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(ord_min_set_nat,X),bot_bot_set_nat) = bot_bot_set_nat ).

% min_bot2
tff(fact_2947_min__bot2,axiom,
    ! [X: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,X),bot_bot_nat) = bot_bot_nat ).

% min_bot2
tff(fact_2948_min__bot,axiom,
    ! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(ord_min_assn,bot_bot_assn),X) = bot_bot_assn ).

% min_bot
tff(fact_2949_min__bot,axiom,
    ! [X: set_o] : ord_min_set_o(bot_bot_set_o,X) = bot_bot_set_o ).

% min_bot
tff(fact_2950_min__bot,axiom,
    ! [X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(ord_min_set_nat,bot_bot_set_nat),X) = bot_bot_set_nat ).

% min_bot
tff(fact_2951_min__bot,axiom,
    ! [X: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,bot_bot_nat),X) = bot_bot_nat ).

% min_bot
tff(fact_2952_max__bot2,axiom,
    ! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(ord_max_assn,X),bot_bot_assn) = X ).

% max_bot2
tff(fact_2953_max__bot2,axiom,
    ! [X: set_o] : ord_max_set_o(X,bot_bot_set_o) = X ).

% max_bot2
tff(fact_2954_max__bot2,axiom,
    ! [X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(ord_max_set_nat,X),bot_bot_set_nat) = X ).

% max_bot2
tff(fact_2955_max__bot,axiom,
    ! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(ord_max_assn,bot_bot_assn),X) = X ).

% max_bot
tff(fact_2956_max__bot,axiom,
    ! [X: set_o] : ord_max_set_o(bot_bot_set_o,X) = X ).

% max_bot
tff(fact_2957_max__bot,axiom,
    ! [X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(ord_max_set_nat,bot_bot_set_nat),X) = X ).

% max_bot
tff(fact_2958_max__min__same_I4_J,axiom,
    ! [Y: nat,X: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,X),Y)) = Y ).

% max_min_same(4)
tff(fact_2959_max__min__same_I3_J,axiom,
    ! [X: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,X),Y)),Y) = Y ).

% max_min_same(3)
tff(fact_2960_max__min__same_I2_J,axiom,
    ! [X: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,X),Y)),X) = X ).

% max_min_same(2)
tff(fact_2961_max__min__same_I1_J,axiom,
    ! [X: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,X),Y)) = X ).

% max_min_same(1)
tff(fact_2962_Max__insert,axiom,
    ! [A2: set_o,X: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ( A2 != bot_bot_set_o )
       => ( aa_set_o_o(lattic1921953407002678535_Max_o,insert_o((X),A2))
        <=> aa_o_o(aa_o_fun_o_o(ord_max_o,(X)),aa_set_o_o(lattic1921953407002678535_Max_o,A2)) ) ) ) ).

% Max_insert
tff(fact_2963_Max__insert,axiom,
    ! [A2: set_nat,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( A2 != bot_bot_set_nat )
       => ( lattic8265883725875713057ax_nat(insert_nat2(X,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),lattic8265883725875713057ax_nat(A2)) ) ) ) ).

% Max_insert
tff(fact_2964_min_Oassoc,axiom,
    ! [A: nat,B: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A),B)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,B),C2)) ).

% min.assoc
tff(fact_2965_min_Ocommute,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,B),A) ).

% min.commute
tff(fact_2966_max__min__distrib1,axiom,
    ! [B: nat,C2: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,B),C2)),A) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B),A)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,C2),A)) ).

% max_min_distrib1
tff(fact_2967_max__min__distrib2,axiom,
    ! [A: nat,B: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,B),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),C2)) ).

% max_min_distrib2
tff(fact_2968_min_Oleft__commute,axiom,
    ! [B: nat,A: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,B),C2)) ).

% min.left_commute
tff(fact_2969_min__max__distrib1,axiom,
    ! [B: nat,C2: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B),C2)),A) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,B),A)),aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,C2),A)) ).

% min_max_distrib1
tff(fact_2970_min__max__distrib2,axiom,
    ! [A: nat,B: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A),C2)) ).

% min_max_distrib2
tff(fact_2971_max_Omono,axiom,
    ! [C2: nat,A: nat,D2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,D2),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,C2),D2)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B)) ) ) ).

% max.mono
tff(fact_2972_max_Omono,axiom,
    ! [C2: int,A: int,D2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,D2),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(ord_max_int,C2),D2)),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B)) ) ) ).

% max.mono
tff(fact_2973_max_OorderE,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( A = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) ) ) ).

% max.orderE
tff(fact_2974_max_OorderE,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( A = aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) ) ) ).

% max.orderE
tff(fact_2975_max_OorderI,axiom,
    ! [A: nat,B: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A) ) ).

% max.orderI
tff(fact_2976_max_OorderI,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ).

% max.orderI
tff(fact_2977_max_OboundedE,axiom,
    ! [B: nat,C2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B),C2)),A)
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),A) ) ) ).

% max.boundedE
tff(fact_2978_max_OboundedE,axiom,
    ! [B: int,C2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(ord_max_int,B),C2)),A)
     => ~ ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
         => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),A) ) ) ).

% max.boundedE
tff(fact_2979_max_OboundedI,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B),C2)),A) ) ) ).

% max.boundedI
tff(fact_2980_max_OboundedI,axiom,
    ! [B: int,A: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(ord_max_int,B),C2)),A) ) ) ).

% max.boundedI
tff(fact_2981_max_Oorder__iff,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
    <=> ( A = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) ) ) ).

% max.order_iff
tff(fact_2982_max_Oorder__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
    <=> ( A = aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) ) ) ).

% max.order_iff
tff(fact_2983_max_Ocobounded1,axiom,
    ! [A: nat,B: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B)) ).

% max.cobounded1
tff(fact_2984_max_Ocobounded1,axiom,
    ! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B)) ).

% max.cobounded1
tff(fact_2985_max_Ocobounded2,axiom,
    ! [B: nat,A: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B)) ).

% max.cobounded2
tff(fact_2986_max_Ocobounded2,axiom,
    ! [B: int,A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B)) ).

% max.cobounded2
tff(fact_2987_le__max__iff__disj,axiom,
    ! [Z: nat,X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Y))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),X)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),Y) ) ) ).

% le_max_iff_disj
tff(fact_2988_le__max__iff__disj,axiom,
    ! [Z: int,X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_int_int(aa_int_fun_int_int(ord_max_int,X),Y))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),X)
        | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),Y) ) ) ).

% le_max_iff_disj
tff(fact_2989_max_Oabsorb__iff1,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = A ) ) ).

% max.absorb_iff1
tff(fact_2990_max_Oabsorb__iff1,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
    <=> ( aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) = A ) ) ).

% max.absorb_iff1
tff(fact_2991_max_Oabsorb__iff2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = B ) ) ).

% max.absorb_iff2
tff(fact_2992_max_Oabsorb__iff2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
    <=> ( aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) = B ) ) ).

% max.absorb_iff2
tff(fact_2993_max_OcoboundedI1,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B)) ) ).

% max.coboundedI1
tff(fact_2994_max_OcoboundedI1,axiom,
    ! [C2: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B)) ) ).

% max.coboundedI1
tff(fact_2995_max_OcoboundedI2,axiom,
    ! [C2: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B)) ) ).

% max.coboundedI2
tff(fact_2996_max_OcoboundedI2,axiom,
    ! [C2: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B)) ) ).

% max.coboundedI2
tff(fact_2997_max__def,axiom,
    ! [A: assn,B: assn] :
      aa_assn_assn(aa_ass954072293579886259n_assn(ord_max_assn,A),B) = $ite(aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B),B,A) ).

% max_def
tff(fact_2998_max__def,axiom,
    ! [A: nat,B: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B),B,A) ).

% max_def
tff(fact_2999_max__def,axiom,
    ! [A: int,B: int] :
      aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B),B,A) ).

% max_def
tff(fact_3000_max__def,axiom,
    ! [A: set_nat,B: set_nat] :
      aa_set_nat_set_nat(aa_set2309701074089104077et_nat(ord_max_set_nat,A),B) = $ite(aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B),B,A) ).

% max_def
tff(fact_3001_max__absorb1,axiom,
    ! [Y: assn,X: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,Y),X)
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(ord_max_assn,X),Y) = X ) ) ).

% max_absorb1
tff(fact_3002_max__absorb1,axiom,
    ! [Y: nat,X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Y) = X ) ) ).

% max_absorb1
tff(fact_3003_max__absorb1,axiom,
    ! [Y: int,X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),X)
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,X),Y) = X ) ) ).

% max_absorb1
tff(fact_3004_max__absorb1,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Y),X)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(ord_max_set_nat,X),Y) = X ) ) ).

% max_absorb1
tff(fact_3005_max__absorb2,axiom,
    ! [X: assn,Y: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),Y)
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(ord_max_assn,X),Y) = Y ) ) ).

% max_absorb2
tff(fact_3006_max__absorb2,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Y) = Y ) ) ).

% max_absorb2
tff(fact_3007_max__absorb2,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,X),Y) = Y ) ) ).

% max_absorb2
tff(fact_3008_max__absorb2,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Y)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(ord_max_set_nat,X),Y) = Y ) ) ).

% max_absorb2
tff(fact_3009_less__max__iff__disj,axiom,
    ! [Z: nat,X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Y))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z),X)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z),Y) ) ) ).

% less_max_iff_disj
tff(fact_3010_less__max__iff__disj,axiom,
    ! [Z: int,X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_int_int(aa_int_fun_int_int(ord_max_int,X),Y))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z),X)
        | aa_int_o(aa_int_fun_int_o(ord_less_int,Z),Y) ) ) ).

% less_max_iff_disj
tff(fact_3011_max_Ostrict__boundedE,axiom,
    ! [B: nat,C2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B),C2)),A)
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),A) ) ) ).

% max.strict_boundedE
tff(fact_3012_max_Ostrict__boundedE,axiom,
    ! [B: int,C2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(ord_max_int,B),C2)),A)
     => ~ ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
         => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,C2),A) ) ) ).

% max.strict_boundedE
tff(fact_3013_max_Ostrict__order__iff,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
    <=> ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) )
        & ( A != B ) ) ) ).

% max.strict_order_iff
tff(fact_3014_max_Ostrict__order__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
    <=> ( ( A = aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) )
        & ( A != B ) ) ) ).

% max.strict_order_iff
tff(fact_3015_max_Ostrict__coboundedI1,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B)) ) ).

% max.strict_coboundedI1
tff(fact_3016_max_Ostrict__coboundedI1,axiom,
    ! [C2: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,C2),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B)) ) ).

% max.strict_coboundedI1
tff(fact_3017_max_Ostrict__coboundedI2,axiom,
    ! [C2: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B)) ) ).

% max.strict_coboundedI2
tff(fact_3018_max_Ostrict__coboundedI2,axiom,
    ! [C2: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,C2),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B)) ) ).

% max.strict_coboundedI2
tff(fact_3019_min_Omono,axiom,
    ! [A: nat,C2: nat,B: nat,D2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),D2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,C2),D2)) ) ) ).

% min.mono
tff(fact_3020_min_Omono,axiom,
    ! [A: int,C2: int,B: int,D2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),D2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(ord_min_int,A),B)),aa_int_int(aa_int_fun_int_int(ord_min_int,C2),D2)) ) ) ).

% min.mono
tff(fact_3021_min_OorderE,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( A = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A),B) ) ) ).

% min.orderE
tff(fact_3022_min_OorderE,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( A = aa_int_int(aa_int_fun_int_int(ord_min_int,A),B) ) ) ).

% min.orderE
tff(fact_3023_min_OorderI,axiom,
    ! [A: nat,B: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A),B) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% min.orderI
tff(fact_3024_min_OorderI,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(ord_min_int,A),B) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% min.orderI
tff(fact_3025_min_OboundedE,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,B),C2))
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C2) ) ) ).

% min.boundedE
tff(fact_3026_min_OboundedE,axiom,
    ! [A: int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(ord_min_int,B),C2))
     => ~ ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
         => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C2) ) ) ).

% min.boundedE
tff(fact_3027_min_OboundedI,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,B),C2)) ) ) ).

% min.boundedI
tff(fact_3028_min_OboundedI,axiom,
    ! [A: int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(ord_min_int,B),C2)) ) ) ).

% min.boundedI
tff(fact_3029_min_Oorder__iff,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
    <=> ( A = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A),B) ) ) ).

% min.order_iff
tff(fact_3030_min_Oorder__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
    <=> ( A = aa_int_int(aa_int_fun_int_int(ord_min_int,A),B) ) ) ).

% min.order_iff
tff(fact_3031_min_Ocobounded1,axiom,
    ! [A: nat,B: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A),B)),A) ).

% min.cobounded1
tff(fact_3032_min_Ocobounded1,axiom,
    ! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(ord_min_int,A),B)),A) ).

% min.cobounded1
tff(fact_3033_min_Ocobounded2,axiom,
    ! [A: nat,B: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A),B)),B) ).

% min.cobounded2
tff(fact_3034_min_Ocobounded2,axiom,
    ! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(ord_min_int,A),B)),B) ).

% min.cobounded2
tff(fact_3035_min_Oabsorb__iff1,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A),B) = A ) ) ).

% min.absorb_iff1
tff(fact_3036_min_Oabsorb__iff1,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
    <=> ( aa_int_int(aa_int_fun_int_int(ord_min_int,A),B) = A ) ) ).

% min.absorb_iff1
tff(fact_3037_min_Oabsorb__iff2,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A),B) = B ) ) ).

% min.absorb_iff2
tff(fact_3038_min_Oabsorb__iff2,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
    <=> ( aa_int_int(aa_int_fun_int_int(ord_min_int,A),B) = B ) ) ).

% min.absorb_iff2
tff(fact_3039_min_OcoboundedI1,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A),B)),C2) ) ).

% min.coboundedI1
tff(fact_3040_min_OcoboundedI1,axiom,
    ! [A: int,C2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C2)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(ord_min_int,A),B)),C2) ) ).

% min.coboundedI1
tff(fact_3041_min_OcoboundedI2,axiom,
    ! [B: nat,C2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A),B)),C2) ) ).

% min.coboundedI2
tff(fact_3042_min_OcoboundedI2,axiom,
    ! [B: int,C2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C2)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(ord_min_int,A),B)),C2) ) ).

% min.coboundedI2
tff(fact_3043_min__le__iff__disj,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,X),Y)),Z)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Z)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Z) ) ) ).

% min_le_iff_disj
tff(fact_3044_min__le__iff__disj,axiom,
    ! [X: int,Y: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(ord_min_int,X),Y)),Z)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Z)
        | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Z) ) ) ).

% min_le_iff_disj
tff(fact_3045_min__def,axiom,
    ! [A: assn,B: assn] :
      aa_assn_assn(aa_ass954072293579886259n_assn(ord_min_assn,A),B) = $ite(aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B),A,B) ).

% min_def
tff(fact_3046_min__def,axiom,
    ! [A: nat,B: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A),B) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B),A,B) ).

% min_def
tff(fact_3047_min__def,axiom,
    ! [A: int,B: int] :
      aa_int_int(aa_int_fun_int_int(ord_min_int,A),B) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B),A,B) ).

% min_def
tff(fact_3048_min__def,axiom,
    ! [A: set_nat,B: set_nat] :
      aa_set_nat_set_nat(aa_set2309701074089104077et_nat(ord_min_set_nat,A),B) = $ite(aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B),A,B) ).

% min_def
tff(fact_3049_min__absorb1,axiom,
    ! [X: assn,Y: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),Y)
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(ord_min_assn,X),Y) = X ) ) ).

% min_absorb1
tff(fact_3050_min__absorb1,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,X),Y) = X ) ) ).

% min_absorb1
tff(fact_3051_min__absorb1,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
     => ( aa_int_int(aa_int_fun_int_int(ord_min_int,X),Y) = X ) ) ).

% min_absorb1
tff(fact_3052_min__absorb1,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Y)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(ord_min_set_nat,X),Y) = X ) ) ).

% min_absorb1
tff(fact_3053_min__absorb2,axiom,
    ! [Y: assn,X: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,Y),X)
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(ord_min_assn,X),Y) = Y ) ) ).

% min_absorb2
tff(fact_3054_min__absorb2,axiom,
    ! [Y: nat,X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,X),Y) = Y ) ) ).

% min_absorb2
tff(fact_3055_min__absorb2,axiom,
    ! [Y: int,X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),X)
     => ( aa_int_int(aa_int_fun_int_int(ord_min_int,X),Y) = Y ) ) ).

% min_absorb2
tff(fact_3056_min__absorb2,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Y),X)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(ord_min_set_nat,X),Y) = Y ) ) ).

% min_absorb2
tff(fact_3057_min__less__iff__disj,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,X),Y)),Z)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Z) ) ) ).

% min_less_iff_disj
tff(fact_3058_min__less__iff__disj,axiom,
    ! [X: int,Y: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(ord_min_int,X),Y)),Z)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z)
        | aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Z) ) ) ).

% min_less_iff_disj
tff(fact_3059_min_Ostrict__boundedE,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,B),C2))
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),C2) ) ) ).

% min.strict_boundedE
tff(fact_3060_min_Ostrict__boundedE,axiom,
    ! [A: int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(aa_int_fun_int_int(ord_min_int,B),C2))
     => ~ ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
         => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A),C2) ) ) ).

% min.strict_boundedE
tff(fact_3061_min_Ostrict__order__iff,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
    <=> ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A),B) )
        & ( A != B ) ) ) ).

% min.strict_order_iff
tff(fact_3062_min_Ostrict__order__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
    <=> ( ( A = aa_int_int(aa_int_fun_int_int(ord_min_int,A),B) )
        & ( A != B ) ) ) ).

% min.strict_order_iff
tff(fact_3063_min_Ostrict__coboundedI1,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),C2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A),B)),C2) ) ).

% min.strict_coboundedI1
tff(fact_3064_min_Ostrict__coboundedI1,axiom,
    ! [A: int,C2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),C2)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(ord_min_int,A),B)),C2) ) ).

% min.strict_coboundedI1
tff(fact_3065_min_Ostrict__coboundedI2,axiom,
    ! [B: nat,C2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A),B)),C2) ) ).

% min.strict_coboundedI2
tff(fact_3066_min_Ostrict__coboundedI2,axiom,
    ! [B: int,C2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C2)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(ord_min_int,A),B)),C2) ) ).

% min.strict_coboundedI2
tff(fact_3067_max__diff__distrib__left,axiom,
    ! [X: int,Y: int,Z: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(ord_max_int,X),Y)),Z) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Z)),aa_int_int(aa_int_fun_int_int(minus_minus_int,Y),Z)) ).

% max_diff_distrib_left
tff(fact_3068_min__diff__distrib__left,axiom,
    ! [X: int,Y: int,Z: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(ord_min_int,X),Y)),Z) = aa_int_int(aa_int_fun_int_int(ord_min_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Z)),aa_int_int(aa_int_fun_int_int(minus_minus_int,Y),Z)) ).

% min_diff_distrib_left
tff(fact_3069_inf__min,axiom,
    inf_inf_Product_unit = ord_min_Product_unit ).

% inf_min
tff(fact_3070_inf__min,axiom,
    inf_inf_nat = ord_min_nat ).

% inf_min
tff(fact_3071_remove1_Osimps_I2_J,axiom,
    ! [X: nat,Y: nat,Xs: list_nat] :
      remove1_nat(X,aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Y),Xs)) = $ite(X = Y,Xs,aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Y),remove1_nat(X,Xs))) ).

% remove1.simps(2)
tff(fact_3072_remove1_Osimps_I2_J,axiom,
    ! [X: int,Y: int,Xs: list_int] :
      remove1_int(X,aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Y),Xs)) = $ite(X = Y,Xs,aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Y),remove1_int(X,Xs))) ).

% remove1.simps(2)
tff(fact_3073_remove1_Osimps_I2_J,axiom,
    ! [X: produc6575502325842934193n_assn,Y: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] :
      remove1670527618125605709n_assn(X,aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,Y),Xs)) = $ite(X = Y,Xs,aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,Y),remove1670527618125605709n_assn(X,Xs))) ).

% remove1.simps(2)
tff(fact_3074_remove1_Osimps_I1_J,axiom,
    ! [X: a] : remove1_a(X,nil_a) = nil_a ).

% remove1.simps(1)
tff(fact_3075_remove1_Osimps_I1_J,axiom,
    ! [X: b] : remove1_b(X,nil_b) = nil_b ).

% remove1.simps(1)
tff(fact_3076_remove1_Osimps_I1_J,axiom,
    ! [X: produc6575502325842934193n_assn] : remove1670527618125605709n_assn(X,nil_Pr5671120429643327159n_assn) = nil_Pr5671120429643327159n_assn ).

% remove1.simps(1)
tff(fact_3077_remove1_Osimps_I1_J,axiom,
    ! [X: nat] : remove1_nat(X,nil_nat) = nil_nat ).

% remove1.simps(1)
tff(fact_3078_remove1_Osimps_I1_J,axiom,
    ! [X: int] : remove1_int(X,nil_int) = nil_int ).

% remove1.simps(1)
tff(fact_3079_Max_Osubset,axiom,
    ! [A2: set_o,B2: set_o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ( B2 != bot_bot_set_o )
       => ( aa_set_o_o(ord_less_eq_set_o(B2),A2)
         => ( aa_o_o(aa_o_fun_o_o(ord_max_o,aa_set_o_o(lattic1921953407002678535_Max_o,B2)),aa_set_o_o(lattic1921953407002678535_Max_o,A2))
          <=> aa_set_o_o(lattic1921953407002678535_Max_o,A2) ) ) ) ) ).

% Max.subset
tff(fact_3080_Max_Osubset,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( B2 != bot_bot_set_nat )
       => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B2),A2)
         => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,lattic8265883725875713057ax_nat(B2)),lattic8265883725875713057ax_nat(A2)) = lattic8265883725875713057ax_nat(A2) ) ) ) ) ).

% Max.subset
tff(fact_3081_Max_Oclosed,axiom,
    ! [A2: set_o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ( A2 != bot_bot_set_o )
       => ( ! [X2: $o,Y2: $o] : aa_set_o_o(member_o(aa_o_o(aa_o_fun_o_o(ord_max_o,(X2)),(Y2))),insert_o((X2),insert_o((Y2),bot_bot_set_o)))
         => aa_set_o_o(member_o(aa_set_o_o(lattic1921953407002678535_Max_o,A2)),A2) ) ) ) ).

% Max.closed
tff(fact_3082_Max_Oclosed,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( A2 != bot_bot_set_nat )
       => ( ! [X2: nat,Y2: nat] : aa_set_nat_o(member_nat2(aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X2),Y2)),insert_nat2(X2,insert_nat2(Y2,bot_bot_set_nat)))
         => aa_set_nat_o(member_nat2(lattic8265883725875713057ax_nat(A2)),A2) ) ) ) ).

% Max.closed
tff(fact_3083_Max_Oinsert__not__elem,axiom,
    ! [A2: set_o,X: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ~ aa_set_o_o(member_o((X)),A2)
       => ( ( A2 != bot_bot_set_o )
         => ( aa_set_o_o(lattic1921953407002678535_Max_o,insert_o((X),A2))
          <=> aa_o_o(aa_o_fun_o_o(ord_max_o,(X)),aa_set_o_o(lattic1921953407002678535_Max_o,A2)) ) ) ) ) ).

% Max.insert_not_elem
tff(fact_3084_Max_Oinsert__not__elem,axiom,
    ! [A2: set_nat,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ~ aa_set_nat_o(member_nat2(X),A2)
       => ( ( A2 != bot_bot_set_nat )
         => ( lattic8265883725875713057ax_nat(insert_nat2(X,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),lattic8265883725875713057ax_nat(A2)) ) ) ) ) ).

% Max.insert_not_elem
tff(fact_3085_Max_Ounion,axiom,
    ! [A2: set_o,B2: set_o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ( A2 != bot_bot_set_o )
       => ( aa_set_o_o(finite_finite_o,B2)
         => ( ( B2 != bot_bot_set_o )
           => ( aa_set_o_o(lattic1921953407002678535_Max_o,sup_sup_set_o(A2,B2))
            <=> aa_o_o(aa_o_fun_o_o(ord_max_o,aa_set_o_o(lattic1921953407002678535_Max_o,A2)),aa_set_o_o(lattic1921953407002678535_Max_o,B2)) ) ) ) ) ) ).

% Max.union
tff(fact_3086_Max_Ounion,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( A2 != bot_bot_set_nat )
       => ( aa_set_nat_o(finite_finite_nat,B2)
         => ( ( B2 != bot_bot_set_nat )
           => ( lattic8265883725875713057ax_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,lattic8265883725875713057ax_nat(A2)),lattic8265883725875713057ax_nat(B2)) ) ) ) ) ) ).

% Max.union
tff(fact_3087_Min_Osubset,axiom,
    ! [A2: set_o,B2: set_o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ( B2 != bot_bot_set_o )
       => ( aa_set_o_o(ord_less_eq_set_o(B2),A2)
         => ( aa_o_o(aa_o_fun_o_o(ord_min_o,aa_set_o_o(lattic1973801136483472281_Min_o,B2)),aa_set_o_o(lattic1973801136483472281_Min_o,A2))
          <=> aa_set_o_o(lattic1973801136483472281_Min_o,A2) ) ) ) ) ).

% Min.subset
tff(fact_3088_Min_Osubset,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( B2 != bot_bot_set_nat )
       => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B2),A2)
         => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,lattic8721135487736765967in_nat(B2)),lattic8721135487736765967in_nat(A2)) = lattic8721135487736765967in_nat(A2) ) ) ) ) ).

% Min.subset
tff(fact_3089_Min_Oclosed,axiom,
    ! [A2: set_o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ( A2 != bot_bot_set_o )
       => ( ! [X2: $o,Y2: $o] : aa_set_o_o(member_o(aa_o_o(aa_o_fun_o_o(ord_min_o,(X2)),(Y2))),insert_o((X2),insert_o((Y2),bot_bot_set_o)))
         => aa_set_o_o(member_o(aa_set_o_o(lattic1973801136483472281_Min_o,A2)),A2) ) ) ) ).

% Min.closed
tff(fact_3090_Min_Oclosed,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( A2 != bot_bot_set_nat )
       => ( ! [X2: nat,Y2: nat] : aa_set_nat_o(member_nat2(aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,X2),Y2)),insert_nat2(X2,insert_nat2(Y2,bot_bot_set_nat)))
         => aa_set_nat_o(member_nat2(lattic8721135487736765967in_nat(A2)),A2) ) ) ) ).

% Min.closed
tff(fact_3091_Min_Oinsert__not__elem,axiom,
    ! [A2: set_o,X: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ~ aa_set_o_o(member_o((X)),A2)
       => ( ( A2 != bot_bot_set_o )
         => ( aa_set_o_o(lattic1973801136483472281_Min_o,insert_o((X),A2))
          <=> aa_o_o(aa_o_fun_o_o(ord_min_o,(X)),aa_set_o_o(lattic1973801136483472281_Min_o,A2)) ) ) ) ) ).

% Min.insert_not_elem
tff(fact_3092_Min_Oinsert__not__elem,axiom,
    ! [A2: set_nat,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ~ aa_set_nat_o(member_nat2(X),A2)
       => ( ( A2 != bot_bot_set_nat )
         => ( lattic8721135487736765967in_nat(insert_nat2(X,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,X),lattic8721135487736765967in_nat(A2)) ) ) ) ) ).

% Min.insert_not_elem
tff(fact_3093_Min_Ounion,axiom,
    ! [A2: set_o,B2: set_o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ( A2 != bot_bot_set_o )
       => ( aa_set_o_o(finite_finite_o,B2)
         => ( ( B2 != bot_bot_set_o )
           => ( aa_set_o_o(lattic1973801136483472281_Min_o,sup_sup_set_o(A2,B2))
            <=> aa_o_o(aa_o_fun_o_o(ord_min_o,aa_set_o_o(lattic1973801136483472281_Min_o,A2)),aa_set_o_o(lattic1973801136483472281_Min_o,B2)) ) ) ) ) ) ).

% Min.union
tff(fact_3094_Min_Ounion,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( A2 != bot_bot_set_nat )
       => ( aa_set_nat_o(finite_finite_nat,B2)
         => ( ( B2 != bot_bot_set_nat )
           => ( lattic8721135487736765967in_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,lattic8721135487736765967in_nat(A2)),lattic8721135487736765967in_nat(B2)) ) ) ) ) ) ).

% Min.union
tff(fact_3095_complete__linorder__inf__min,axiom,
    inf_inf_Product_unit = ord_min_Product_unit ).

% complete_linorder_inf_min
tff(fact_3096_semilattice__order__set_Osubset__imp,axiom,
    ! [F: fun_o_fun_o_o,Less_eq: fun_o_fun_o_o,Less: fun_o_fun_o_o,A2: set_o,B2: set_o] :
      ( lattic5087519243920114290_set_o(F,Less_eq,Less)
     => ( aa_set_o_o(ord_less_eq_set_o(A2),B2)
       => ( ( A2 != bot_bot_set_o )
         => ( aa_set_o_o(finite_finite_o,B2)
           => aa_o_o(aa_o_fun_o_o(Less_eq,aa_set_o_o(lattic3100085485418696562ce_F_o(F),B2)),aa_set_o_o(lattic3100085485418696562ce_F_o(F),A2)) ) ) ) ) ).

% semilattice_order_set.subset_imp
tff(fact_3097_semilattice__order__set_Osubset__imp,axiom,
    ! [F: fun_nat_fun_nat_nat,Less_eq: fun_nat_fun_nat_o,Less: fun_nat_fun_nat_o,A2: set_nat,B2: set_nat] :
      ( lattic6009151579333465974et_nat(F,Less_eq,Less)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B2)
       => ( ( A2 != bot_bot_set_nat )
         => ( aa_set_nat_o(finite_finite_nat,B2)
           => aa_nat_o(aa_nat_fun_nat_o(Less_eq,aa_set_nat_nat(lattic7742739596368939638_F_nat(F),B2)),aa_set_nat_nat(lattic7742739596368939638_F_nat(F),A2)) ) ) ) ) ).

% semilattice_order_set.subset_imp
tff(fact_3098_mult__le__cancel__left1,axiom,
    ! [C2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C2)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),one_one_int) ) ) ) ).

% mult_le_cancel_left1
tff(fact_3099_mult__le__cancel__left2,axiom,
    ! [C2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),C2)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C2)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A) ) ) ) ).

% mult_le_cancel_left2
tff(fact_3100_mult__le__cancel__right1,axiom,
    ! [C2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C2)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),one_one_int) ) ) ) ).

% mult_le_cancel_right1
tff(fact_3101_mult__le__cancel__right2,axiom,
    ! [A: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),C2)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C2)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A) ) ) ) ).

% mult_le_cancel_right2
tff(fact_3102_mult__less__cancel__left1,axiom,
    ! [C2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,B),one_one_int) ) ) ) ).

% mult_less_cancel_left1
tff(fact_3103_mult__less__cancel__left2,axiom,
    ! [C2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),C2)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),one_one_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A) ) ) ) ).

% mult_less_cancel_left2
tff(fact_3104_le__zero__eq,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),zero_zero_nat)
    <=> ( N = zero_zero_nat ) ) ).

% le_zero_eq
tff(fact_3105_not__gr__zero,axiom,
    ! [N: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
    <=> ( N = zero_zero_nat ) ) ).

% not_gr_zero
tff(fact_3106_mult__zero__left,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,zero_zero_nat),A) = zero_zero_nat ).

% mult_zero_left
tff(fact_3107_mult__zero__right,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),zero_zero_nat) = zero_zero_nat ).

% mult_zero_right
tff(fact_3108_mult__eq__0__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) = zero_zero_nat )
    <=> ( ( A = zero_zero_nat )
        | ( B = zero_zero_nat ) ) ) ).

% mult_eq_0_iff
tff(fact_3109_mult__cancel__left,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B) )
    <=> ( ( C2 = zero_zero_nat )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_3110_mult__cancel__right,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C2) )
    <=> ( ( C2 = zero_zero_nat )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_3111_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),A) = zero_zero_nat ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_3112_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),A) = zero_zero_int ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_3113_diff__zero,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),zero_zero_nat) = A ).

% diff_zero
tff(fact_3114_diff__zero,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),zero_zero_int) = A ).

% diff_zero
tff(fact_3115_zero__diff,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,zero_zero_nat),A) = zero_zero_nat ).

% zero_diff
tff(fact_3116_diff__0__right,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),zero_zero_int) = A ).

% diff_0_right
tff(fact_3117_diff__self,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),A) = zero_zero_int ).

% diff_self
tff(fact_3118_diff__ge__0__iff__ge,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ).

% diff_ge_0_iff_ge
tff(fact_3119_zero__comp__diff__simps_I1_J,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ).

% zero_comp_diff_simps(1)
tff(fact_3120_diff__gt__0__iff__gt,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ).

% diff_gt_0_iff_gt
tff(fact_3121_zero__comp__diff__simps_I2_J,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ).

% zero_comp_diff_simps(2)
tff(fact_3122_mult__cancel__right2,axiom,
    ! [A: int,C2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),C2) = C2 )
    <=> ( ( C2 = zero_zero_int )
        | ( A = one_one_int ) ) ) ).

% mult_cancel_right2
tff(fact_3123_mult__cancel__right1,axiom,
    ! [C2: int,B: int] :
      ( ( C2 = aa_int_int(aa_int_fun_int_int(times_times_int,B),C2) )
    <=> ( ( C2 = zero_zero_int )
        | ( B = one_one_int ) ) ) ).

% mult_cancel_right1
tff(fact_3124_mult__cancel__left2,axiom,
    ! [C2: int,A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,C2),A) = C2 )
    <=> ( ( C2 = zero_zero_int )
        | ( A = one_one_int ) ) ) ).

% mult_cancel_left2
tff(fact_3125_mult__cancel__left1,axiom,
    ! [C2: int,B: int] :
      ( ( C2 = aa_int_int(aa_int_fun_int_int(times_times_int,C2),B) )
    <=> ( ( C2 = zero_zero_int )
        | ( B = one_one_int ) ) ) ).

% mult_cancel_left1
tff(fact_3126_diff__numeral__special_I9_J,axiom,
    aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),one_one_int) = zero_zero_int ).

% diff_numeral_special(9)
tff(fact_3127_min__0__1_I1_J,axiom,
    aa_int_int(aa_int_fun_int_int(ord_min_int,zero_zero_int),one_one_int) = zero_zero_int ).

% min_0_1(1)
tff(fact_3128_min__0__1_I1_J,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,zero_zero_nat),one_one_nat) = zero_zero_nat ).

% min_0_1(1)
tff(fact_3129_min__0__1_I2_J,axiom,
    aa_int_int(aa_int_fun_int_int(ord_min_int,one_one_int),zero_zero_int) = zero_zero_int ).

% min_0_1(2)
tff(fact_3130_min__0__1_I2_J,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,one_one_nat),zero_zero_nat) = zero_zero_nat ).

% min_0_1(2)
tff(fact_3131_max__0__1_I1_J,axiom,
    aa_int_int(aa_int_fun_int_int(ord_max_int,zero_zero_int),one_one_int) = one_one_int ).

% max_0_1(1)
tff(fact_3132_max__0__1_I1_J,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,zero_zero_nat),one_one_nat) = one_one_nat ).

% max_0_1(1)
tff(fact_3133_max__0__1_I2_J,axiom,
    aa_int_int(aa_int_fun_int_int(ord_max_int,one_one_int),zero_zero_int) = one_one_int ).

% max_0_1(2)
tff(fact_3134_max__0__1_I2_J,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,one_one_nat),zero_zero_nat) = one_one_nat ).

% max_0_1(2)
tff(fact_3135_zero__reorient,axiom,
    ! [X: nat] :
      ( ( zero_zero_nat = X )
    <=> ( X = zero_zero_nat ) ) ).

% zero_reorient
tff(fact_3136_zero__le,axiom,
    ! [X: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),X) ).

% zero_le
tff(fact_3137_zero__less__iff__neq__zero,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
    <=> ( N != zero_zero_nat ) ) ).

% zero_less_iff_neq_zero
tff(fact_3138_gr__implies__not__zero,axiom,
    ! [M2: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),N)
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not_zero
tff(fact_3139_not__less__zero,axiom,
    ! [N: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),zero_zero_nat) ).

% not_less_zero
tff(fact_3140_gr__zeroI,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) ) ).

% gr_zeroI
tff(fact_3141_mult__not__zero,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) != zero_zero_nat )
     => ( ( A != zero_zero_nat )
        & ( B != zero_zero_nat ) ) ) ).

% mult_not_zero
tff(fact_3142_divisors__zero,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) = zero_zero_nat )
     => ( ( A = zero_zero_nat )
        | ( B = zero_zero_nat ) ) ) ).

% divisors_zero
tff(fact_3143_no__zero__divisors,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( ( B != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) != zero_zero_nat ) ) ) ).

% no_zero_divisors
tff(fact_3144_mult__left__cancel,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( C2 != zero_zero_nat )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_3145_mult__right__cancel,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( C2 != zero_zero_nat )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C2) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_3146_zero__neq__one,axiom,
    zero_zero_int != one_one_int ).

% zero_neq_one
tff(fact_3147_zero__neq__one,axiom,
    zero_zero_nat != one_one_nat ).

% zero_neq_one
tff(fact_3148_eq__iff__diff__eq__0,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
    <=> ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = zero_zero_int ) ) ).

% eq_iff_diff_eq_0
tff(fact_3149_mult__mono,axiom,
    ! [A: nat,B: nat,C2: nat,D2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),D2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D2)) ) ) ) ) ).

% mult_mono
tff(fact_3150_mult__mono,axiom,
    ! [A: int,B: int,C2: int,D2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),D2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D2)) ) ) ) ) ).

% mult_mono
tff(fact_3151_mult__mono_H,axiom,
    ! [A: nat,B: nat,C2: nat,D2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),D2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D2)) ) ) ) ) ).

% mult_mono'
tff(fact_3152_mult__mono_H,axiom,
    ! [A: int,B: int,C2: int,D2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),D2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D2)) ) ) ) ) ).

% mult_mono'
tff(fact_3153_zero__le__square,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),A)) ).

% zero_le_square
tff(fact_3154_split__mult__pos__le,axiom,
    ! [A: int,B: int] :
      ( ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int) ) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ) ).

% split_mult_pos_le
tff(fact_3155_mult__left__mono__neg,axiom,
    ! [B: int,A: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B)) ) ) ).

% mult_left_mono_neg
tff(fact_3156_mult__nonpos__nonpos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ) ) ).

% mult_nonpos_nonpos
tff(fact_3157_mult__left__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B)) ) ) ).

% mult_left_mono
tff(fact_3158_mult__left__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B)) ) ) ).

% mult_left_mono
tff(fact_3159_mult__right__mono__neg,axiom,
    ! [B: int,A: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)) ) ) ).

% mult_right_mono_neg
tff(fact_3160_mult__right__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C2)) ) ) ).

% mult_right_mono
tff(fact_3161_mult__right__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)) ) ) ).

% mult_right_mono
tff(fact_3162_mult__le__0__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B) ) ) ) ).

% mult_le_0_iff
tff(fact_3163_split__mult__neg__le,axiom,
    ! [A: nat,B: nat] :
      ( ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),zero_zero_nat) )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B) ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat) ) ).

% split_mult_neg_le
tff(fact_3164_split__mult__neg__le,axiom,
    ! [A: int,B: int] :
      ( ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B) ) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int) ) ).

% split_mult_neg_le
tff(fact_3165_mult__nonneg__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)) ) ) ).

% mult_nonneg_nonneg
tff(fact_3166_mult__nonneg__nonneg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ) ) ).

% mult_nonneg_nonneg
tff(fact_3167_mult__nonneg__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat) ) ) ).

% mult_nonneg_nonpos
tff(fact_3168_mult__nonneg__nonpos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int) ) ) ).

% mult_nonneg_nonpos
tff(fact_3169_mult__nonpos__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat) ) ) ).

% mult_nonpos_nonneg
tff(fact_3170_mult__nonpos__nonneg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int) ) ) ).

% mult_nonpos_nonneg
tff(fact_3171_mult__nonneg__nonpos2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),zero_zero_nat) ) ) ).

% mult_nonneg_nonpos2
tff(fact_3172_mult__nonneg__nonpos2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),A)),zero_zero_int) ) ) ).

% mult_nonneg_nonpos2
tff(fact_3173_zero__le__mult__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int) ) ) ) ).

% zero_le_mult_iff
tff(fact_3174_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_3175_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_3176_zero__less__one__class_Ozero__le__one,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),one_one_nat) ).

% zero_less_one_class.zero_le_one
tff(fact_3177_zero__less__one__class_Ozero__le__one,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),one_one_int) ).

% zero_less_one_class.zero_le_one
tff(fact_3178_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),one_one_nat) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_3179_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),one_one_int) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_3180_not__one__le__zero,axiom,
    ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),zero_zero_nat) ).

% not_one_le_zero
tff(fact_3181_not__one__le__zero,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),zero_zero_int) ).

% not_one_le_zero
tff(fact_3182_le__iff__diff__le__0,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),zero_zero_int) ) ).

% le_iff_diff_le_0
tff(fact_3183_mult__neg__neg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ) ) ).

% mult_neg_neg
tff(fact_3184_not__square__less__zero,axiom,
    ! [A: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),A)),zero_zero_int) ).

% not_square_less_zero
tff(fact_3185_mult__less__0__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) ) ) ) ).

% mult_less_0_iff
tff(fact_3186_mult__neg__pos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat) ) ) ).

% mult_neg_pos
tff(fact_3187_mult__neg__pos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int) ) ) ).

% mult_neg_pos
tff(fact_3188_mult__pos__neg,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat) ) ) ).

% mult_pos_neg
tff(fact_3189_mult__pos__neg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int) ) ) ).

% mult_pos_neg
tff(fact_3190_mult__pos__pos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)) ) ) ).

% mult_pos_pos
tff(fact_3191_mult__pos__pos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ) ) ).

% mult_pos_pos
tff(fact_3192_mult__pos__neg2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),zero_zero_nat) ) ) ).

% mult_pos_neg2
tff(fact_3193_mult__pos__neg2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),A)),zero_zero_int) ) ) ).

% mult_pos_neg2
tff(fact_3194_zero__less__mult__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int) ) ) ) ).

% zero_less_mult_iff
tff(fact_3195_zero__less__mult__pos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B) ) ) ).

% zero_less_mult_pos
tff(fact_3196_zero__less__mult__pos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) ) ) ).

% zero_less_mult_pos
tff(fact_3197_zero__less__mult__pos2,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B) ) ) ).

% zero_less_mult_pos2
tff(fact_3198_zero__less__mult__pos2,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,B),A))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) ) ) ).

% zero_less_mult_pos2
tff(fact_3199_mult__less__cancel__left__neg,axiom,
    ! [C2: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ).

% mult_less_cancel_left_neg
tff(fact_3200_mult__less__cancel__left__pos,axiom,
    ! [C2: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).

% mult_less_cancel_left_pos
tff(fact_3201_mult__strict__left__mono__neg,axiom,
    ! [B: int,A: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B)) ) ) ).

% mult_strict_left_mono_neg
tff(fact_3202_mult__strict__left__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B)) ) ) ).

% mult_strict_left_mono
tff(fact_3203_mult__strict__left__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B)) ) ) ).

% mult_strict_left_mono
tff(fact_3204_mult__less__cancel__left__disj,axiom,
    ! [C2: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C2)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_3205_mult__strict__right__mono__neg,axiom,
    ! [B: int,A: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)) ) ) ).

% mult_strict_right_mono_neg
tff(fact_3206_mult__strict__right__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C2)) ) ) ).

% mult_strict_right_mono
tff(fact_3207_mult__strict__right__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)) ) ) ).

% mult_strict_right_mono
tff(fact_3208_mult__less__cancel__right__disj,axiom,
    ! [A: int,C2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C2)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_3209_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B)) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_3210_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B)) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_3211_less__numeral__extra_I1_J,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),one_one_nat) ).

% less_numeral_extra(1)
tff(fact_3212_less__numeral__extra_I1_J,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),one_one_int) ).

% less_numeral_extra(1)
tff(fact_3213_zero__less__one,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),one_one_nat) ).

% zero_less_one
tff(fact_3214_zero__less__one,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),one_one_int) ).

% zero_less_one
tff(fact_3215_not__one__less__zero,axiom,
    ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),zero_zero_nat) ).

% not_one_less_zero
tff(fact_3216_not__one__less__zero,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),zero_zero_int) ).

% not_one_less_zero
tff(fact_3217_less__iff__diff__less__0,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),zero_zero_int) ) ).

% less_iff_diff_less_0
tff(fact_3218_Inf__fin__def,axiom,
    lattic47131356835913163n_assn = lattic7606391089408249806F_assn(inf_inf_assn) ).

% Inf_fin_def
tff(fact_3219_Inf__fin__def,axiom,
    lattic5238388535129920115in_nat = lattic7742739596368939638_F_nat(inf_inf_nat) ).

% Inf_fin_def
tff(fact_3220_Inf__fin__def,axiom,
    lattic3014633134055518761et_nat = lattic4908145837437951532et_nat(inf_inf_set_nat) ).

% Inf_fin_def
tff(fact_3221_Inf__fin__def,axiom,
    lattic1263872656861969706t_unit = lattic2430820486025211623t_unit(inf_inf_Product_unit) ).

% Inf_fin_def
tff(fact_3222_Inf__fin__def,axiom,
    lattic30941717366863870at_nat = lattic2063973316643036219at_nat(inf_in2572325071724192079at_nat) ).

% Inf_fin_def
tff(fact_3223_mult__le__cancel__left,axiom,
    ! [C2: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C2)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ) ) ).

% mult_le_cancel_left
tff(fact_3224_mult__le__cancel__right,axiom,
    ! [A: int,C2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C2)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ) ) ).

% mult_le_cancel_right
tff(fact_3225_mult__left__less__imp__less,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ) ).

% mult_left_less_imp_less
tff(fact_3226_mult__left__less__imp__less,axiom,
    ! [C2: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).

% mult_left_less_imp_less
tff(fact_3227_mult__strict__mono,axiom,
    ! [A: nat,B: nat,C2: nat,D2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),D2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D2)) ) ) ) ) ).

% mult_strict_mono
tff(fact_3228_mult__strict__mono,axiom,
    ! [A: int,B: int,C2: int,D2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),D2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D2)) ) ) ) ) ).

% mult_strict_mono
tff(fact_3229_mult__less__cancel__left,axiom,
    ! [C2: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ) ).

% mult_less_cancel_left
tff(fact_3230_mult__right__less__imp__less,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ) ).

% mult_right_less_imp_less
tff(fact_3231_mult__right__less__imp__less,axiom,
    ! [A: int,C2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).

% mult_right_less_imp_less
tff(fact_3232_mult__strict__mono_H,axiom,
    ! [A: nat,B: nat,C2: nat,D2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),D2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D2)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_3233_mult__strict__mono_H,axiom,
    ! [A: int,B: int,C2: int,D2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),D2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D2)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_3234_mult__less__cancel__right,axiom,
    ! [A: int,C2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ) ).

% mult_less_cancel_right
tff(fact_3235_mult__le__cancel__left__neg,axiom,
    ! [C2: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ) ).

% mult_le_cancel_left_neg
tff(fact_3236_mult__le__cancel__left__pos,axiom,
    ! [C2: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).

% mult_le_cancel_left_pos
tff(fact_3237_mult__left__le__imp__le,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ) ).

% mult_left_le_imp_le
tff(fact_3238_mult__left__le__imp__le,axiom,
    ! [C2: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).

% mult_left_le_imp_le
tff(fact_3239_mult__right__le__imp__le,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ) ).

% mult_right_le_imp_le
tff(fact_3240_mult__right__le__imp__le,axiom,
    ! [A: int,C2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).

% mult_right_le_imp_le
tff(fact_3241_mult__le__less__imp__less,axiom,
    ! [A: nat,B: nat,C2: nat,D2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),D2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D2)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_3242_mult__le__less__imp__less,axiom,
    ! [A: int,B: int,C2: int,D2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),D2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D2)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_3243_mult__less__le__imp__less,axiom,
    ! [A: nat,B: nat,C2: nat,D2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),D2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D2)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_3244_mult__less__le__imp__less,axiom,
    ! [A: int,B: int,C2: int,D2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),D2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C2)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D2)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_3245_mult__left__le,axiom,
    ! [C2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)),A) ) ) ).

% mult_left_le
tff(fact_3246_mult__left__le,axiom,
    ! [C2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),A) ) ) ).

% mult_left_le
tff(fact_3247_mult__le__one,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),one_one_nat)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),one_one_nat) ) ) ) ).

% mult_le_one
tff(fact_3248_mult__le__one,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),one_one_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),one_one_int) ) ) ) ).

% mult_le_one
tff(fact_3249_mult__right__le__one__le,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),one_one_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),Y)),X) ) ) ) ).

% mult_right_le_one_le
tff(fact_3250_mult__left__le__one__le,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),one_one_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,Y),X)),X) ) ) ) ).

% mult_left_le_one_le
tff(fact_3251_semilattice__order__set_OboundedE,axiom,
    ! [F: fun_fu1678786105377694981_nat_o,Less_eq: fun_fu4164567611050918574at_o_o,Less: fun_fu4164567611050918574at_o_o,A2: set_fu3555266933746557007_nat_o,X: fun_Pr6460944699146269935_nat_o] :
      ( lattic955679240546196425_nat_o(F,Less_eq,Less)
     => ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,A2)
       => ( ( A2 != bot_bo8632255151191677115_nat_o )
         => ( aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(Less_eq,X),lattic3774668733822265545_nat_o(F,A2))
           => ! [A9: fun_Pr6460944699146269935_nat_o] :
                ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A9),A2)
               => aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(Less_eq,X),A9) ) ) ) ) ) ).

% semilattice_order_set.boundedE
tff(fact_3252_semilattice__order__set_OboundedE,axiom,
    ! [F: fun_o_fun_o_o,Less_eq: fun_o_fun_o_o,Less: fun_o_fun_o_o,A2: set_o,X: $o] :
      ( lattic5087519243920114290_set_o(F,Less_eq,Less)
     => ( aa_set_o_o(finite_finite_o,A2)
       => ( ( A2 != bot_bot_set_o )
         => ( aa_o_o(aa_o_fun_o_o(Less_eq,(X)),aa_set_o_o(lattic3100085485418696562ce_F_o(F),A2))
           => ! [A9: $o] :
                ( aa_set_o_o(member_o((A9)),A2)
               => aa_o_o(aa_o_fun_o_o(Less_eq,(X)),(A9)) ) ) ) ) ) ).

% semilattice_order_set.boundedE
tff(fact_3253_semilattice__order__set_OboundedE,axiom,
    ! [F: fun_nat_fun_nat_nat,Less_eq: fun_nat_fun_nat_o,Less: fun_nat_fun_nat_o,A2: set_nat,X: nat] :
      ( lattic6009151579333465974et_nat(F,Less_eq,Less)
     => ( aa_set_nat_o(finite_finite_nat,A2)
       => ( ( A2 != bot_bot_set_nat )
         => ( aa_nat_o(aa_nat_fun_nat_o(Less_eq,X),aa_set_nat_nat(lattic7742739596368939638_F_nat(F),A2))
           => ! [A9: nat] :
                ( aa_set_nat_o(member_nat2(A9),A2)
               => aa_nat_o(aa_nat_fun_nat_o(Less_eq,X),A9) ) ) ) ) ) ).

% semilattice_order_set.boundedE
tff(fact_3254_semilattice__order__set_OboundedI,axiom,
    ! [F: fun_fu1678786105377694981_nat_o,Less_eq: fun_fu4164567611050918574at_o_o,Less: fun_fu4164567611050918574at_o_o,A2: set_fu3555266933746557007_nat_o,X: fun_Pr6460944699146269935_nat_o] :
      ( lattic955679240546196425_nat_o(F,Less_eq,Less)
     => ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,A2)
       => ( ( A2 != bot_bo8632255151191677115_nat_o )
         => ( ! [A3: fun_Pr6460944699146269935_nat_o] :
                ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A3),A2)
               => aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(Less_eq,X),A3) )
           => aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(Less_eq,X),lattic3774668733822265545_nat_o(F,A2)) ) ) ) ) ).

% semilattice_order_set.boundedI
tff(fact_3255_semilattice__order__set_OboundedI,axiom,
    ! [F: fun_o_fun_o_o,Less_eq: fun_o_fun_o_o,Less: fun_o_fun_o_o,A2: set_o,X: $o] :
      ( lattic5087519243920114290_set_o(F,Less_eq,Less)
     => ( aa_set_o_o(finite_finite_o,A2)
       => ( ( A2 != bot_bot_set_o )
         => ( ! [A3: $o] :
                ( aa_set_o_o(member_o((A3)),A2)
               => aa_o_o(aa_o_fun_o_o(Less_eq,(X)),(A3)) )
           => aa_o_o(aa_o_fun_o_o(Less_eq,(X)),aa_set_o_o(lattic3100085485418696562ce_F_o(F),A2)) ) ) ) ) ).

% semilattice_order_set.boundedI
tff(fact_3256_semilattice__order__set_OboundedI,axiom,
    ! [F: fun_nat_fun_nat_nat,Less_eq: fun_nat_fun_nat_o,Less: fun_nat_fun_nat_o,A2: set_nat,X: nat] :
      ( lattic6009151579333465974et_nat(F,Less_eq,Less)
     => ( aa_set_nat_o(finite_finite_nat,A2)
       => ( ( A2 != bot_bot_set_nat )
         => ( ! [A3: nat] :
                ( aa_set_nat_o(member_nat2(A3),A2)
               => aa_nat_o(aa_nat_fun_nat_o(Less_eq,X),A3) )
           => aa_nat_o(aa_nat_fun_nat_o(Less_eq,X),aa_set_nat_nat(lattic7742739596368939638_F_nat(F),A2)) ) ) ) ) ).

% semilattice_order_set.boundedI
tff(fact_3257_semilattice__order__set_Obounded__iff,axiom,
    ! [F: fun_o_fun_o_o,Less_eq: fun_o_fun_o_o,Less: fun_o_fun_o_o,A2: set_o,X: $o] :
      ( lattic5087519243920114290_set_o(F,Less_eq,Less)
     => ( aa_set_o_o(finite_finite_o,A2)
       => ( ( A2 != bot_bot_set_o )
         => ( aa_o_o(aa_o_fun_o_o(Less_eq,(X)),aa_set_o_o(lattic3100085485418696562ce_F_o(F),A2))
          <=> ! [X3: $o] :
                ( aa_set_o_o(member_o((X3)),A2)
               => aa_o_o(aa_o_fun_o_o(Less_eq,(X)),(X3)) ) ) ) ) ) ).

% semilattice_order_set.bounded_iff
tff(fact_3258_semilattice__order__set_Obounded__iff,axiom,
    ! [F: fun_nat_fun_nat_nat,Less_eq: fun_nat_fun_nat_o,Less: fun_nat_fun_nat_o,A2: set_nat,X: nat] :
      ( lattic6009151579333465974et_nat(F,Less_eq,Less)
     => ( aa_set_nat_o(finite_finite_nat,A2)
       => ( ( A2 != bot_bot_set_nat )
         => ( aa_nat_o(aa_nat_fun_nat_o(Less_eq,X),aa_set_nat_nat(lattic7742739596368939638_F_nat(F),A2))
          <=> ! [X3: nat] :
                ( aa_set_nat_o(member_nat2(X3),A2)
               => aa_nat_o(aa_nat_fun_nat_o(Less_eq,X),X3) ) ) ) ) ) ).

% semilattice_order_set.bounded_iff
tff(fact_3259_mult__less__cancel__right2,axiom,
    ! [A: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),C2)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),one_one_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A) ) ) ) ).

% mult_less_cancel_right2
tff(fact_3260_mult__less__cancel__right1,axiom,
    ! [C2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,B),one_one_int) ) ) ) ).

% mult_less_cancel_right1
tff(fact_3261_max__mult__distrib__left,axiom,
    ! [P3: int,X: int,Y: int] :
      aa_int_int(aa_int_fun_int_int(times_times_int,P3),aa_int_int(aa_int_fun_int_int(ord_max_int,X),Y)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),P3),aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(aa_int_fun_int_int(times_times_int,P3),X)),aa_int_int(aa_int_fun_int_int(times_times_int,P3),Y)),aa_int_int(aa_int_fun_int_int(ord_min_int,aa_int_int(aa_int_fun_int_int(times_times_int,P3),X)),aa_int_int(aa_int_fun_int_int(times_times_int,P3),Y))) ).

% max_mult_distrib_left
tff(fact_3262_min__mult__distrib__left,axiom,
    ! [P3: int,X: int,Y: int] :
      aa_int_int(aa_int_fun_int_int(times_times_int,P3),aa_int_int(aa_int_fun_int_int(ord_min_int,X),Y)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),P3),aa_int_int(aa_int_fun_int_int(ord_min_int,aa_int_int(aa_int_fun_int_int(times_times_int,P3),X)),aa_int_int(aa_int_fun_int_int(times_times_int,P3),Y)),aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(aa_int_fun_int_int(times_times_int,P3),X)),aa_int_int(aa_int_fun_int_int(times_times_int,P3),Y))) ).

% min_mult_distrib_left
tff(fact_3263_max__mult__distrib__right,axiom,
    ! [X: int,Y: int,P3: int] :
      aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(ord_max_int,X),Y)),P3) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),P3),aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),P3)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),P3)),aa_int_int(aa_int_fun_int_int(ord_min_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),P3)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),P3))) ).

% max_mult_distrib_right
tff(fact_3264_min__mult__distrib__right,axiom,
    ! [X: int,Y: int,P3: int] :
      aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(ord_min_int,X),Y)),P3) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),P3),aa_int_int(aa_int_fun_int_int(ord_min_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),P3)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),P3)),aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),P3)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),P3))) ).

% min_mult_distrib_right
tff(fact_3265_mult__le__cancel__iff1,axiom,
    ! [Z: int,X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),Z)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Z))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y) ) ) ).

% mult_le_cancel_iff1
tff(fact_3266_mult__le__cancel__iff2,axiom,
    ! [Z: int,X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,Z),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Z),Y))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y) ) ) ).

% mult_le_cancel_iff2
tff(fact_3267_mult__less__iff1,axiom,
    ! [Z: int,X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),Z)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Z))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y) ) ) ).

% mult_less_iff1
tff(fact_3268_n__lists__Nil,axiom,
    ! [N: nat] :
      n_lists_a(N,nil_a) = $ite(N = zero_zero_nat,cons_list_a(nil_a,nil_list_a),nil_list_a) ).

% n_lists_Nil
tff(fact_3269_n__lists__Nil,axiom,
    ! [N: nat] :
      n_lists_b(N,nil_b) = $ite(N = zero_zero_nat,cons_list_b(nil_b,nil_list_b),nil_list_b) ).

% n_lists_Nil
tff(fact_3270_n__lists__Nil,axiom,
    ! [N: nat] :
      n_list679713369942834879n_assn(N,nil_Pr5671120429643327159n_assn) = $ite(N = zero_zero_nat,cons_l2423627976422276333n_assn(nil_Pr5671120429643327159n_assn,nil_li5476096274760905021n_assn),nil_li5476096274760905021n_assn) ).

% n_lists_Nil
tff(fact_3271_n__lists__Nil,axiom,
    ! [N: nat] :
      n_lists_nat(N,nil_nat) = $ite(N = zero_zero_nat,cons_list_nat(nil_nat,nil_list_nat),nil_list_nat) ).

% n_lists_Nil
tff(fact_3272_n__lists__Nil,axiom,
    ! [N: nat] :
      n_lists_int(N,nil_int) = $ite(N = zero_zero_nat,cons_list_int(nil_int,nil_list_int),nil_list_int) ).

% n_lists_Nil
tff(fact_3273_nat__geq__1__eq__neqz,axiom,
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),X)
    <=> ( X != zero_zero_nat ) ) ).

% nat_geq_1_eq_neqz
tff(fact_3274_n__lists_Osimps_I1_J,axiom,
    ! [Xs: list_a] : n_lists_a(zero_zero_nat,Xs) = cons_list_a(nil_a,nil_list_a) ).

% n_lists.simps(1)
tff(fact_3275_n__lists_Osimps_I1_J,axiom,
    ! [Xs: list_b] : n_lists_b(zero_zero_nat,Xs) = cons_list_b(nil_b,nil_list_b) ).

% n_lists.simps(1)
tff(fact_3276_n__lists_Osimps_I1_J,axiom,
    ! [Xs: list_P8527749157015355191n_assn] : n_list679713369942834879n_assn(zero_zero_nat,Xs) = cons_l2423627976422276333n_assn(nil_Pr5671120429643327159n_assn,nil_li5476096274760905021n_assn) ).

% n_lists.simps(1)
tff(fact_3277_n__lists_Osimps_I1_J,axiom,
    ! [Xs: list_nat] : n_lists_nat(zero_zero_nat,Xs) = cons_list_nat(nil_nat,nil_list_nat) ).

% n_lists.simps(1)
tff(fact_3278_n__lists_Osimps_I1_J,axiom,
    ! [Xs: list_int] : n_lists_int(zero_zero_nat,Xs) = cons_list_int(nil_int,nil_list_int) ).

% n_lists.simps(1)
tff(fact_3279_semilattice__set_Oinsert__remove,axiom,
    ! [F: fun_o_fun_o_o,A2: set_o,X: $o] :
      ( lattic7139874143898644262_set_o(F)
     => ( aa_set_o_o(finite_finite_o,A2)
       => ( aa_set_o_o(lattic3100085485418696562ce_F_o(F),insert_o((X),A2))
        <=> $ite(minus_minus_set_o(A2,insert_o((X),bot_bot_set_o)) = bot_bot_set_o,(X),aa_o_o(aa_o_fun_o_o(F,(X)),aa_set_o_o(lattic3100085485418696562ce_F_o(F),minus_minus_set_o(A2,insert_o((X),bot_bot_set_o))))) ) ) ) ).

% semilattice_set.insert_remove
tff(fact_3280_semilattice__set_Oinsert__remove,axiom,
    ! [F: fun_nat_fun_nat_nat,A2: set_nat,X: nat] :
      ( lattic1029310888574255042et_nat(F)
     => ( aa_set_nat_o(finite_finite_nat,A2)
       => ( aa_set_nat_nat(lattic7742739596368939638_F_nat(F),insert_nat2(X,A2)) = $ite(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),insert_nat2(X,bot_bot_set_nat)) = bot_bot_set_nat,X,aa_nat_nat(aa_nat_fun_nat_nat(F,X),aa_set_nat_nat(lattic7742739596368939638_F_nat(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),insert_nat2(X,bot_bot_set_nat))))) ) ) ) ).

% semilattice_set.insert_remove
tff(fact_3281_semilattice__set_Oremove,axiom,
    ! [F: fun_fu1678786105377694981_nat_o,A2: set_fu3555266933746557007_nat_o,X: fun_Pr6460944699146269935_nat_o] :
      ( lattic8372730991425538173_nat_o(F)
     => ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,A2)
       => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),A2)
         => ( lattic3774668733822265545_nat_o(F,A2) = $ite(minus_6456595905716665654_nat_o(A2,insert4923026906511862463_nat_o(X,bot_bo8632255151191677115_nat_o)) = bot_bo8632255151191677115_nat_o,X,aa_fun2638059832795814268_nat_o(aa_fun6000889628616002998_nat_o(F,X),lattic3774668733822265545_nat_o(F,minus_6456595905716665654_nat_o(A2,insert4923026906511862463_nat_o(X,bot_bo8632255151191677115_nat_o))))) ) ) ) ) ).

% semilattice_set.remove
tff(fact_3282_semilattice__set_Oremove,axiom,
    ! [F: fun_o_fun_o_o,A2: set_o,X: $o] :
      ( lattic7139874143898644262_set_o(F)
     => ( aa_set_o_o(finite_finite_o,A2)
       => ( aa_set_o_o(member_o((X)),A2)
         => ( aa_set_o_o(lattic3100085485418696562ce_F_o(F),A2)
          <=> $ite(minus_minus_set_o(A2,insert_o((X),bot_bot_set_o)) = bot_bot_set_o,(X),aa_o_o(aa_o_fun_o_o(F,(X)),aa_set_o_o(lattic3100085485418696562ce_F_o(F),minus_minus_set_o(A2,insert_o((X),bot_bot_set_o))))) ) ) ) ) ).

% semilattice_set.remove
tff(fact_3283_semilattice__set_Oremove,axiom,
    ! [F: fun_nat_fun_nat_nat,A2: set_nat,X: nat] :
      ( lattic1029310888574255042et_nat(F)
     => ( aa_set_nat_o(finite_finite_nat,A2)
       => ( aa_set_nat_o(member_nat2(X),A2)
         => ( aa_set_nat_nat(lattic7742739596368939638_F_nat(F),A2) = $ite(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),insert_nat2(X,bot_bot_set_nat)) = bot_bot_set_nat,X,aa_nat_nat(aa_nat_fun_nat_nat(F,X),aa_set_nat_nat(lattic7742739596368939638_F_nat(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),insert_nat2(X,bot_bot_set_nat))))) ) ) ) ) ).

% semilattice_set.remove
tff(fact_3284_dbl__inc__simps_I2_J,axiom,
    aa_int_int(neg_nu5851722552734809277nc_int,zero_zero_int) = one_one_int ).

% dbl_inc_simps(2)
tff(fact_3285_semilattice__set_Ounion,axiom,
    ! [F: fun_o_fun_o_o,A2: set_o,B2: set_o] :
      ( lattic7139874143898644262_set_o(F)
     => ( aa_set_o_o(finite_finite_o,A2)
       => ( ( A2 != bot_bot_set_o )
         => ( aa_set_o_o(finite_finite_o,B2)
           => ( ( B2 != bot_bot_set_o )
             => ( aa_set_o_o(lattic3100085485418696562ce_F_o(F),sup_sup_set_o(A2,B2))
              <=> aa_o_o(aa_o_fun_o_o(F,aa_set_o_o(lattic3100085485418696562ce_F_o(F),A2)),aa_set_o_o(lattic3100085485418696562ce_F_o(F),B2)) ) ) ) ) ) ) ).

% semilattice_set.union
tff(fact_3286_semilattice__set_Ounion,axiom,
    ! [F: fun_nat_fun_nat_nat,A2: set_nat,B2: set_nat] :
      ( lattic1029310888574255042et_nat(F)
     => ( aa_set_nat_o(finite_finite_nat,A2)
       => ( ( A2 != bot_bot_set_nat )
         => ( aa_set_nat_o(finite_finite_nat,B2)
           => ( ( B2 != bot_bot_set_nat )
             => ( aa_set_nat_nat(lattic7742739596368939638_F_nat(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(F,aa_set_nat_nat(lattic7742739596368939638_F_nat(F),A2)),aa_set_nat_nat(lattic7742739596368939638_F_nat(F),B2)) ) ) ) ) ) ) ).

% semilattice_set.union
tff(fact_3287_semilattice__set_Oinsert__not__elem,axiom,
    ! [F: fun_fu1678786105377694981_nat_o,A2: set_fu3555266933746557007_nat_o,X: fun_Pr6460944699146269935_nat_o] :
      ( lattic8372730991425538173_nat_o(F)
     => ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,A2)
       => ( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),A2)
         => ( ( A2 != bot_bo8632255151191677115_nat_o )
           => ( lattic3774668733822265545_nat_o(F,insert4923026906511862463_nat_o(X,A2)) = aa_fun2638059832795814268_nat_o(aa_fun6000889628616002998_nat_o(F,X),lattic3774668733822265545_nat_o(F,A2)) ) ) ) ) ) ).

% semilattice_set.insert_not_elem
tff(fact_3288_semilattice__set_Oinsert__not__elem,axiom,
    ! [F: fun_o_fun_o_o,A2: set_o,X: $o] :
      ( lattic7139874143898644262_set_o(F)
     => ( aa_set_o_o(finite_finite_o,A2)
       => ( ~ aa_set_o_o(member_o((X)),A2)
         => ( ( A2 != bot_bot_set_o )
           => ( aa_set_o_o(lattic3100085485418696562ce_F_o(F),insert_o((X),A2))
            <=> aa_o_o(aa_o_fun_o_o(F,(X)),aa_set_o_o(lattic3100085485418696562ce_F_o(F),A2)) ) ) ) ) ) ).

% semilattice_set.insert_not_elem
tff(fact_3289_semilattice__set_Oinsert__not__elem,axiom,
    ! [F: fun_nat_fun_nat_nat,A2: set_nat,X: nat] :
      ( lattic1029310888574255042et_nat(F)
     => ( aa_set_nat_o(finite_finite_nat,A2)
       => ( ~ aa_set_nat_o(member_nat2(X),A2)
         => ( ( A2 != bot_bot_set_nat )
           => ( aa_set_nat_nat(lattic7742739596368939638_F_nat(F),insert_nat2(X,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(F,X),aa_set_nat_nat(lattic7742739596368939638_F_nat(F),A2)) ) ) ) ) ) ).

% semilattice_set.insert_not_elem
tff(fact_3290_semilattice__set_Oinsert,axiom,
    ! [F: fun_o_fun_o_o,A2: set_o,X: $o] :
      ( lattic7139874143898644262_set_o(F)
     => ( aa_set_o_o(finite_finite_o,A2)
       => ( ( A2 != bot_bot_set_o )
         => ( aa_set_o_o(lattic3100085485418696562ce_F_o(F),insert_o((X),A2))
          <=> aa_o_o(aa_o_fun_o_o(F,(X)),aa_set_o_o(lattic3100085485418696562ce_F_o(F),A2)) ) ) ) ) ).

% semilattice_set.insert
tff(fact_3291_semilattice__set_Oinsert,axiom,
    ! [F: fun_nat_fun_nat_nat,A2: set_nat,X: nat] :
      ( lattic1029310888574255042et_nat(F)
     => ( aa_set_nat_o(finite_finite_nat,A2)
       => ( ( A2 != bot_bot_set_nat )
         => ( aa_set_nat_nat(lattic7742739596368939638_F_nat(F),insert_nat2(X,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(F,X),aa_set_nat_nat(lattic7742739596368939638_F_nat(F),A2)) ) ) ) ) ).

% semilattice_set.insert
tff(fact_3292_semilattice__set_Oclosed,axiom,
    ! [F: fun_fu1678786105377694981_nat_o,A2: set_fu3555266933746557007_nat_o] :
      ( lattic8372730991425538173_nat_o(F)
     => ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,A2)
       => ( ( A2 != bot_bo8632255151191677115_nat_o )
         => ( ! [X2: fun_Pr6460944699146269935_nat_o,Y2: fun_Pr6460944699146269935_nat_o] : aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(aa_fun2638059832795814268_nat_o(aa_fun6000889628616002998_nat_o(F,X2),Y2)),insert4923026906511862463_nat_o(X2,insert4923026906511862463_nat_o(Y2,bot_bo8632255151191677115_nat_o)))
           => aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(lattic3774668733822265545_nat_o(F,A2)),A2) ) ) ) ) ).

% semilattice_set.closed
tff(fact_3293_semilattice__set_Oclosed,axiom,
    ! [F: fun_o_fun_o_o,A2: set_o] :
      ( lattic7139874143898644262_set_o(F)
     => ( aa_set_o_o(finite_finite_o,A2)
       => ( ( A2 != bot_bot_set_o )
         => ( ! [X2: $o,Y2: $o] : aa_set_o_o(member_o(aa_o_o(aa_o_fun_o_o(F,(X2)),(Y2))),insert_o((X2),insert_o((Y2),bot_bot_set_o)))
           => aa_set_o_o(member_o(aa_set_o_o(lattic3100085485418696562ce_F_o(F),A2)),A2) ) ) ) ) ).

% semilattice_set.closed
tff(fact_3294_semilattice__set_Oclosed,axiom,
    ! [F: fun_nat_fun_nat_nat,A2: set_nat] :
      ( lattic1029310888574255042et_nat(F)
     => ( aa_set_nat_o(finite_finite_nat,A2)
       => ( ( A2 != bot_bot_set_nat )
         => ( ! [X2: nat,Y2: nat] : aa_set_nat_o(member_nat2(aa_nat_nat(aa_nat_fun_nat_nat(F,X2),Y2)),insert_nat2(X2,insert_nat2(Y2,bot_bot_set_nat)))
           => aa_set_nat_o(member_nat2(aa_set_nat_nat(lattic7742739596368939638_F_nat(F),A2)),A2) ) ) ) ) ).

% semilattice_set.closed
tff(fact_3295_semilattice__set_Osubset,axiom,
    ! [F: fun_o_fun_o_o,A2: set_o,B2: set_o] :
      ( lattic7139874143898644262_set_o(F)
     => ( aa_set_o_o(finite_finite_o,A2)
       => ( ( B2 != bot_bot_set_o )
         => ( aa_set_o_o(ord_less_eq_set_o(B2),A2)
           => ( aa_o_o(aa_o_fun_o_o(F,aa_set_o_o(lattic3100085485418696562ce_F_o(F),B2)),aa_set_o_o(lattic3100085485418696562ce_F_o(F),A2))
            <=> aa_set_o_o(lattic3100085485418696562ce_F_o(F),A2) ) ) ) ) ) ).

% semilattice_set.subset
tff(fact_3296_semilattice__set_Osubset,axiom,
    ! [F: fun_nat_fun_nat_nat,A2: set_nat,B2: set_nat] :
      ( lattic1029310888574255042et_nat(F)
     => ( aa_set_nat_o(finite_finite_nat,A2)
       => ( ( B2 != bot_bot_set_nat )
         => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B2),A2)
           => ( aa_nat_nat(aa_nat_fun_nat_nat(F,aa_set_nat_nat(lattic7742739596368939638_F_nat(F),B2)),aa_set_nat_nat(lattic7742739596368939638_F_nat(F),A2)) = aa_set_nat_nat(lattic7742739596368939638_F_nat(F),A2) ) ) ) ) ) ).

% semilattice_set.subset
tff(fact_3297_convex__bound__lt,axiom,
    ! [X: int,A: int,Y: int,U: int,V: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),U)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),V)
           => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,U),V) = one_one_int )
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,U),X)),aa_int_int(aa_int_fun_int_int(times_times_int,V),Y))),A) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_3298_add__right__cancel,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),A) )
    <=> ( B = C2 ) ) ).

% add_right_cancel
tff(fact_3299_add__right__cancel,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),A) )
    <=> ( B = C2 ) ) ).

% add_right_cancel
tff(fact_3300_add__left__cancel,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2) )
    <=> ( B = C2 ) ) ).

% add_left_cancel
tff(fact_3301_add__left__cancel,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2) )
    <=> ( B = C2 ) ) ).

% add_left_cancel
tff(fact_3302_add__le__cancel__right,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% add_le_cancel_right
tff(fact_3303_add__le__cancel__right,axiom,
    ! [A: int,C2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% add_le_cancel_right
tff(fact_3304_add__le__cancel__left,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% add_le_cancel_left
tff(fact_3305_add__le__cancel__left,axiom,
    ! [C2: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% add_le_cancel_left
tff(fact_3306_add__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,zero_zero_int),A) = A ).

% add_0
tff(fact_3307_add__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),A) = A ).

% add_0
tff(fact_3308_zero__eq__add__iff__both__eq__0,axiom,
    ! [X: nat,Y: nat] :
      ( ( zero_zero_nat = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y) )
    <=> ( ( X = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% zero_eq_add_iff_both_eq_0
tff(fact_3309_add__eq__0__iff__both__eq__0,axiom,
    ! [X: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y) = zero_zero_nat )
    <=> ( ( X = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% add_eq_0_iff_both_eq_0
tff(fact_3310_add__cancel__right__right,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) )
    <=> ( B = zero_zero_int ) ) ).

% add_cancel_right_right
tff(fact_3311_add__cancel__right__right,axiom,
    ! [A: nat,B: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) )
    <=> ( B = zero_zero_nat ) ) ).

% add_cancel_right_right
tff(fact_3312_add__cancel__right__left,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) )
    <=> ( B = zero_zero_int ) ) ).

% add_cancel_right_left
tff(fact_3313_add__cancel__right__left,axiom,
    ! [A: nat,B: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) )
    <=> ( B = zero_zero_nat ) ) ).

% add_cancel_right_left
tff(fact_3314_add__cancel__left__right,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = A )
    <=> ( B = zero_zero_int ) ) ).

% add_cancel_left_right
tff(fact_3315_add__cancel__left__right,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) = A )
    <=> ( B = zero_zero_nat ) ) ).

% add_cancel_left_right
tff(fact_3316_add__cancel__left__left,axiom,
    ! [B: int,A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) = A )
    <=> ( B = zero_zero_int ) ) ).

% add_cancel_left_left
tff(fact_3317_add__cancel__left__left,axiom,
    ! [B: nat,A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) = A )
    <=> ( B = zero_zero_nat ) ) ).

% add_cancel_left_left
tff(fact_3318_double__zero__sym,axiom,
    ! [A: int] :
      ( ( zero_zero_int = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A) )
    <=> ( A = zero_zero_int ) ) ).

% double_zero_sym
tff(fact_3319_add_Oright__neutral,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),zero_zero_int) = A ).

% add.right_neutral
tff(fact_3320_add_Oright__neutral,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),zero_zero_nat) = A ).

% add.right_neutral
tff(fact_3321_add__less__cancel__right,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% add_less_cancel_right
tff(fact_3322_add__less__cancel__right,axiom,
    ! [A: int,C2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% add_less_cancel_right
tff(fact_3323_add__less__cancel__left,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% add_less_cancel_left
tff(fact_3324_add__less__cancel__left,axiom,
    ! [C2: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% add_less_cancel_left
tff(fact_3325_add__diff__cancel__right_H,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),B) = A ).

% add_diff_cancel_right'
tff(fact_3326_add__diff__cancel__right_H,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),B) = A ).

% add_diff_cancel_right'
tff(fact_3327_add__diff__cancel__right,axiom,
    ! [A: nat,C2: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B) ).

% add_diff_cancel_right
tff(fact_3328_add__diff__cancel__right,axiom,
    ! [A: int,C2: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) ).

% add_diff_cancel_right
tff(fact_3329_add__diff__cancel__left_H,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),A) = B ).

% add_diff_cancel_left'
tff(fact_3330_add__diff__cancel__left_H,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),A) = B ).

% add_diff_cancel_left'
tff(fact_3331_add__diff__cancel__left,axiom,
    ! [C2: nat,A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),B)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B) ).

% add_diff_cancel_left
tff(fact_3332_add__diff__cancel__left,axiom,
    ! [C2: int,A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),B)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) ).

% add_diff_cancel_left
tff(fact_3333_diff__add__cancel,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),B) = A ).

% diff_add_cancel
tff(fact_3334_add__diff__cancel,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),B) = A ).

% add_diff_cancel
tff(fact_3335_add__le__same__cancel1,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A)),B)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat) ) ).

% add_le_same_cancel1
tff(fact_3336_add__le__same__cancel1,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A)),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ).

% add_le_same_cancel1
tff(fact_3337_add__le__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),B)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat) ) ).

% add_le_same_cancel2
tff(fact_3338_add__le__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ).

% add_le_same_cancel2
tff(fact_3339_le__add__same__cancel1,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B) ) ).

% le_add_same_cancel1
tff(fact_3340_le__add__same__cancel1,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B) ) ).

% le_add_same_cancel1
tff(fact_3341_le__add__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B) ) ).

% le_add_same_cancel2
tff(fact_3342_le__add__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B) ) ).

% le_add_same_cancel2
tff(fact_3343_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_3344_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_3345_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_3346_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_3347_less__add__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B) ) ).

% less_add_same_cancel2
tff(fact_3348_less__add__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) ) ).

% less_add_same_cancel2
tff(fact_3349_less__add__same__cancel1,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B) ) ).

% less_add_same_cancel1
tff(fact_3350_less__add__same__cancel1,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) ) ).

% less_add_same_cancel1
tff(fact_3351_add__less__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),B)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),zero_zero_nat) ) ).

% add_less_same_cancel2
tff(fact_3352_add__less__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ).

% add_less_same_cancel2
tff(fact_3353_add__less__same__cancel1,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A)),B)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),zero_zero_nat) ) ).

% add_less_same_cancel1
tff(fact_3354_add__less__same__cancel1,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A)),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ).

% add_less_same_cancel1
tff(fact_3355_diff__add__zero,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) = zero_zero_nat ).

% diff_add_zero
tff(fact_3356_exists__leI,axiom,
    ! [N: nat,Pa: fun_nat_o] :
      ( ( ! [N5: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N5),N)
           => ~ aa_nat_o(Pa,N5) )
       => aa_nat_o(Pa,N) )
     => ? [N6: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N6),N)
          & aa_nat_o(Pa,N6) ) ) ).

% exists_leI
tff(fact_3357_add_Oright__commute,axiom,
    ! [A: nat,B: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2)),B) ).

% add.right_commute
tff(fact_3358_add_Oright__commute,axiom,
    ! [A: int,B: int,C2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)),B) ).

% add.right_commute
tff(fact_3359_add_Oright__assoc,axiom,
    ! [A: nat,B: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C2)) ).

% add.right_assoc
tff(fact_3360_add_Oright__assoc,axiom,
    ! [A: int,B: int,C2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C2)) ).

% add.right_assoc
tff(fact_3361_add__right__imp__eq,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),A) )
     => ( B = C2 ) ) ).

% add_right_imp_eq
tff(fact_3362_add__right__imp__eq,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),A) )
     => ( B = C2 ) ) ).

% add_right_imp_eq
tff(fact_3363_add__left__imp__eq,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2) )
     => ( B = C2 ) ) ).

% add_left_imp_eq
tff(fact_3364_add__left__imp__eq,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2) )
     => ( B = C2 ) ) ).

% add_left_imp_eq
tff(fact_3365_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B: nat,A: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C2)) ).

% ab_semigroup_add_class.add.left_commute
tff(fact_3366_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B: int,A: int,C2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C2)) ).

% ab_semigroup_add_class.add.left_commute
tff(fact_3367_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) ).

% ab_semigroup_add_class.add.commute
tff(fact_3368_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) ).

% ab_semigroup_add_class.add.commute
tff(fact_3369_add_Oright__cancel,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),A) )
    <=> ( B = C2 ) ) ).

% add.right_cancel
tff(fact_3370_add_Oleft__cancel,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2) )
    <=> ( B = C2 ) ) ).

% add.left_cancel
tff(fact_3371_add_Oassoc,axiom,
    ! [A: nat,B: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C2)) ).

% add.assoc
tff(fact_3372_add_Oassoc,axiom,
    ! [A: int,B: int,C2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C2)) ).

% add.assoc
tff(fact_3373_group__cancel_Oadd2,axiom,
    ! [B2: nat,K: nat,B: nat,A: nat] :
      ( ( B2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),B) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% group_cancel.add2
tff(fact_3374_group__cancel_Oadd2,axiom,
    ! [B2: int,K: int,B: int,A: int] :
      ( ( B2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),B) )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% group_cancel.add2
tff(fact_3375_group__cancel_Oadd1,axiom,
    ! [A2: nat,K: nat,A: nat,B: nat] :
      ( ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),A) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% group_cancel.add1
tff(fact_3376_group__cancel_Oadd1,axiom,
    ! [A2: int,K: int,A: int,B: int] :
      ( ( A2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),A) )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% group_cancel.add1
tff(fact_3377_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: nat,J: nat,K: nat,La: nat] :
      ( ( ( I = J )
        & ( K = La ) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),La) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_3378_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: int,J: int,K: int,La: int] :
      ( ( ( I = J )
        & ( K = La ) )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K) = aa_int_int(aa_int_fun_int_int(plus_plus_int,J),La) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_3379_mlex__leI,axiom,
    ! [A: nat,A8: nat,B: nat,B7: nat,N4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),A8)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),B7)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),N4)),B)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A8),N4)),B7)) ) ) ).

% mlex_leI
tff(fact_3380_dbl__inc__def,axiom,
    ! [X: int] : aa_int_int(neg_nu5851722552734809277nc_int,X) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),X)),one_one_int) ).

% dbl_inc_def
tff(fact_3381_add__le__imp__le__right,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C2))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% add_le_imp_le_right
tff(fact_3382_add__le__imp__le__right,axiom,
    ! [A: int,C2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C2))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% add_le_imp_le_right
tff(fact_3383_add__le__imp__le__left,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),B))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% add_le_imp_le_left
tff(fact_3384_add__le__imp__le__left,axiom,
    ! [C2: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),B))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% add_le_imp_le_left
tff(fact_3385_le__iff__add,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
    <=> ? [C5: nat] : B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C5) ) ).

% le_iff_add
tff(fact_3386_add__right__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C2)) ) ).

% add_right_mono
tff(fact_3387_add__right__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C2)) ) ).

% add_right_mono
tff(fact_3388_less__eqE,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ~ ! [C: nat] : B != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C) ) ).

% less_eqE
tff(fact_3389_add__left__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),B)) ) ).

% add_left_mono
tff(fact_3390_add__left__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),B)) ) ).

% add_left_mono
tff(fact_3391_add__mono,axiom,
    ! [A: nat,B: nat,C2: nat,D2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),D2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D2)) ) ) ).

% add_mono
tff(fact_3392_add__mono,axiom,
    ! [A: int,B: int,C2: int,D2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),D2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D2)) ) ) ).

% add_mono
tff(fact_3393_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: nat,J: nat,K: nat,La: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),La) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),La)) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_3394_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: int,J: int,K: int,La: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),La) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),La)) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_3395_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: nat,J: nat,K: nat,La: nat] :
      ( ( ( I = J )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),La) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),La)) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_3396_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: int,J: int,K: int,La: int] :
      ( ( ( I = J )
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),La) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),La)) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_3397_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: nat,J: nat,K: nat,La: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
        & ( K = La ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),La)) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_3398_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: int,J: int,K: int,La: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
        & ( K = La ) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),La)) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_3399_add_Ogroup__left__neutral,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,zero_zero_int),A) = A ).

% add.group_left_neutral
tff(fact_3400_add_Ocomm__neutral,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),zero_zero_int) = A ).

% add.comm_neutral
tff(fact_3401_add_Ocomm__neutral,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),zero_zero_nat) = A ).

% add.comm_neutral
tff(fact_3402_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,zero_zero_int),A) = A ).

% comm_monoid_add_class.add_0
tff(fact_3403_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),A) = A ).

% comm_monoid_add_class.add_0
tff(fact_3404_add__less__imp__less__right,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C2))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% add_less_imp_less_right
tff(fact_3405_add__less__imp__less__right,axiom,
    ! [A: int,C2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C2))
     => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% add_less_imp_less_right
tff(fact_3406_add__less__imp__less__left,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),B))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% add_less_imp_less_left
tff(fact_3407_add__less__imp__less__left,axiom,
    ! [C2: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),B))
     => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% add_less_imp_less_left
tff(fact_3408_add__strict__right__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C2)) ) ).

% add_strict_right_mono
tff(fact_3409_add__strict__right__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C2)) ) ).

% add_strict_right_mono
tff(fact_3410_add__strict__left__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),B)) ) ).

% add_strict_left_mono
tff(fact_3411_add__strict__left__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),B)) ) ).

% add_strict_left_mono
tff(fact_3412_add__strict__mono,axiom,
    ! [A: nat,B: nat,C2: nat,D2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),D2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D2)) ) ) ).

% add_strict_mono
tff(fact_3413_add__strict__mono,axiom,
    ! [A: int,B: int,C2: int,D2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),D2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D2)) ) ) ).

% add_strict_mono
tff(fact_3414_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: nat,J: nat,K: nat,La: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
        & ( K = La ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),La)) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_3415_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: int,J: int,K: int,La: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),J)
        & ( K = La ) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),La)) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_3416_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: nat,J: nat,K: nat,La: nat] :
      ( ( ( I = J )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),La) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),La)) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_3417_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: int,J: int,K: int,La: int] :
      ( ( ( I = J )
        & aa_int_o(aa_int_fun_int_o(ord_less_int,K),La) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),La)) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_3418_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: nat,J: nat,K: nat,La: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),La) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),La)) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_3419_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: int,J: int,K: int,La: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),J)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,K),La) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),La)) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_3420_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: int,B: int,C2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)) ).

% ring_class.ring_distribs(2)
tff(fact_3421_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: int,B: int,C2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)) ).

% ring_class.ring_distribs(1)
tff(fact_3422_comm__semiring__class_Odistrib,axiom,
    ! [A: int,B: int,C2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)) ).

% comm_semiring_class.distrib
tff(fact_3423_comm__semiring__class_Odistrib,axiom,
    ! [A: nat,B: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C2)) ).

% comm_semiring_class.distrib
tff(fact_3424_distrib__left,axiom,
    ! [A: int,B: int,C2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)) ).

% distrib_left
tff(fact_3425_distrib__left,axiom,
    ! [A: nat,B: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)) ).

% distrib_left
tff(fact_3426_distrib__right,axiom,
    ! [A: int,B: int,C2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)) ).

% distrib_right
tff(fact_3427_distrib__right,axiom,
    ! [A: nat,B: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C2)) ).

% distrib_right
tff(fact_3428_combine__common__factor,axiom,
    ! [A: int,E3: int,B: int,C2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E3)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E3)),C2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),E3)),C2) ).

% combine_common_factor
tff(fact_3429_combine__common__factor,axiom,
    ! [A: nat,E3: nat,B: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),E3)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),E3)),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),E3)),C2) ).

% combine_common_factor
tff(fact_3430_diff__diff__eq,axiom,
    ! [A: nat,B: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C2)) ).

% diff_diff_eq
tff(fact_3431_diff__diff__eq,axiom,
    ! [A: int,B: int,C2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C2) = aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C2)) ).

% diff_diff_eq
tff(fact_3432_add__implies__diff,axiom,
    ! [C2: nat,B: nat,A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),B) = A )
     => ( C2 = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B) ) ) ).

% add_implies_diff
tff(fact_3433_add__implies__diff,axiom,
    ! [C2: int,B: int,A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),B) = A )
     => ( C2 = aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) ) ) ).

% add_implies_diff
tff(fact_3434_diff__add__eq__diff__diff__swap,axiom,
    ! [A: int,B: int,C2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C2)),B) ).

% diff_add_eq_diff_diff_swap
tff(fact_3435_diff__add__eq,axiom,
    ! [A: int,B: int,C2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C2) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)),B) ).

% diff_add_eq
tff(fact_3436_diff__diff__eq2,axiom,
    ! [A: int,B: int,C2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)),B) ).

% diff_diff_eq2
tff(fact_3437_add__diff__eq,axiom,
    ! [A: int,B: int,C2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C2) ).

% add_diff_eq
tff(fact_3438_eq__diff__eq,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(minus_minus_int,C2),B) )
    <=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = C2 ) ) ).

% eq_diff_eq
tff(fact_3439_diff__eq__eq,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = C2 )
    <=> ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),B) ) ) ).

% diff_eq_eq
tff(fact_3440_group__cancel_Osub1,axiom,
    ! [A2: int,K: int,A: int,B: int] :
      ( ( A2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),A) )
     => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) ) ) ).

% group_cancel.sub1
tff(fact_3441_min__add__distrib__right,axiom,
    ! [X: int,Y: int,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,X),aa_int_int(aa_int_fun_int_int(ord_min_int,Y),Z)) = aa_int_int(aa_int_fun_int_int(ord_min_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Y)),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Z)) ).

% min_add_distrib_right
tff(fact_3442_min__add__distrib__right,axiom,
    ! [X: nat,Y: nat,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Y),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Z)) ).

% min_add_distrib_right
tff(fact_3443_min__add__distrib__left,axiom,
    ! [X: int,Y: int,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(ord_min_int,X),Y)),Z) = aa_int_int(aa_int_fun_int_int(ord_min_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Z)),aa_int_int(aa_int_fun_int_int(plus_plus_int,Y),Z)) ).

% min_add_distrib_left
tff(fact_3444_min__add__distrib__left,axiom,
    ! [X: nat,Y: nat,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,X),Y)),Z) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Z)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Y),Z)) ).

% min_add_distrib_left
tff(fact_3445_max__add__distrib__right,axiom,
    ! [X: nat,Y: nat,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Y),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Z)) ).

% max_add_distrib_right
tff(fact_3446_max__add__distrib__right,axiom,
    ! [X: int,Y: int,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,X),aa_int_int(aa_int_fun_int_int(ord_max_int,Y),Z)) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Y)),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Z)) ).

% max_add_distrib_right
tff(fact_3447_max__add__distrib__left,axiom,
    ! [X: nat,Y: nat,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Y)),Z) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Z)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Y),Z)) ).

% max_add_distrib_left
tff(fact_3448_max__add__distrib__left,axiom,
    ! [X: int,Y: int,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(ord_max_int,X),Y)),Z) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Z)),aa_int_int(aa_int_fun_int_int(plus_plus_int,Y),Z)) ).

% max_add_distrib_left
tff(fact_3449_Inf__fin_Osemilattice__set__axioms,axiom,
    lattic7983604933768736026t_assn(inf_inf_assn) ).

% Inf_fin.semilattice_set_axioms
tff(fact_3450_Inf__fin_Osemilattice__set__axioms,axiom,
    lattic1029310888574255042et_nat(inf_inf_nat) ).

% Inf_fin.semilattice_set_axioms
tff(fact_3451_Inf__fin_Osemilattice__set__axioms,axiom,
    lattic6452893353811829624et_nat(inf_inf_set_nat) ).

% Inf_fin.semilattice_set_axioms
tff(fact_3452_Inf__fin_Osemilattice__set__axioms,axiom,
    lattic356177490429882523t_unit(inf_inf_Product_unit) ).

% Inf_fin.semilattice_set_axioms
tff(fact_3453_Inf__fin_Osemilattice__set__axioms,axiom,
    lattic6581507972311975919at_nat(inf_in2572325071724192079at_nat) ).

% Inf_fin.semilattice_set_axioms
tff(fact_3454_add_Osafe__commute,axiom,
    ! [X: nat,Y: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(syntax4682126007086162916at_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y)),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) ) ) ).

% add.safe_commute
tff(fact_3455_add_Osafe__commute,axiom,
    ! [X: int,Y: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(syntax5678989248478167196nt_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Y)),A)
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) ) ) ).

% add.safe_commute
tff(fact_3456_add__decreasing,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2)),B) ) ) ).

% add_decreasing
tff(fact_3457_add__decreasing,axiom,
    ! [A: int,C2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)),B) ) ) ).

% add_decreasing
tff(fact_3458_add__increasing,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2)) ) ) ).

% add_increasing
tff(fact_3459_add__increasing,axiom,
    ! [A: int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)) ) ) ).

% add_increasing
tff(fact_3460_add__decreasing2,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2)),B) ) ) ).

% add_decreasing2
tff(fact_3461_add__decreasing2,axiom,
    ! [C2: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)),B) ) ) ).

% add_decreasing2
tff(fact_3462_add__increasing2,axiom,
    ! [C2: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2)) ) ) ).

% add_increasing2
tff(fact_3463_add__increasing2,axiom,
    ! [C2: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)) ) ) ).

% add_increasing2
tff(fact_3464_add__nonneg__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% add_nonneg_nonneg
tff(fact_3465_add__nonneg__nonneg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% add_nonneg_nonneg
tff(fact_3466_add__nonpos__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),zero_zero_nat) ) ) ).

% add_nonpos_nonpos
tff(fact_3467_add__nonpos__nonpos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),zero_zero_int) ) ) ).

% add_nonpos_nonpos
tff(fact_3468_add__nonneg__eq__0__iff,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),X)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Y)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y) = zero_zero_nat )
        <=> ( ( X = zero_zero_nat )
            & ( Y = zero_zero_nat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_3469_add__nonneg__eq__0__iff,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Y) = zero_zero_int )
        <=> ( ( X = zero_zero_int )
            & ( Y = zero_zero_int ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_3470_add__nonpos__eq__0__iff,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),zero_zero_nat)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y) = zero_zero_nat )
        <=> ( ( X = zero_zero_nat )
            & ( Y = zero_zero_nat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_3471_add__nonpos__eq__0__iff,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),zero_zero_int)
       => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Y) = zero_zero_int )
        <=> ( ( X = zero_zero_int )
            & ( Y = zero_zero_int ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_3472_add__less__le__mono,axiom,
    ! [A: nat,B: nat,C2: nat,D2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),D2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D2)) ) ) ).

% add_less_le_mono
tff(fact_3473_add__less__le__mono,axiom,
    ! [A: int,B: int,C2: int,D2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),D2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D2)) ) ) ).

% add_less_le_mono
tff(fact_3474_add__le__less__mono,axiom,
    ! [A: nat,B: nat,C2: nat,D2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),D2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D2)) ) ) ).

% add_le_less_mono
tff(fact_3475_add__le__less__mono,axiom,
    ! [A: int,B: int,C2: int,D2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),D2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D2)) ) ) ).

% add_le_less_mono
tff(fact_3476_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: nat,J: nat,K: nat,La: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),La) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),La)) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_3477_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: int,J: int,K: int,La: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),J)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),La) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),La)) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_3478_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: nat,J: nat,K: nat,La: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),La) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),La)) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_3479_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: int,J: int,K: int,La: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,K),La) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),La)) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_3480_pos__add__strict,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2)) ) ) ).

% pos_add_strict
tff(fact_3481_pos__add__strict,axiom,
    ! [A: int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)) ) ) ).

% pos_add_strict
tff(fact_3482_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ~ ! [C: nat] :
            ( ( B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C) )
           => ( C = zero_zero_nat ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
tff(fact_3483_add__pos__pos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% add_pos_pos
tff(fact_3484_add__pos__pos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% add_pos_pos
tff(fact_3485_add__neg__neg,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),zero_zero_nat) ) ) ).

% add_neg_neg
tff(fact_3486_add__neg__neg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),zero_zero_int) ) ) ).

% add_neg_neg
tff(fact_3487_diff__le__eq,axiom,
    ! [A: int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),B)) ) ).

% diff_le_eq
tff(fact_3488_le__diff__eq,axiom,
    ! [A: int,C2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,C2),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C2) ) ).

% le_diff_eq
tff(fact_3489_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)),A) = B ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add
tff(fact_3490_le__add__diff,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C2)),A)) ) ).

% le_add_diff
tff(fact_3491_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),A)),B) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
tff(fact_3492_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),B)),A) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
tff(fact_3493_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),B)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
tff(fact_3494_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C2)),A) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
tff(fact_3495_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C2)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)),C2) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
tff(fact_3496_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),A)),B) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_diff_right
tff(fact_3497_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)) = B ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
tff(fact_3498_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A) = C2 )
        <=> ( B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),A) ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
tff(fact_3499_less__add__one,axiom,
    ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)) ).

% less_add_one
tff(fact_3500_less__add__one,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)) ).

% less_add_one
tff(fact_3501_add__mono1,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),one_one_nat)) ) ).

% add_mono1
tff(fact_3502_add__mono1,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),one_one_int)) ) ).

% add_mono1
tff(fact_3503_diff__less__eq,axiom,
    ! [A: int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),B)) ) ).

% diff_less_eq
tff(fact_3504_less__diff__eq,axiom,
    ! [A: int,C2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,C2),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C2) ) ).

% less_diff_eq
tff(fact_3505_square__diff__square__factored,axiom,
    ! [X: int,Y: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Y)),aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Y)) ).

% square_diff_square_factored
tff(fact_3506_eq__add__iff2,axiom,
    ! [A: int,E3: int,C2: int,B: int,D2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E3)),C2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E3)),D2) )
    <=> ( C2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A)),E3)),D2) ) ) ).

% eq_add_iff2
tff(fact_3507_eq__add__iff1,axiom,
    ! [A: int,E3: int,C2: int,B: int,D2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E3)),C2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E3)),D2) )
    <=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),E3)),C2) = D2 ) ) ).

% eq_add_iff1
tff(fact_3508_add__strict__increasing2,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2)) ) ) ).

% add_strict_increasing2
tff(fact_3509_add__strict__increasing2,axiom,
    ! [A: int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)) ) ) ).

% add_strict_increasing2
tff(fact_3510_add__strict__increasing,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2)) ) ) ).

% add_strict_increasing
tff(fact_3511_add__strict__increasing,axiom,
    ! [A: int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)) ) ) ).

% add_strict_increasing
tff(fact_3512_add__pos__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% add_pos_nonneg
tff(fact_3513_add__pos__nonneg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% add_pos_nonneg
tff(fact_3514_add__nonpos__neg,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),zero_zero_nat) ) ) ).

% add_nonpos_neg
tff(fact_3515_add__nonpos__neg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),zero_zero_int) ) ) ).

% add_nonpos_neg
tff(fact_3516_add__nonneg__pos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% add_nonneg_pos
tff(fact_3517_add__nonneg__pos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% add_nonneg_pos
tff(fact_3518_add__neg__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),zero_zero_nat) ) ) ).

% add_neg_nonpos
tff(fact_3519_add__neg__nonpos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),zero_zero_int) ) ) ).

% add_neg_nonpos
tff(fact_3520_sum__squares__ge__zero,axiom,
    ! [X: int,Y: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y))) ).

% sum_squares_ge_zero
tff(fact_3521_not__sum__squares__lt__zero,axiom,
    ! [X: int,Y: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y))),zero_zero_int) ).

% not_sum_squares_lt_zero
tff(fact_3522_zero__less__two,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat)) ).

% zero_less_two
tff(fact_3523_zero__less__two,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),one_one_int)) ).

% zero_less_two
tff(fact_3524_le__add__iff2,axiom,
    ! [A: int,E3: int,C2: int,B: int,D2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E3)),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E3)),D2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A)),E3)),D2)) ) ).

% le_add_iff2
tff(fact_3525_le__add__iff1,axiom,
    ! [A: int,E3: int,C2: int,B: int,D2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E3)),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E3)),D2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),E3)),C2)),D2) ) ).

% le_add_iff1
tff(fact_3526_less__add__iff1,axiom,
    ! [A: int,E3: int,C2: int,B: int,D2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E3)),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E3)),D2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),E3)),C2)),D2) ) ).

% less_add_iff1
tff(fact_3527_less__add__iff2,axiom,
    ! [A: int,E3: int,C2: int,B: int,D2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E3)),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E3)),D2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,C2),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A)),E3)),D2)) ) ).

% less_add_iff2
tff(fact_3528_square__diff__one__factored,axiom,
    ! [X: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),one_one_int) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),one_one_int)),aa_int_int(aa_int_fun_int_int(minus_minus_int,X),one_one_int)) ).

% square_diff_one_factored
tff(fact_3529_convex__bound__le,axiom,
    ! [X: int,A: int,Y: int,U: int,V: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),U)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),V)
           => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,U),V) = one_one_int )
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,U),X)),aa_int_int(aa_int_fun_int_int(times_times_int,V),Y))),A) ) ) ) ) ) ).

% convex_bound_le
tff(fact_3530_semilattice__set_Osingleton,axiom,
    ! [F: fun_o_fun_o_o,X: $o] :
      ( lattic7139874143898644262_set_o(F)
     => ( aa_set_o_o(lattic3100085485418696562ce_F_o(F),insert_o((X),bot_bot_set_o))
      <=> (X) ) ) ).

% semilattice_set.singleton
tff(fact_3531_semilattice__set_Osingleton,axiom,
    ! [F: fun_nat_fun_nat_nat,X: nat] :
      ( lattic1029310888574255042et_nat(F)
     => ( aa_set_nat_nat(lattic7742739596368939638_F_nat(F),insert_nat2(X,bot_bot_set_nat)) = X ) ) ).

% semilattice_set.singleton
tff(fact_3532_discrete,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),B) ) ).

% discrete
tff(fact_3533_discrete,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)),B) ) ).

% discrete
tff(fact_3534_sum__squares__gt__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y)))
    <=> ( ( X != zero_zero_int )
        | ( Y != zero_zero_int ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_3535_sum__squares__le__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y))),zero_zero_int)
    <=> ( ( X = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_3536_sum__squares__eq__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y)) = zero_zero_int )
    <=> ( ( X = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_3537_add__scale__eq__noteq,axiom,
    ! [R3: int,A: int,B: int,C2: int,D2: int] :
      ( ( R3 != zero_zero_int )
     => ( ( ( A = B )
          & ( C2 != D2 ) )
       => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,R3),C2)) != aa_int_int(aa_int_fun_int_int(plus_plus_int,B),aa_int_int(aa_int_fun_int_int(times_times_int,R3),D2)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_3538_add__scale__eq__noteq,axiom,
    ! [R3: nat,A: nat,B: nat,C2: nat,D2: nat] :
      ( ( R3 != zero_zero_nat )
     => ( ( ( A = B )
          & ( C2 != D2 ) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R3),C2)) != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R3),D2)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_3539_crossproduct__noteq,axiom,
    ! [A: int,B: int,C2: int,D2: int] :
      ( ( ( A != B )
        & ( C2 != D2 ) )
    <=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D2)) != aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),D2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)) ) ) ).

% crossproduct_noteq
tff(fact_3540_crossproduct__noteq,axiom,
    ! [A: nat,B: nat,C2: nat,D2: nat] :
      ( ( ( A != B )
        & ( C2 != D2 ) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D2)) != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),D2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C2)) ) ) ).

% crossproduct_noteq
tff(fact_3541_crossproduct__eq,axiom,
    ! [W: int,Y: int,X: int,Z: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,W),Y)),aa_int_int(aa_int_fun_int_int(times_times_int,X),Z)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,W),Z)),aa_int_int(aa_int_fun_int_int(times_times_int,X),Y)) )
    <=> ( ( W = X )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
tff(fact_3542_crossproduct__eq,axiom,
    ! [W: nat,Y: nat,X: nat,Z: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,W),Y)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,W),Z)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X),Y)) )
    <=> ( ( W = X )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
tff(fact_3543_dbl__dec__def,axiom,
    ! [X: int] : aa_int_int(neg_nu3811975205180677377ec_int,X) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),X)),one_one_int) ).

% dbl_dec_def
tff(fact_3544_slice__Cons,axiom,
    ! [Begin: nat,End: nat,X: nat,Xs: list_nat] :
      slice_nat(Begin,End,aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X),Xs)) = $ite(
        ( ( Begin = zero_zero_nat )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),End) ),
        aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X),slice_nat(Begin,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,End),one_one_nat),Xs)),
        slice_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Begin),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,End),one_one_nat),Xs) ) ).

% slice_Cons
tff(fact_3545_slice__Cons,axiom,
    ! [Begin: nat,End: nat,X: int,Xs: list_int] :
      slice_int(Begin,End,aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X),Xs)) = $ite(
        ( ( Begin = zero_zero_nat )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),End) ),
        aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X),slice_int(Begin,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,End),one_one_nat),Xs)),
        slice_int(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Begin),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,End),one_one_nat),Xs) ) ).

% slice_Cons
tff(fact_3546_slice__Cons,axiom,
    ! [Begin: nat,End: nat,X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] :
      slice_7964966981652229398n_assn(Begin,End,aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X),Xs)) = $ite(
        ( ( Begin = zero_zero_nat )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),End) ),
        aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X),slice_7964966981652229398n_assn(Begin,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,End),one_one_nat),Xs)),
        slice_7964966981652229398n_assn(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Begin),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,End),one_one_nat),Xs) ) ).

% slice_Cons
tff(fact_3547_slice__Nil,axiom,
    ! [Begin: nat,End: nat] : slice_a(Begin,End,nil_a) = nil_a ).

% slice_Nil
tff(fact_3548_slice__Nil,axiom,
    ! [Begin: nat,End: nat] : slice_b(Begin,End,nil_b) = nil_b ).

% slice_Nil
tff(fact_3549_slice__Nil,axiom,
    ! [Begin: nat,End: nat] : slice_7964966981652229398n_assn(Begin,End,nil_Pr5671120429643327159n_assn) = nil_Pr5671120429643327159n_assn ).

% slice_Nil
tff(fact_3550_slice__Nil,axiom,
    ! [Begin: nat,End: nat] : slice_nat(Begin,End,nil_nat) = nil_nat ).

% slice_Nil
tff(fact_3551_slice__Nil,axiom,
    ! [Begin: nat,End: nat] : slice_int(Begin,End,nil_int) = nil_int ).

% slice_Nil
tff(fact_3552_slice__eq__bounds__empty,axiom,
    ! [I: nat,Xs: list_a] : slice_a(I,I,Xs) = nil_a ).

% slice_eq_bounds_empty
tff(fact_3553_slice__eq__bounds__empty,axiom,
    ! [I: nat,Xs: list_b] : slice_b(I,I,Xs) = nil_b ).

% slice_eq_bounds_empty
tff(fact_3554_slice__eq__bounds__empty,axiom,
    ! [I: nat,Xs: list_P8527749157015355191n_assn] : slice_7964966981652229398n_assn(I,I,Xs) = nil_Pr5671120429643327159n_assn ).

% slice_eq_bounds_empty
tff(fact_3555_slice__eq__bounds__empty,axiom,
    ! [I: nat,Xs: list_nat] : slice_nat(I,I,Xs) = nil_nat ).

% slice_eq_bounds_empty
tff(fact_3556_slice__eq__bounds__empty,axiom,
    ! [I: nat,Xs: list_int] : slice_int(I,I,Xs) = nil_int ).

% slice_eq_bounds_empty
tff(fact_3557_dbl__dec__simps_I3_J,axiom,
    aa_int_int(neg_nu3811975205180677377ec_int,one_one_int) = one_one_int ).

% dbl_dec_simps(3)
tff(fact_3558_mlex__snd__decrI,axiom,
    ! [A: nat,A8: nat,B: nat,B7: nat,N4: nat] :
      ( ( A = A8 )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),B7)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),N4)),B)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A8),N4)),B7)) ) ) ).

% mlex_snd_decrI
tff(fact_3559_mlex__fst__decrI,axiom,
    ! [A: nat,A8: nat,B: nat,N4: nat,B7: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),A8)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),N4)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B7),N4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),N4)),B)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A8),N4)),B7)) ) ) ) ).

% mlex_fst_decrI
tff(fact_3560_mlex__bound,axiom,
    ! [A: nat,A2: nat,B: nat,N4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),N4)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),N4)),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),N4)) ) ) ).

% mlex_bound
tff(fact_3561_count__list_Osimps_I2_J,axiom,
    ! [X: nat,Xs: list_nat,Y: nat] :
      aa_nat_nat(count_list_nat(aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X),Xs)),Y) = $ite(X = Y,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(count_list_nat(Xs),Y)),one_one_nat),aa_nat_nat(count_list_nat(Xs),Y)) ).

% count_list.simps(2)
tff(fact_3562_count__list_Osimps_I2_J,axiom,
    ! [X: int,Xs: list_int,Y: int] :
      aa_int_nat(count_list_int(aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X),Xs)),Y) = $ite(X = Y,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(count_list_int(Xs),Y)),one_one_nat),aa_int_nat(count_list_int(Xs),Y)) ).

% count_list.simps(2)
tff(fact_3563_count__list_Osimps_I2_J,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn,Y: produc6575502325842934193n_assn] :
      aa_Pro8055486230010607173sn_nat(count_2530312006313534765n_assn(aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X),Xs)),Y) = $ite(X = Y,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_Pro8055486230010607173sn_nat(count_2530312006313534765n_assn(Xs),Y)),one_one_nat),aa_Pro8055486230010607173sn_nat(count_2530312006313534765n_assn(Xs),Y)) ).

% count_list.simps(2)
tff(fact_3564_power__decreasing__iff,axiom,
    ! [B: nat,M2: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),one_one_nat)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),M2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M2) ) ) ) ).

% power_decreasing_iff
tff(fact_3565_power__decreasing__iff,axiom,
    ! [B: int,M2: nat,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),one_one_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(B),M2)),aa_nat_int(power_power_int(B),N))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M2) ) ) ) ).

% power_decreasing_iff
tff(fact_3566_nth__Cons__pos,axiom,
    ! [N: nat,X: nat,Xs: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_nat_nat(nth_nat(aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X),Xs)),N) = aa_nat_nat(nth_nat(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) ) ) ).

% nth_Cons_pos
tff(fact_3567_nth__Cons__pos,axiom,
    ! [N: nat,X: int,Xs: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_nat_int(nth_int(aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X),Xs)),N) = aa_nat_int(nth_int(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) ) ) ).

% nth_Cons_pos
tff(fact_3568_nth__Cons__pos,axiom,
    ! [N: nat,X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_nat3152888703114873891n_assn(nth_Pr1769885009046257848n_assn(aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X),Xs)),N) = aa_nat3152888703114873891n_assn(nth_Pr1769885009046257848n_assn(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) ) ) ).

% nth_Cons_pos
tff(fact_3569_power__minus__mult,axiom,
    ! [N: nat,A: assn] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(power_power_assn(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),A) = aa_nat_assn(power_power_assn(A),N) ) ) ).

% power_minus_mult
tff(fact_3570_power__minus__mult,axiom,
    ! [N: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),A) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N) ) ) ).

% power_minus_mult
tff(fact_3571_power__eq__if,axiom,
    ! [P3: int,M2: nat] :
      aa_nat_int(power_power_int(P3),M2) = $ite(M2 = zero_zero_nat,one_one_int,aa_int_int(aa_int_fun_int_int(times_times_int,P3),aa_nat_int(power_power_int(P3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),one_one_nat)))) ).

% power_eq_if
tff(fact_3572_power__eq__if,axiom,
    ! [P3: assn,M2: nat] :
      aa_nat_assn(power_power_assn(P3),M2) = $ite(M2 = zero_zero_nat,one_one_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P3),aa_nat_assn(power_power_assn(P3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),one_one_nat)))) ).

% power_eq_if
tff(fact_3573_power__eq__if,axiom,
    ! [P3: nat,M2: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,P3),M2) = $ite(M2 = zero_zero_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,P3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,P3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),one_one_nat)))) ).

% power_eq_if
tff(fact_3574_card__Diff__singleton,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,A2: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),A2)
     => ( finite5795495896937551127_nat_o(minus_6456595905716665654_nat_o(A2,insert4923026906511862463_nat_o(X,bot_bo8632255151191677115_nat_o))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite5795495896937551127_nat_o(A2)),one_one_nat) ) ) ).

% card_Diff_singleton
tff(fact_3575_card__Diff__singleton,axiom,
    ! [X: $o,A2: set_o] :
      ( aa_set_o_o(member_o((X)),A2)
     => ( finite_card_o(minus_minus_set_o(A2,insert_o((X),bot_bot_set_o))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_o(A2)),one_one_nat) ) ) ).

% card_Diff_singleton
tff(fact_3576_card__Diff__singleton,axiom,
    ! [X: nat,A2: set_nat] :
      ( aa_set_nat_o(member_nat2(X),A2)
     => ( finite_card_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),insert_nat2(X,bot_bot_set_nat))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_nat(A2)),one_one_nat) ) ) ).

% card_Diff_singleton
tff(fact_3577_card__Diff__singleton__if,axiom,
    ! [A2: set_fu3555266933746557007_nat_o,X: fun_Pr6460944699146269935_nat_o] :
      finite5795495896937551127_nat_o(minus_6456595905716665654_nat_o(A2,insert4923026906511862463_nat_o(X,bot_bo8632255151191677115_nat_o))) = $ite(aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite5795495896937551127_nat_o(A2)),one_one_nat),finite5795495896937551127_nat_o(A2)) ).

% card_Diff_singleton_if
tff(fact_3578_card__Diff__singleton__if,axiom,
    ! [A2: set_o,X: $o] :
      finite_card_o(minus_minus_set_o(A2,insert_o((X),bot_bot_set_o))) = $ite(aa_set_o_o(member_o((X)),A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_o(A2)),one_one_nat),finite_card_o(A2)) ).

% card_Diff_singleton_if
tff(fact_3579_card__Diff__singleton__if,axiom,
    ! [A2: set_nat,X: nat] :
      finite_card_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),insert_nat2(X,bot_bot_set_nat))) = $ite(aa_set_nat_o(member_nat2(X),A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_nat(A2)),one_one_nat),finite_card_nat(A2)) ).

% card_Diff_singleton_if
tff(fact_3580_power__one,axiom,
    ! [N: nat] : aa_nat_assn(power_power_assn(one_one_assn),N) = one_one_assn ).

% power_one
tff(fact_3581_power__one,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,one_one_nat),N) = one_one_nat ).

% power_one
tff(fact_3582_power__one,axiom,
    ! [N: nat] : aa_nat_int(power_power_int(one_one_int),N) = one_one_int ).

% power_one
tff(fact_3583_power__inject__exp,axiom,
    ! [A: nat,M2: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N) )
      <=> ( M2 = N ) ) ) ).

% power_inject_exp
tff(fact_3584_power__inject__exp,axiom,
    ! [A: int,M2: nat,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => ( ( aa_nat_int(power_power_int(A),M2) = aa_nat_int(power_power_int(A),N) )
      <=> ( M2 = N ) ) ) ).

% power_inject_exp
tff(fact_3585_card_Oempty,axiom,
    finite_card_o(bot_bot_set_o) = zero_zero_nat ).

% card.empty
tff(fact_3586_card_Oempty,axiom,
    finite_card_nat(bot_bot_set_nat) = zero_zero_nat ).

% card.empty
tff(fact_3587_nth__Cons__0,axiom,
    ! [X: nat,Xs: list_nat] : aa_nat_nat(nth_nat(aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X),Xs)),zero_zero_nat) = X ).

% nth_Cons_0
tff(fact_3588_nth__Cons__0,axiom,
    ! [X: int,Xs: list_int] : aa_nat_int(nth_int(aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X),Xs)),zero_zero_nat) = X ).

% nth_Cons_0
tff(fact_3589_nth__Cons__0,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] : aa_nat3152888703114873891n_assn(nth_Pr1769885009046257848n_assn(aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X),Xs)),zero_zero_nat) = X ).

% nth_Cons_0
tff(fact_3590_gbinomial__0_I1_J,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gbinomial_nat,A),zero_zero_nat) = one_one_nat ).

% gbinomial_0(1)
tff(fact_3591_gbinomial__0_I1_J,axiom,
    ! [A: int] : aa_nat_int(gbinomial_int(A),zero_zero_nat) = one_one_int ).

% gbinomial_0(1)
tff(fact_3592_power__strict__increasing__iff,axiom,
    ! [B: nat,X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),X)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y) ) ) ).

% power_strict_increasing_iff
tff(fact_3593_power__strict__increasing__iff,axiom,
    ! [B: int,X: nat,Y: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(B),X)),aa_nat_int(power_power_int(B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y) ) ) ).

% power_strict_increasing_iff
tff(fact_3594_card__0__eq,axiom,
    ! [A2: set_o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ( finite_card_o(A2) = zero_zero_nat )
      <=> ( A2 = bot_bot_set_o ) ) ) ).

% card_0_eq
tff(fact_3595_card__0__eq,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( finite_card_nat(A2) = zero_zero_nat )
      <=> ( A2 = bot_bot_set_nat ) ) ) ).

% card_0_eq
tff(fact_3596_power__strict__decreasing__iff,axiom,
    ! [B: nat,M2: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),one_one_nat)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),M2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M2) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_3597_power__strict__decreasing__iff,axiom,
    ! [B: int,M2: nat,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),one_one_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(B),M2)),aa_nat_int(power_power_int(B),N))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M2) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_3598_power__increasing__iff,axiom,
    ! [B: nat,X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),X)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y) ) ) ).

% power_increasing_iff
tff(fact_3599_power__increasing__iff,axiom,
    ! [B: int,X: nat,Y: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(B),X)),aa_nat_int(power_power_int(B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y) ) ) ).

% power_increasing_iff
tff(fact_3600_power__commuting__commutes,axiom,
    ! [X: assn,Y: assn,N: nat] :
      ( ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,X),Y) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Y),X) )
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(power_power_assn(X),N)),Y) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Y),aa_nat_assn(power_power_assn(X),N)) ) ) ).

% power_commuting_commutes
tff(fact_3601_power__commuting__commutes,axiom,
    ! [X: nat,Y: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X),Y) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),X) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),N)),Y) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),N)) ) ) ).

% power_commuting_commutes
tff(fact_3602_power__mult__distrib,axiom,
    ! [A: assn,B: assn,N: nat] : aa_nat_assn(power_power_assn(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),B)),N) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(power_power_assn(A),N)),aa_nat_assn(power_power_assn(B),N)) ).

% power_mult_distrib
tff(fact_3603_power__mult__distrib,axiom,
    ! [A: nat,B: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),N) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N)) ).

% power_mult_distrib
tff(fact_3604_power__commutes,axiom,
    ! [A: assn,N: nat] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(power_power_assn(A),N)),A) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),aa_nat_assn(power_power_assn(A),N)) ).

% power_commutes
tff(fact_3605_power__commutes,axiom,
    ! [A: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)) ).

% power_commutes
tff(fact_3606_map__eq__nth__eq,axiom,
    ! [F: fun_nat_nat,La: list_nat,L_a: list_nat,I: nat] :
      ( ( aa_list_nat_list_nat(aa_fun4736978287202700568st_nat(map_nat_nat,F),La) = aa_list_nat_list_nat(aa_fun4736978287202700568st_nat(map_nat_nat,F),L_a) )
     => ( aa_nat_nat(F,aa_nat_nat(nth_nat(La),I)) = aa_nat_nat(F,aa_nat_nat(nth_nat(L_a),I)) ) ) ).

% map_eq_nth_eq
tff(fact_3607_map__eq__nth__eq,axiom,
    ! [F: fun_Pr3816910165196019820n_assn,La: list_P8527749157015355191n_assn,L_a: list_P8527749157015355191n_assn,I: nat] :
      ( ( aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,F),La) = aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,F),L_a) )
     => ( aa_Pro2979818503705714077n_assn(F,aa_nat3152888703114873891n_assn(nth_Pr1769885009046257848n_assn(La),I)) = aa_Pro2979818503705714077n_assn(F,aa_nat3152888703114873891n_assn(nth_Pr1769885009046257848n_assn(L_a),I)) ) ) ).

% map_eq_nth_eq
tff(fact_3608_one__le__power,axiom,
    ! [A: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)) ) ).

% one_le_power
tff(fact_3609_one__le__power,axiom,
    ! [A: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_nat_int(power_power_int(A),N)) ) ).

% one_le_power
tff(fact_3610_left__right__inverse__power,axiom,
    ! [X: int,Y: int,N: nat] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,X),Y) = one_one_int )
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(X),N)),aa_nat_int(power_power_int(Y),N)) = one_one_int ) ) ).

% left_right_inverse_power
tff(fact_3611_left__right__inverse__power,axiom,
    ! [X: assn,Y: assn,N: nat] :
      ( ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,X),Y) = one_one_assn )
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(power_power_assn(X),N)),aa_nat_assn(power_power_assn(Y),N)) = one_one_assn ) ) ).

% left_right_inverse_power
tff(fact_3612_left__right__inverse__power,axiom,
    ! [X: nat,Y: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X),Y) = one_one_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),N)) = one_one_nat ) ) ).

% left_right_inverse_power
tff(fact_3613_power__0,axiom,
    ! [A: assn] : aa_nat_assn(power_power_assn(A),zero_zero_nat) = one_one_assn ).

% power_0
tff(fact_3614_power__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),zero_zero_nat) = one_one_nat ).

% power_0
tff(fact_3615_power__0,axiom,
    ! [A: int] : aa_nat_int(power_power_int(A),zero_zero_nat) = one_one_int ).

% power_0
tff(fact_3616_power__add,axiom,
    ! [A: assn,M2: nat,N: nat] : aa_nat_assn(power_power_assn(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(power_power_assn(A),M2)),aa_nat_assn(power_power_assn(A),N)) ).

% power_add
tff(fact_3617_power__add,axiom,
    ! [A: nat,M2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)) ).

% power_add
tff(fact_3618_card__eq__0__iff,axiom,
    ! [A2: set_o] :
      ( ( finite_card_o(A2) = zero_zero_nat )
    <=> ( ( A2 = bot_bot_set_o )
        | ~ aa_set_o_o(finite_finite_o,A2) ) ) ).

% card_eq_0_iff
tff(fact_3619_card__eq__0__iff,axiom,
    ! [A2: set_nat] :
      ( ( finite_card_nat(A2) = zero_zero_nat )
    <=> ( ( A2 = bot_bot_set_nat )
        | ~ aa_set_nat_o(finite_finite_nat,A2) ) ) ).

% card_eq_0_iff
tff(fact_3620_power__le__one,axiom,
    ! [A: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),one_one_nat) ) ) ).

% power_le_one
tff(fact_3621_power__le__one,axiom,
    ! [A: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),N)),one_one_int) ) ) ).

% power_le_one
tff(fact_3622_power__less__power__Suc,axiom,
    ! [A: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))) ) ).

% power_less_power_Suc
tff(fact_3623_power__less__power__Suc,axiom,
    ! [A: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),N)),aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(A),N))) ) ).

% power_less_power_Suc
tff(fact_3624_power__gt1__lemma,axiom,
    ! [A: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))) ) ).

% power_gt1_lemma
tff(fact_3625_power__gt1__lemma,axiom,
    ! [A: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(A),N))) ) ).

% power_gt1_lemma
tff(fact_3626_power__0__left,axiom,
    ! [N: nat] :
      aa_nat_int(power_power_int(zero_zero_int),N) = $ite(N = zero_zero_nat,one_one_int,zero_zero_int) ).

% power_0_left
tff(fact_3627_power__0__left,axiom,
    ! [N: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),N) = $ite(N = zero_zero_nat,one_one_nat,zero_zero_nat) ).

% power_0_left
tff(fact_3628_card__1__singletonE,axiom,
    ! [A2: set_o] :
      ( ( finite_card_o(A2) = one_one_nat )
     => ~ ! [X2: $o] : A2 != insert_o((X2),bot_bot_set_o) ) ).

% card_1_singletonE
tff(fact_3629_card__1__singletonE,axiom,
    ! [A2: set_nat] :
      ( ( finite_card_nat(A2) = one_one_nat )
     => ~ ! [X2: nat] : A2 != insert_nat2(X2,bot_bot_set_nat) ) ).

% card_1_singletonE
tff(fact_3630_power__less__imp__less__exp,axiom,
    ! [A: nat,M2: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),N) ) ) ).

% power_less_imp_less_exp
tff(fact_3631_power__less__imp__less__exp,axiom,
    ! [A: int,M2: nat,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),M2)),aa_nat_int(power_power_int(A),N))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),N) ) ) ).

% power_less_imp_less_exp
tff(fact_3632_power__strict__increasing,axiom,
    ! [N: nat,N4: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),N4)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N4)) ) ) ).

% power_strict_increasing
tff(fact_3633_power__strict__increasing,axiom,
    ! [N: nat,N4: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),N4)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),N)),aa_nat_int(power_power_int(A),N4)) ) ) ).

% power_strict_increasing
tff(fact_3634_power__increasing,axiom,
    ! [N: nat,N4: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),N4)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N4)) ) ) ).

% power_increasing
tff(fact_3635_power__increasing,axiom,
    ! [N: nat,N4: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),N4)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),N)),aa_nat_int(power_power_int(A),N4)) ) ) ).

% power_increasing
tff(fact_3636_card__gt__0__iff,axiom,
    ! [A2: set_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),finite_card_o(A2))
    <=> ( ( A2 != bot_bot_set_o )
        & aa_set_o_o(finite_finite_o,A2) ) ) ).

% card_gt_0_iff
tff(fact_3637_card__gt__0__iff,axiom,
    ! [A2: set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),finite_card_nat(A2))
    <=> ( ( A2 != bot_bot_set_nat )
        & aa_set_nat_o(finite_finite_nat,A2) ) ) ).

% card_gt_0_iff
tff(fact_3638_power__Suc__less,axiom,
    ! [A: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)) ) ) ).

% power_Suc_less
tff(fact_3639_power__Suc__less,axiom,
    ! [A: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),one_one_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(A),N))),aa_nat_int(power_power_int(A),N)) ) ) ).

% power_Suc_less
tff(fact_3640_card__1__singletonI,axiom,
    ! [S: set_fu3555266933746557007_nat_o,X: fun_Pr6460944699146269935_nat_o] :
      ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,S)
     => ( ( finite5795495896937551127_nat_o(S) = one_one_nat )
       => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),S)
         => ( S = insert4923026906511862463_nat_o(X,bot_bo8632255151191677115_nat_o) ) ) ) ) ).

% card_1_singletonI
tff(fact_3641_card__1__singletonI,axiom,
    ! [S: set_o,X: $o] :
      ( aa_set_o_o(finite_finite_o,S)
     => ( ( finite_card_o(S) = one_one_nat )
       => ( aa_set_o_o(member_o((X)),S)
         => ( S = insert_o((X),bot_bot_set_o) ) ) ) ) ).

% card_1_singletonI
tff(fact_3642_card__1__singletonI,axiom,
    ! [S: set_nat,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( ( finite_card_nat(S) = one_one_nat )
       => ( aa_set_nat_o(member_nat2(X),S)
         => ( S = insert_nat2(X,bot_bot_set_nat) ) ) ) ) ).

% card_1_singletonI
tff(fact_3643_power__strict__decreasing,axiom,
    ! [N: nat,N4: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),N4)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),one_one_nat)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N4)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)) ) ) ) ).

% power_strict_decreasing
tff(fact_3644_power__strict__decreasing,axiom,
    ! [N: nat,N4: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),N4)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),one_one_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),N4)),aa_nat_int(power_power_int(A),N)) ) ) ) ).

% power_strict_decreasing
tff(fact_3645_card__Diff1__le,axiom,
    ! [A2: set_o,X: $o] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_o(minus_minus_set_o(A2,insert_o((X),bot_bot_set_o)))),finite_card_o(A2)) ).

% card_Diff1_le
tff(fact_3646_card__Diff1__le,axiom,
    ! [A2: set_nat,X: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),insert_nat2(X,bot_bot_set_nat)))),finite_card_nat(A2)) ).

% card_Diff1_le
tff(fact_3647_power__decreasing,axiom,
    ! [N: nat,N4: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),N4)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),one_one_nat)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N4)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)) ) ) ) ).

% power_decreasing
tff(fact_3648_power__decreasing,axiom,
    ! [N: nat,N4: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),N4)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),N4)),aa_nat_int(power_power_int(A),N)) ) ) ) ).

% power_decreasing
tff(fact_3649_power__le__imp__le__exp,axiom,
    ! [A: nat,M2: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N) ) ) ).

% power_le_imp_le_exp
tff(fact_3650_power__le__imp__le__exp,axiom,
    ! [A: int,M2: nat,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),M2)),aa_nat_int(power_power_int(A),N))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N) ) ) ).

% power_le_imp_le_exp
tff(fact_3651_nth__Cons_H,axiom,
    ! [X: nat,Xs: list_nat,N: nat] :
      aa_nat_nat(nth_nat(aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X),Xs)),N) = $ite(N = zero_zero_nat,X,aa_nat_nat(nth_nat(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ).

% nth_Cons'
tff(fact_3652_nth__Cons_H,axiom,
    ! [X: int,Xs: list_int,N: nat] :
      aa_nat_int(nth_int(aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X),Xs)),N) = $ite(N = zero_zero_nat,X,aa_nat_int(nth_int(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ).

% nth_Cons'
tff(fact_3653_nth__Cons_H,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn,N: nat] :
      aa_nat3152888703114873891n_assn(nth_Pr1769885009046257848n_assn(aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X),Xs)),N) = $ite(N = zero_zero_nat,X,aa_nat3152888703114873891n_assn(nth_Pr1769885009046257848n_assn(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ).

% nth_Cons'
tff(fact_3654_self__le__power,axiom,
    ! [A: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)) ) ) ).

% self_le_power
tff(fact_3655_self__le__power,axiom,
    ! [A: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(power_power_int(A),N)) ) ) ).

% self_le_power
tff(fact_3656_card__Un__Int,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( finite6177210948735845034at_nat(A2)
     => ( finite6177210948735845034at_nat(B2)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite711546835091564841at_nat(A2)),finite711546835091564841at_nat(B2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite711546835091564841at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A2),B2))),finite711546835091564841at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2))) ) ) ) ).

% card_Un_Int
tff(fact_3657_card__Un__Int,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(finite_finite_nat,B2)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite_card_nat(A2)),finite_card_nat(B2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite_card_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2))),finite_card_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2))) ) ) ) ).

% card_Un_Int
tff(fact_3658_one__less__power,axiom,
    ! [A: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)) ) ) ).

% one_less_power
tff(fact_3659_one__less__power,axiom,
    ! [A: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_nat_int(power_power_int(A),N)) ) ) ).

% one_less_power
tff(fact_3660_card__Diff__subset__Int,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2))
     => ( finite_card_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_nat(A2)),finite_card_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2))) ) ) ).

% card_Diff_subset_Int
tff(fact_3661_card__Diff__subset__Int,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( finite6177210948735845034at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2))
     => ( finite711546835091564841at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite711546835091564841at_nat(A2)),finite711546835091564841at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2))) ) ) ).

% card_Diff_subset_Int
tff(fact_3662_count__list_Osimps_I1_J,axiom,
    ! [Y: a] : aa_a_nat(count_list_a(nil_a),Y) = zero_zero_nat ).

% count_list.simps(1)
tff(fact_3663_count__list_Osimps_I1_J,axiom,
    ! [Y: b] : aa_b_nat(count_list_b(nil_b),Y) = zero_zero_nat ).

% count_list.simps(1)
tff(fact_3664_count__list_Osimps_I1_J,axiom,
    ! [Y: produc6575502325842934193n_assn] : aa_Pro8055486230010607173sn_nat(count_2530312006313534765n_assn(nil_Pr5671120429643327159n_assn),Y) = zero_zero_nat ).

% count_list.simps(1)
tff(fact_3665_count__list_Osimps_I1_J,axiom,
    ! [Y: nat] : aa_nat_nat(count_list_nat(nil_nat),Y) = zero_zero_nat ).

% count_list.simps(1)
tff(fact_3666_count__list_Osimps_I1_J,axiom,
    ! [Y: int] : aa_int_nat(count_list_int(nil_int),Y) = zero_zero_nat ).

% count_list.simps(1)
tff(fact_3667_card__Diff1__less__iff,axiom,
    ! [A2: set_fu3555266933746557007_nat_o,X: fun_Pr6460944699146269935_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite5795495896937551127_nat_o(minus_6456595905716665654_nat_o(A2,insert4923026906511862463_nat_o(X,bot_bo8632255151191677115_nat_o)))),finite5795495896937551127_nat_o(A2))
    <=> ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,A2)
        & aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),A2) ) ) ).

% card_Diff1_less_iff
tff(fact_3668_card__Diff1__less__iff,axiom,
    ! [A2: set_o,X: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_o(minus_minus_set_o(A2,insert_o((X),bot_bot_set_o)))),finite_card_o(A2))
    <=> ( aa_set_o_o(finite_finite_o,A2)
        & aa_set_o_o(member_o((X)),A2) ) ) ).

% card_Diff1_less_iff
tff(fact_3669_card__Diff1__less__iff,axiom,
    ! [A2: set_nat,X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),insert_nat2(X,bot_bot_set_nat)))),finite_card_nat(A2))
    <=> ( aa_set_nat_o(finite_finite_nat,A2)
        & aa_set_nat_o(member_nat2(X),A2) ) ) ).

% card_Diff1_less_iff
tff(fact_3670_card__Diff2__less,axiom,
    ! [A2: set_fu3555266933746557007_nat_o,X: fun_Pr6460944699146269935_nat_o,Y: fun_Pr6460944699146269935_nat_o] :
      ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,A2)
     => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),A2)
       => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Y),A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite5795495896937551127_nat_o(minus_6456595905716665654_nat_o(minus_6456595905716665654_nat_o(A2,insert4923026906511862463_nat_o(X,bot_bo8632255151191677115_nat_o)),insert4923026906511862463_nat_o(Y,bot_bo8632255151191677115_nat_o)))),finite5795495896937551127_nat_o(A2)) ) ) ) ).

% card_Diff2_less
tff(fact_3671_card__Diff2__less,axiom,
    ! [A2: set_o,X: $o,Y: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( aa_set_o_o(member_o((X)),A2)
       => ( aa_set_o_o(member_o((Y)),A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_o(minus_minus_set_o(minus_minus_set_o(A2,insert_o((X),bot_bot_set_o)),insert_o((Y),bot_bot_set_o)))),finite_card_o(A2)) ) ) ) ).

% card_Diff2_less
tff(fact_3672_card__Diff2__less,axiom,
    ! [A2: set_nat,X: nat,Y: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(member_nat2(X),A2)
       => ( aa_set_nat_o(member_nat2(Y),A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),insert_nat2(X,bot_bot_set_nat))),insert_nat2(Y,bot_bot_set_nat)))),finite_card_nat(A2)) ) ) ) ).

% card_Diff2_less
tff(fact_3673_card__Diff1__less,axiom,
    ! [A2: set_fu3555266933746557007_nat_o,X: fun_Pr6460944699146269935_nat_o] :
      ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,A2)
     => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),A2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite5795495896937551127_nat_o(minus_6456595905716665654_nat_o(A2,insert4923026906511862463_nat_o(X,bot_bo8632255151191677115_nat_o)))),finite5795495896937551127_nat_o(A2)) ) ) ).

% card_Diff1_less
tff(fact_3674_card__Diff1__less,axiom,
    ! [A2: set_o,X: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( aa_set_o_o(member_o((X)),A2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_o(minus_minus_set_o(A2,insert_o((X),bot_bot_set_o)))),finite_card_o(A2)) ) ) ).

% card_Diff1_less
tff(fact_3675_card__Diff1__less,axiom,
    ! [A2: set_nat,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(member_nat2(X),A2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),insert_nat2(X,bot_bot_set_nat)))),finite_card_nat(A2)) ) ) ).

% card_Diff1_less
tff(fact_3676_card__Un__disjoint,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( finite6177210948735845034at_nat(A2)
     => ( finite6177210948735845034at_nat(B2)
       => ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2) = bot_bo2099793752762293965at_nat )
         => ( finite711546835091564841at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A2),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite711546835091564841at_nat(A2)),finite711546835091564841at_nat(B2)) ) ) ) ) ).

% card_Un_disjoint
tff(fact_3677_card__Un__disjoint,axiom,
    ! [A2: set_o,B2: set_o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( aa_set_o_o(finite_finite_o,B2)
       => ( ( inf_inf_set_o(A2,B2) = bot_bot_set_o )
         => ( finite_card_o(sup_sup_set_o(A2,B2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite_card_o(A2)),finite_card_o(B2)) ) ) ) ) ).

% card_Un_disjoint
tff(fact_3678_card__Un__disjoint,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(finite_finite_nat,B2)
       => ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) = bot_bot_set_nat )
         => ( finite_card_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite_card_nat(A2)),finite_card_nat(B2)) ) ) ) ) ).

% card_Un_disjoint
tff(fact_3679_nth__non__equal__first__eq,axiom,
    ! [X: nat,Y: nat,Xs: list_nat,N: nat] :
      ( ( X != Y )
     => ( ( aa_nat_nat(nth_nat(aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X),Xs)),N) = Y )
      <=> ( ( aa_nat_nat(nth_nat(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) = Y )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) ) ) ) ).

% nth_non_equal_first_eq
tff(fact_3680_nth__non__equal__first__eq,axiom,
    ! [X: int,Y: int,Xs: list_int,N: nat] :
      ( ( X != Y )
     => ( ( aa_nat_int(nth_int(aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X),Xs)),N) = Y )
      <=> ( ( aa_nat_int(nth_int(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) = Y )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) ) ) ) ).

% nth_non_equal_first_eq
tff(fact_3681_nth__non__equal__first__eq,axiom,
    ! [X: produc6575502325842934193n_assn,Y: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn,N: nat] :
      ( ( X != Y )
     => ( ( aa_nat3152888703114873891n_assn(nth_Pr1769885009046257848n_assn(aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X),Xs)),N) = Y )
      <=> ( ( aa_nat3152888703114873891n_assn(nth_Pr1769885009046257848n_assn(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) = Y )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) ) ) ) ).

% nth_non_equal_first_eq
tff(fact_3682_foldr__max__sorted,axiom,
    ! [Xs: list_nat,Y: nat] :
      ( aa_list_nat_o(sorted_wrt_nat(ord_less_eq_nat),rev_nat(Xs))
     => ( aa_nat_nat(foldr_nat_nat(ord_max_nat,Xs),Y) = $ite(Xs = nil_nat,Y,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(nth_nat(Xs),zero_zero_nat)),Y)) ) ) ).

% foldr_max_sorted
tff(fact_3683_foldr__max__sorted,axiom,
    ! [Xs: list_int,Y: int] :
      ( aa_list_int_o(sorted_wrt_int(ord_less_eq_int),rev_int(Xs))
     => ( aa_int_int(foldr_int_int(ord_max_int,Xs),Y) = $ite(Xs = nil_int,Y,aa_int_int(aa_int_fun_int_int(ord_max_int,aa_nat_int(nth_int(Xs),zero_zero_nat)),Y)) ) ) ).

% foldr_max_sorted
tff(fact_3684_slice__nth,axiom,
    ! [From: nat,To: nat,Xs: list_nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,From),To)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,To),size_size_list_nat(Xs))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,To),From))
         => ( aa_nat_nat(nth_nat(slice_nat(From,To,Xs)),I) = aa_nat_nat(nth_nat(Xs),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,From),I)) ) ) ) ) ).

% slice_nth
tff(fact_3685_slice__nth,axiom,
    ! [From: nat,To: nat,Xs: list_int,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,From),To)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,To),size_size_list_int(Xs))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,To),From))
         => ( aa_nat_int(nth_int(slice_int(From,To,Xs)),I) = aa_nat_int(nth_int(Xs),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,From),I)) ) ) ) ) ).

% slice_nth
tff(fact_3686_last__conv__nth,axiom,
    ! [Xs: list_a] :
      ( ( Xs != nil_a )
     => ( last_a(Xs) = aa_nat_a(nth_a(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,size_size_list_a(Xs)),one_one_nat)) ) ) ).

% last_conv_nth
tff(fact_3687_last__conv__nth,axiom,
    ! [Xs: list_b] :
      ( ( Xs != nil_b )
     => ( last_b(Xs) = aa_nat_b(nth_b(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,size_size_list_b(Xs)),one_one_nat)) ) ) ).

% last_conv_nth
tff(fact_3688_last__conv__nth,axiom,
    ! [Xs: list_P8527749157015355191n_assn] :
      ( ( Xs != nil_Pr5671120429643327159n_assn )
     => ( last_P8723976779861936080n_assn(Xs) = aa_nat3152888703114873891n_assn(nth_Pr1769885009046257848n_assn(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,size_s6829681357464350627n_assn(Xs)),one_one_nat)) ) ) ).

% last_conv_nth
tff(fact_3689_last__conv__nth,axiom,
    ! [Xs: list_nat] :
      ( ( Xs != nil_nat )
     => ( last_nat(Xs) = aa_nat_nat(nth_nat(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,size_size_list_nat(Xs)),one_one_nat)) ) ) ).

% last_conv_nth
tff(fact_3690_last__conv__nth,axiom,
    ! [Xs: list_int] :
      ( ( Xs != nil_int )
     => ( last_int(Xs) = aa_nat_int(nth_int(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,size_size_list_int(Xs)),one_one_nat)) ) ) ).

% last_conv_nth
tff(fact_3691_nth__Cons__numeral,axiom,
    ! [X: nat,Xs: list_nat,V: num] : aa_nat_nat(nth_nat(aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X),Xs)),numeral_numeral_nat(V)) = aa_nat_nat(nth_nat(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,numeral_numeral_nat(V)),one_one_nat)) ).

% nth_Cons_numeral
tff(fact_3692_nth__Cons__numeral,axiom,
    ! [X: int,Xs: list_int,V: num] : aa_nat_int(nth_int(aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X),Xs)),numeral_numeral_nat(V)) = aa_nat_int(nth_int(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,numeral_numeral_nat(V)),one_one_nat)) ).

% nth_Cons_numeral
tff(fact_3693_nth__Cons__numeral,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn,V: num] : aa_nat3152888703114873891n_assn(nth_Pr1769885009046257848n_assn(aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X),Xs)),numeral_numeral_nat(V)) = aa_nat3152888703114873891n_assn(nth_Pr1769885009046257848n_assn(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,numeral_numeral_nat(V)),one_one_nat)) ).

% nth_Cons_numeral
tff(fact_3694_card__insert__disjoint_H,axiom,
    ! [A2: set_o,X: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ~ aa_set_o_o(member_o((X)),A2)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_o(insert_o((X),A2))),aa_nat_nat(suc,zero_zero_nat)) = finite_card_o(A2) ) ) ) ).

% card_insert_disjoint'
tff(fact_3695_card__insert__disjoint_H,axiom,
    ! [A2: set_fu3555266933746557007_nat_o,X: fun_Pr6460944699146269935_nat_o] :
      ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,A2)
     => ( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),A2)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite5795495896937551127_nat_o(insert4923026906511862463_nat_o(X,A2))),aa_nat_nat(suc,zero_zero_nat)) = finite5795495896937551127_nat_o(A2) ) ) ) ).

% card_insert_disjoint'
tff(fact_3696_card_Oremove,axiom,
    ! [A2: set_fu3555266933746557007_nat_o,X: fun_Pr6460944699146269935_nat_o] :
      ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,A2)
     => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),A2)
       => ( finite5795495896937551127_nat_o(A2) = aa_nat_nat(suc,finite5795495896937551127_nat_o(minus_6456595905716665654_nat_o(A2,insert4923026906511862463_nat_o(X,bot_bo8632255151191677115_nat_o)))) ) ) ) ).

% card.remove
tff(fact_3697_card_Oremove,axiom,
    ! [A2: set_o,X: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( aa_set_o_o(member_o((X)),A2)
       => ( finite_card_o(A2) = aa_nat_nat(suc,finite_card_o(minus_minus_set_o(A2,insert_o((X),bot_bot_set_o)))) ) ) ) ).

% card.remove
tff(fact_3698_card_Oremove,axiom,
    ! [A2: set_nat,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(member_nat2(X),A2)
       => ( finite_card_nat(A2) = aa_nat_nat(suc,finite_card_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),insert_nat2(X,bot_bot_set_nat)))) ) ) ) ).

% card.remove
tff(fact_3699_card_Oinsert__remove,axiom,
    ! [A2: set_o,X: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( finite_card_o(insert_o((X),A2)) = aa_nat_nat(suc,finite_card_o(minus_minus_set_o(A2,insert_o((X),bot_bot_set_o)))) ) ) ).

% card.insert_remove
tff(fact_3700_card_Oinsert__remove,axiom,
    ! [A2: set_nat,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( finite_card_nat(insert_nat2(X,A2)) = aa_nat_nat(suc,finite_card_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),insert_nat2(X,bot_bot_set_nat)))) ) ) ).

% card.insert_remove
tff(fact_3701_card__Suc__Diff1,axiom,
    ! [A2: set_fu3555266933746557007_nat_o,X: fun_Pr6460944699146269935_nat_o] :
      ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,A2)
     => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),A2)
       => ( aa_nat_nat(suc,finite5795495896937551127_nat_o(minus_6456595905716665654_nat_o(A2,insert4923026906511862463_nat_o(X,bot_bo8632255151191677115_nat_o)))) = finite5795495896937551127_nat_o(A2) ) ) ) ).

% card_Suc_Diff1
tff(fact_3702_card__Suc__Diff1,axiom,
    ! [A2: set_o,X: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( aa_set_o_o(member_o((X)),A2)
       => ( aa_nat_nat(suc,finite_card_o(minus_minus_set_o(A2,insert_o((X),bot_bot_set_o)))) = finite_card_o(A2) ) ) ) ).

% card_Suc_Diff1
tff(fact_3703_card__Suc__Diff1,axiom,
    ! [A2: set_nat,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(member_nat2(X),A2)
       => ( aa_nat_nat(suc,finite_card_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),insert_nat2(X,bot_bot_set_nat)))) = finite_card_nat(A2) ) ) ) ).

% card_Suc_Diff1
tff(fact_3704_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(V)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(W)),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(times_times_num(V,W))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_3705_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: int] : aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(V)),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(W)),Z)) = aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(times_times_num(V,W))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_3706_numeral__times__numeral,axiom,
    ! [M2: num,N: num] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(M2)),numeral_numeral_nat(N)) = numeral_numeral_nat(times_times_num(M2,N)) ).

% numeral_times_numeral
tff(fact_3707_numeral__times__numeral,axiom,
    ! [M2: num,N: num] : aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(M2)),numeral_numeral_int(N)) = numeral_numeral_int(times_times_num(M2,N)) ).

% numeral_times_numeral
tff(fact_3708_min__Suc__gt_I1_J,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,A)),B) = aa_nat_nat(suc,A) ) ) ).

% min_Suc_gt(1)
tff(fact_3709_min__Suc__gt_I2_J,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,B),aa_nat_nat(suc,A)) = aa_nat_nat(suc,A) ) ) ).

% min_Suc_gt(2)
tff(fact_3710_length__map,axiom,
    ! [F: fun_Pr3816910165196019820n_assn,Xs: list_P8527749157015355191n_assn] : size_size_list_assn(aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,F),Xs)) = size_s6829681357464350627n_assn(Xs) ).

% length_map
tff(fact_3711_length__map,axiom,
    ! [F: fun_nat_nat,Xs: list_nat] : size_size_list_nat(aa_list_nat_list_nat(aa_fun4736978287202700568st_nat(map_nat_nat,F),Xs)) = size_size_list_nat(Xs) ).

% length_map
tff(fact_3712_length__map,axiom,
    ! [F: fun_int_nat,Xs: list_int] : size_size_list_nat(map_int_nat(F,Xs)) = size_size_list_int(Xs) ).

% length_map
tff(fact_3713_length__map,axiom,
    ! [F: fun_nat_int,Xs: list_nat] : size_size_list_int(map_nat_int(F,Xs)) = size_size_list_nat(Xs) ).

% length_map
tff(fact_3714_length__map,axiom,
    ! [F: fun_int_int,Xs: list_int] : size_size_list_int(map_int_int(F,Xs)) = size_size_list_int(Xs) ).

% length_map
tff(fact_3715_append__eq__append__conv,axiom,
    ! [Xs: list_nat,Ys: list_nat,Us2: list_nat,Vs: list_nat] :
      ( ( ( size_size_list_nat(Xs) = size_size_list_nat(Ys) )
        | ( size_size_list_nat(Us2) = size_size_list_nat(Vs) ) )
     => ( ( aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Xs),Us2) = aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Ys),Vs) )
      <=> ( ( Xs = Ys )
          & ( Us2 = Vs ) ) ) ) ).

% append_eq_append_conv
tff(fact_3716_append__eq__append__conv,axiom,
    ! [Xs: list_int,Ys: list_int,Us2: list_int,Vs: list_int] :
      ( ( ( size_size_list_int(Xs) = size_size_list_int(Ys) )
        | ( size_size_list_int(Us2) = size_size_list_int(Vs) ) )
     => ( ( aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Xs),Us2) = aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Ys),Vs) )
      <=> ( ( Xs = Ys )
          & ( Us2 = Vs ) ) ) ) ).

% append_eq_append_conv
tff(fact_3717_length__rev,axiom,
    ! [Xs: list_nat] : size_size_list_nat(rev_nat(Xs)) = size_size_list_nat(Xs) ).

% length_rev
tff(fact_3718_length__rev,axiom,
    ! [Xs: list_int] : size_size_list_int(rev_int(Xs)) = size_size_list_int(Xs) ).

% length_rev
tff(fact_3719_length__rotate1,axiom,
    ! [Xs: list_nat] : size_size_list_nat(rotate1_nat(Xs)) = size_size_list_nat(Xs) ).

% length_rotate1
tff(fact_3720_length__rotate1,axiom,
    ! [Xs: list_int] : size_size_list_int(rotate1_int(Xs)) = size_size_list_int(Xs) ).

% length_rotate1
tff(fact_3721_distrib__left__numeral,axiom,
    ! [V: num,B: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(V)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(V)),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(V)),C2)) ).

% distrib_left_numeral
tff(fact_3722_distrib__left__numeral,axiom,
    ! [V: num,B: int,C2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(V)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(V)),B)),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(V)),C2)) ).

% distrib_left_numeral
tff(fact_3723_distrib__right__numeral,axiom,
    ! [A: nat,B: nat,V: num] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),numeral_numeral_nat(V)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),numeral_numeral_nat(V))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),numeral_numeral_nat(V))) ).

% distrib_right_numeral
tff(fact_3724_distrib__right__numeral,axiom,
    ! [A: int,B: int,V: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),numeral_numeral_int(V)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),numeral_numeral_int(V))),aa_int_int(aa_int_fun_int_int(times_times_int,B),numeral_numeral_int(V))) ).

% distrib_right_numeral
tff(fact_3725_right__diff__distrib__numeral,axiom,
    ! [V: num,B: int,C2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(V)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(V)),B)),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(V)),C2)) ).

% right_diff_distrib_numeral
tff(fact_3726_left__diff__distrib__numeral,axiom,
    ! [A: int,B: int,V: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),numeral_numeral_int(V)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),numeral_numeral_int(V))),aa_int_int(aa_int_fun_int_int(times_times_int,B),numeral_numeral_int(V))) ).

% left_diff_distrib_numeral
tff(fact_3727_length__0__conv,axiom,
    ! [Xs: list_a] :
      ( ( size_size_list_a(Xs) = zero_zero_nat )
    <=> ( Xs = nil_a ) ) ).

% length_0_conv
tff(fact_3728_length__0__conv,axiom,
    ! [Xs: list_b] :
      ( ( size_size_list_b(Xs) = zero_zero_nat )
    <=> ( Xs = nil_b ) ) ).

% length_0_conv
tff(fact_3729_length__0__conv,axiom,
    ! [Xs: list_P8527749157015355191n_assn] :
      ( ( size_s6829681357464350627n_assn(Xs) = zero_zero_nat )
    <=> ( Xs = nil_Pr5671120429643327159n_assn ) ) ).

% length_0_conv
tff(fact_3730_length__0__conv,axiom,
    ! [Xs: list_nat] :
      ( ( size_size_list_nat(Xs) = zero_zero_nat )
    <=> ( Xs = nil_nat ) ) ).

% length_0_conv
tff(fact_3731_length__0__conv,axiom,
    ! [Xs: list_int] :
      ( ( size_size_list_int(Xs) = zero_zero_nat )
    <=> ( Xs = nil_int ) ) ).

% length_0_conv
tff(fact_3732_power__add__numeral2,axiom,
    ! [A: assn,M2: num,N: num,B: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(power_power_assn(A),numeral_numeral_nat(M2))),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(power_power_assn(A),numeral_numeral_nat(N))),B)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(power_power_assn(A),numeral_numeral_nat(plus_plus_num(M2,N)))),B) ).

% power_add_numeral2
tff(fact_3733_power__add__numeral2,axiom,
    ! [A: nat,M2: num,N: num,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),numeral_numeral_nat(M2))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),numeral_numeral_nat(N))),B)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),numeral_numeral_nat(plus_plus_num(M2,N)))),B) ).

% power_add_numeral2
tff(fact_3734_power__add__numeral,axiom,
    ! [A: assn,M2: num,N: num] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(power_power_assn(A),numeral_numeral_nat(M2))),aa_nat_assn(power_power_assn(A),numeral_numeral_nat(N))) = aa_nat_assn(power_power_assn(A),numeral_numeral_nat(plus_plus_num(M2,N))) ).

% power_add_numeral
tff(fact_3735_power__add__numeral,axiom,
    ! [A: nat,M2: num,N: num] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),numeral_numeral_nat(M2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),numeral_numeral_nat(N))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),numeral_numeral_nat(plus_plus_num(M2,N))) ).

% power_add_numeral
tff(fact_3736_min__0__1_I6_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,numeral_numeral_nat(X)),one_one_nat) = one_one_nat ).

% min_0_1(6)
tff(fact_3737_min__0__1_I6_J,axiom,
    ! [X: num] : aa_int_int(aa_int_fun_int_int(ord_min_int,numeral_numeral_int(X)),one_one_int) = one_one_int ).

% min_0_1(6)
tff(fact_3738_min__0__1_I5_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,one_one_nat),numeral_numeral_nat(X)) = one_one_nat ).

% min_0_1(5)
tff(fact_3739_min__0__1_I5_J,axiom,
    ! [X: num] : aa_int_int(aa_int_fun_int_int(ord_min_int,one_one_int),numeral_numeral_int(X)) = one_one_int ).

% min_0_1(5)
tff(fact_3740_max__0__1_I6_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,numeral_numeral_nat(X)),one_one_nat) = numeral_numeral_nat(X) ).

% max_0_1(6)
tff(fact_3741_max__0__1_I6_J,axiom,
    ! [X: num] : aa_int_int(aa_int_fun_int_int(ord_max_int,numeral_numeral_int(X)),one_one_int) = numeral_numeral_int(X) ).

% max_0_1(6)
tff(fact_3742_max__0__1_I5_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,one_one_nat),numeral_numeral_nat(X)) = numeral_numeral_nat(X) ).

% max_0_1(5)
tff(fact_3743_max__0__1_I5_J,axiom,
    ! [X: num] : aa_int_int(aa_int_fun_int_int(ord_max_int,one_one_int),numeral_numeral_int(X)) = numeral_numeral_int(X) ).

% max_0_1(5)
tff(fact_3744_nth__Cons__Suc,axiom,
    ! [X: nat,Xs: list_nat,N: nat] : aa_nat_nat(nth_nat(aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X),Xs)),aa_nat_nat(suc,N)) = aa_nat_nat(nth_nat(Xs),N) ).

% nth_Cons_Suc
tff(fact_3745_nth__Cons__Suc,axiom,
    ! [X: int,Xs: list_int,N: nat] : aa_nat_int(nth_int(aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X),Xs)),aa_nat_nat(suc,N)) = aa_nat_int(nth_int(Xs),N) ).

% nth_Cons_Suc
tff(fact_3746_nth__Cons__Suc,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn,N: nat] : aa_nat3152888703114873891n_assn(nth_Pr1769885009046257848n_assn(aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X),Xs)),aa_nat_nat(suc,N)) = aa_nat3152888703114873891n_assn(nth_Pr1769885009046257848n_assn(Xs),N) ).

% nth_Cons_Suc
tff(fact_3747_length__append,axiom,
    ! [Xs: list_nat,Ys: list_nat] : size_size_list_nat(aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Xs),Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_size_list_nat(Xs)),size_size_list_nat(Ys)) ).

% length_append
tff(fact_3748_length__append,axiom,
    ! [Xs: list_int,Ys: list_int] : size_size_list_int(aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Xs),Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_size_list_int(Xs)),size_size_list_int(Ys)) ).

% length_append
tff(fact_3749_sorted__list__of__set_Olength__sorted__key__list__of__set,axiom,
    ! [A2: set_int] : size_size_list_int(linord2612477271533052124et_int(A2)) = finite_card_int(A2) ).

% sorted_list_of_set.length_sorted_key_list_of_set
tff(fact_3750_sorted__list__of__set_Olength__sorted__key__list__of__set,axiom,
    ! [A2: set_nat] : size_size_list_nat(linord2614967742042102400et_nat(A2)) = finite_card_nat(A2) ).

% sorted_list_of_set.length_sorted_key_list_of_set
tff(fact_3751_slice__complete,axiom,
    ! [Xs: list_nat] : slice_nat(zero_zero_nat,size_size_list_nat(Xs),Xs) = Xs ).

% slice_complete
tff(fact_3752_slice__complete,axiom,
    ! [Xs: list_int] : slice_int(zero_zero_nat,size_size_list_int(Xs),Xs) = Xs ).

% slice_complete
tff(fact_3753_length__product,axiom,
    ! [Xs: list_nat,Ys: list_nat] : size_s5460976970255530739at_nat(product_nat_nat(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_nat(Xs)),size_size_list_nat(Ys)) ).

% length_product
tff(fact_3754_length__product,axiom,
    ! [Xs: list_nat,Ys: list_int] : size_s2970893825323803983at_int(product_nat_int(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_nat(Xs)),size_size_list_int(Ys)) ).

% length_product
tff(fact_3755_length__product,axiom,
    ! [Xs: list_int,Ys: list_nat] : size_s7647898544948552527nt_nat(product_int_nat(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_int(Xs)),size_size_list_nat(Ys)) ).

% length_product
tff(fact_3756_length__product,axiom,
    ! [Xs: list_int,Ys: list_int] : size_s5157815400016825771nt_int(product_int_int(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_int(Xs)),size_size_list_int(Ys)) ).

% length_product
tff(fact_3757_length__greater__0__conv,axiom,
    ! [Xs: list_a] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_size_list_a(Xs))
    <=> ( Xs != nil_a ) ) ).

% length_greater_0_conv
tff(fact_3758_length__greater__0__conv,axiom,
    ! [Xs: list_b] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_size_list_b(Xs))
    <=> ( Xs != nil_b ) ) ).

% length_greater_0_conv
tff(fact_3759_length__greater__0__conv,axiom,
    ! [Xs: list_P8527749157015355191n_assn] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_s6829681357464350627n_assn(Xs))
    <=> ( Xs != nil_Pr5671120429643327159n_assn ) ) ).

% length_greater_0_conv
tff(fact_3760_length__greater__0__conv,axiom,
    ! [Xs: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_size_list_nat(Xs))
    <=> ( Xs != nil_nat ) ) ).

% length_greater_0_conv
tff(fact_3761_length__greater__0__conv,axiom,
    ! [Xs: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_size_list_int(Xs))
    <=> ( Xs != nil_int ) ) ).

% length_greater_0_conv
tff(fact_3762_length__concat__rev,axiom,
    ! [Xs: list_list_nat] : size_size_list_nat(concat_nat(rev_list_nat(Xs))) = size_size_list_nat(concat_nat(Xs)) ).

% length_concat_rev
tff(fact_3763_length__concat__rev,axiom,
    ! [Xs: list_list_int] : size_size_list_int(concat_int(rev_list_int(Xs))) = size_size_list_int(concat_int(Xs)) ).

% length_concat_rev
tff(fact_3764_nth__append__length,axiom,
    ! [Xs: list_P8527749157015355191n_assn,X: produc6575502325842934193n_assn,Ys: list_P8527749157015355191n_assn] : aa_nat3152888703114873891n_assn(nth_Pr1769885009046257848n_assn(aa_lis2797772165808192636n_assn(aa_lis4738365772350288766n_assn(append282499809098378956n_assn,Xs),aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X),Ys))),size_s6829681357464350627n_assn(Xs)) = X ).

% nth_append_length
tff(fact_3765_nth__append__length,axiom,
    ! [Xs: list_nat,X: nat,Ys: list_nat] : aa_nat_nat(nth_nat(aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Xs),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X),Ys))),size_size_list_nat(Xs)) = X ).

% nth_append_length
tff(fact_3766_nth__append__length,axiom,
    ! [Xs: list_int,X: int,Ys: list_int] : aa_nat_int(nth_int(aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Xs),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X),Ys))),size_size_list_int(Xs)) = X ).

% nth_append_length
tff(fact_3767_Suc__diff,axiom,
    ! [M2: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),M2)
       => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),one_one_nat)) ) ) ) ).

% Suc_diff
tff(fact_3768_nth__map,axiom,
    ! [N: nat,Xs: list_P8527749157015355191n_assn,F: fun_Pr3816910165196019820n_assn] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),size_s6829681357464350627n_assn(Xs))
     => ( aa_nat_assn(nth_assn(aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,F),Xs)),N) = aa_Pro2979818503705714077n_assn(F,aa_nat3152888703114873891n_assn(nth_Pr1769885009046257848n_assn(Xs),N)) ) ) ).

% nth_map
tff(fact_3769_nth__map,axiom,
    ! [N: nat,Xs: list_nat,F: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),size_size_list_nat(Xs))
     => ( aa_nat_int(nth_int(map_nat_int(F,Xs)),N) = aa_nat_int(F,aa_nat_nat(nth_nat(Xs),N)) ) ) ).

% nth_map
tff(fact_3770_nth__map,axiom,
    ! [N: nat,Xs: list_nat,F: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),size_size_list_nat(Xs))
     => ( aa_nat_nat(nth_nat(aa_list_nat_list_nat(aa_fun4736978287202700568st_nat(map_nat_nat,F),Xs)),N) = aa_nat_nat(F,aa_nat_nat(nth_nat(Xs),N)) ) ) ).

% nth_map
tff(fact_3771_nth__map,axiom,
    ! [N: nat,Xs: list_int,F: fun_int_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),size_size_list_int(Xs))
     => ( aa_nat_nat(nth_nat(map_int_nat(F,Xs)),N) = aa_int_nat(F,aa_nat_int(nth_int(Xs),N)) ) ) ).

% nth_map
tff(fact_3772_nth__map,axiom,
    ! [N: nat,Xs: list_int,F: fun_int_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),size_size_list_int(Xs))
     => ( aa_nat_int(nth_int(map_int_int(F,Xs)),N) = aa_int_int(F,aa_nat_int(nth_int(Xs),N)) ) ) ).

% nth_map
tff(fact_3773_nth__append__first,axiom,
    ! [I: nat,La: list_nat,L_a: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_nat(La))
     => ( aa_nat_nat(nth_nat(aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,La),L_a)),I) = aa_nat_nat(nth_nat(La),I) ) ) ).

% nth_append_first
tff(fact_3774_nth__append__first,axiom,
    ! [I: nat,La: list_int,L_a: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(La))
     => ( aa_nat_int(nth_int(aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,La),L_a)),I) = aa_nat_int(nth_int(La),I) ) ) ).

% nth_append_first
tff(fact_3775_nth__append__length__plus,axiom,
    ! [Xs: list_nat,Ys: list_nat,N: nat] : aa_nat_nat(nth_nat(aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Xs),Ys)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_size_list_nat(Xs)),N)) = aa_nat_nat(nth_nat(Ys),N) ).

% nth_append_length_plus
tff(fact_3776_nth__append__length__plus,axiom,
    ! [Xs: list_int,Ys: list_int,N: nat] : aa_nat_int(nth_int(aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Xs),Ys)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_size_list_int(Xs)),N)) = aa_nat_int(nth_int(Ys),N) ).

% nth_append_length_plus
tff(fact_3777_length__butlast,axiom,
    ! [Xs: list_nat] : size_size_list_nat(butlast_nat(Xs)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,size_size_list_nat(Xs)),one_one_nat) ).

% length_butlast
tff(fact_3778_length__butlast,axiom,
    ! [Xs: list_int] : size_size_list_int(butlast_int(Xs)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,size_size_list_int(Xs)),one_one_nat) ).

% length_butlast
tff(fact_3779_rotate1__length01,axiom,
    ! [Xs: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_nat(Xs)),one_one_nat)
     => ( rotate1_nat(Xs) = Xs ) ) ).

% rotate1_length01
tff(fact_3780_rotate1__length01,axiom,
    ! [Xs: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_int(Xs)),one_one_nat)
     => ( rotate1_int(Xs) = Xs ) ) ).

% rotate1_length01
tff(fact_3781_slice__len,axiom,
    ! [From: nat,To: nat,Xs: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,From),To)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,To),size_size_list_nat(Xs))
       => ( size_size_list_nat(slice_nat(From,To,Xs)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,To),From) ) ) ) ).

% slice_len
tff(fact_3782_slice__len,axiom,
    ! [From: nat,To: nat,Xs: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,From),To)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,To),size_size_list_int(Xs))
       => ( size_size_list_int(slice_int(From,To,Xs)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,To),From) ) ) ) ).

% slice_len
tff(fact_3783_length__ge__1__conv,axiom,
    ! [La: list_a] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),size_size_list_a(La))
    <=> ( La != nil_a ) ) ).

% length_ge_1_conv
tff(fact_3784_length__ge__1__conv,axiom,
    ! [La: list_b] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),size_size_list_b(La))
    <=> ( La != nil_b ) ) ).

% length_ge_1_conv
tff(fact_3785_length__ge__1__conv,axiom,
    ! [La: list_P8527749157015355191n_assn] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),size_s6829681357464350627n_assn(La))
    <=> ( La != nil_Pr5671120429643327159n_assn ) ) ).

% length_ge_1_conv
tff(fact_3786_length__ge__1__conv,axiom,
    ! [La: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),size_size_list_nat(La))
    <=> ( La != nil_nat ) ) ).

% length_ge_1_conv
tff(fact_3787_length__ge__1__conv,axiom,
    ! [La: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),size_size_list_int(La))
    <=> ( La != nil_int ) ) ).

% length_ge_1_conv
tff(fact_3788_length__Cons,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] : size_s6829681357464350627n_assn(aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X),Xs)) = aa_nat_nat(suc,size_s6829681357464350627n_assn(Xs)) ).

% length_Cons
tff(fact_3789_length__Cons,axiom,
    ! [X: nat,Xs: list_nat] : size_size_list_nat(aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X),Xs)) = aa_nat_nat(suc,size_size_list_nat(Xs)) ).

% length_Cons
tff(fact_3790_length__Cons,axiom,
    ! [X: int,Xs: list_int] : size_size_list_int(aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X),Xs)) = aa_nat_nat(suc,size_size_list_int(Xs)) ).

% length_Cons
tff(fact_3791_Suc__length__conv,axiom,
    ! [N: nat,Xs: list_P8527749157015355191n_assn] :
      ( ( aa_nat_nat(suc,N) = size_s6829681357464350627n_assn(Xs) )
    <=> ? [Y3: produc6575502325842934193n_assn,Ys3: list_P8527749157015355191n_assn] :
          ( ( Xs = aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,Y3),Ys3) )
          & ( size_s6829681357464350627n_assn(Ys3) = N ) ) ) ).

% Suc_length_conv
tff(fact_3792_Suc__length__conv,axiom,
    ! [N: nat,Xs: list_nat] :
      ( ( aa_nat_nat(suc,N) = size_size_list_nat(Xs) )
    <=> ? [Y3: nat,Ys3: list_nat] :
          ( ( Xs = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Y3),Ys3) )
          & ( size_size_list_nat(Ys3) = N ) ) ) ).

% Suc_length_conv
tff(fact_3793_Suc__length__conv,axiom,
    ! [N: nat,Xs: list_int] :
      ( ( aa_nat_nat(suc,N) = size_size_list_int(Xs) )
    <=> ? [Y3: int,Ys3: list_int] :
          ( ( Xs = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Y3),Ys3) )
          & ( size_size_list_int(Ys3) = N ) ) ) ).

% Suc_length_conv
tff(fact_3794_length__Suc__conv,axiom,
    ! [Xs: list_P8527749157015355191n_assn,N: nat] :
      ( ( size_s6829681357464350627n_assn(Xs) = aa_nat_nat(suc,N) )
    <=> ? [Y3: produc6575502325842934193n_assn,Ys3: list_P8527749157015355191n_assn] :
          ( ( Xs = aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,Y3),Ys3) )
          & ( size_s6829681357464350627n_assn(Ys3) = N ) ) ) ).

% length_Suc_conv
tff(fact_3795_length__Suc__conv,axiom,
    ! [Xs: list_nat,N: nat] :
      ( ( size_size_list_nat(Xs) = aa_nat_nat(suc,N) )
    <=> ? [Y3: nat,Ys3: list_nat] :
          ( ( Xs = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Y3),Ys3) )
          & ( size_size_list_nat(Ys3) = N ) ) ) ).

% length_Suc_conv
tff(fact_3796_length__Suc__conv,axiom,
    ! [Xs: list_int,N: nat] :
      ( ( size_size_list_int(Xs) = aa_nat_nat(suc,N) )
    <=> ? [Y3: int,Ys3: list_int] :
          ( ( Xs = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Y3),Ys3) )
          & ( size_size_list_int(Ys3) = N ) ) ) ).

% length_Suc_conv
tff(fact_3797_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs2: list_nat] : size_size_list_nat(Xs2) = N ).

% Ex_list_of_length
tff(fact_3798_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs2: list_int] : size_size_list_int(Xs2) = N ).

% Ex_list_of_length
tff(fact_3799_neq__if__length__neq,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( size_size_list_nat(Xs) != size_size_list_nat(Ys) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
tff(fact_3800_neq__if__length__neq,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( size_size_list_int(Xs) != size_size_list_int(Ys) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
tff(fact_3801_sorted__iff__nth__Suc,axiom,
    ! [Xs: list_nat] :
      ( aa_list_nat_o(sorted_wrt_nat(ord_less_eq_nat),Xs)
    <=> ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,I2)),size_size_list_nat(Xs))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(nth_nat(Xs),I2)),aa_nat_nat(nth_nat(Xs),aa_nat_nat(suc,I2))) ) ) ).

% sorted_iff_nth_Suc
tff(fact_3802_sorted__iff__nth__Suc,axiom,
    ! [Xs: list_int] :
      ( aa_list_int_o(sorted_wrt_int(ord_less_eq_int),Xs)
    <=> ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,I2)),size_size_list_int(Xs))
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(nth_int(Xs),I2)),aa_nat_int(nth_int(Xs),aa_nat_nat(suc,I2))) ) ) ).

% sorted_iff_nth_Suc
tff(fact_3803_sorted__wrt__less__idx,axiom,
    ! [Ns: list_nat,I: nat] :
      ( aa_list_nat_o(sorted_wrt_nat(ord_less_nat),Ns)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_nat(Ns))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),aa_nat_nat(nth_nat(Ns),I)) ) ) ).

% sorted_wrt_less_idx
tff(fact_3804_sorted__wrt__nth__less,axiom,
    ! [Pa: fun_nat_fun_nat_o,Xs: list_nat,I: nat,J: nat] :
      ( aa_list_nat_o(sorted_wrt_nat(Pa),Xs)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_size_list_nat(Xs))
         => aa_nat_o(aa_nat_fun_nat_o(Pa,aa_nat_nat(nth_nat(Xs),I)),aa_nat_nat(nth_nat(Xs),J)) ) ) ) ).

% sorted_wrt_nth_less
tff(fact_3805_sorted__wrt__nth__less,axiom,
    ! [Pa: fun_int_fun_int_o,Xs: list_int,I: nat,J: nat] :
      ( aa_list_int_o(sorted_wrt_int(Pa),Xs)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_size_list_int(Xs))
         => aa_int_o(aa_int_fun_int_o(Pa,aa_nat_int(nth_int(Xs),I)),aa_nat_int(nth_int(Xs),J)) ) ) ) ).

% sorted_wrt_nth_less
tff(fact_3806_sorted__wrt__iff__nth__less,axiom,
    ! [Pa: fun_nat_fun_nat_o,Xs: list_nat] :
      ( aa_list_nat_o(sorted_wrt_nat(Pa),Xs)
    <=> ! [I2: nat,J2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J2)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J2),size_size_list_nat(Xs))
           => aa_nat_o(aa_nat_fun_nat_o(Pa,aa_nat_nat(nth_nat(Xs),I2)),aa_nat_nat(nth_nat(Xs),J2)) ) ) ) ).

% sorted_wrt_iff_nth_less
tff(fact_3807_sorted__wrt__iff__nth__less,axiom,
    ! [Pa: fun_int_fun_int_o,Xs: list_int] :
      ( aa_list_int_o(sorted_wrt_int(Pa),Xs)
    <=> ! [I2: nat,J2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J2)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J2),size_size_list_int(Xs))
           => aa_int_o(aa_int_fun_int_o(Pa,aa_nat_int(nth_int(Xs),I2)),aa_nat_int(nth_int(Xs),J2)) ) ) ) ).

% sorted_wrt_iff_nth_less
tff(fact_3808_sorted__wrt01,axiom,
    ! [Xs: list_nat,Pa: fun_nat_fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_nat(Xs)),one_one_nat)
     => aa_list_nat_o(sorted_wrt_nat(Pa),Xs) ) ).

% sorted_wrt01
tff(fact_3809_sorted__wrt01,axiom,
    ! [Xs: list_int,Pa: fun_int_fun_int_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_int(Xs)),one_one_nat)
     => aa_list_int_o(sorted_wrt_int(Pa),Xs) ) ).

% sorted_wrt01
tff(fact_3810_Suc__le__length__iff,axiom,
    ! [N: nat,Xs: list_P8527749157015355191n_assn] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,N)),size_s6829681357464350627n_assn(Xs))
    <=> ? [X3: produc6575502325842934193n_assn,Ys3: list_P8527749157015355191n_assn] :
          ( ( Xs = aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X3),Ys3) )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),size_s6829681357464350627n_assn(Ys3)) ) ) ).

% Suc_le_length_iff
tff(fact_3811_Suc__le__length__iff,axiom,
    ! [N: nat,Xs: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,N)),size_size_list_nat(Xs))
    <=> ? [X3: nat,Ys3: list_nat] :
          ( ( Xs = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X3),Ys3) )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),size_size_list_nat(Ys3)) ) ) ).

% Suc_le_length_iff
tff(fact_3812_Suc__le__length__iff,axiom,
    ! [N: nat,Xs: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,N)),size_size_list_int(Xs))
    <=> ? [X3: int,Ys3: list_int] :
          ( ( Xs = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X3),Ys3) )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),size_size_list_int(Ys3)) ) ) ).

% Suc_le_length_iff
tff(fact_3813_sorted__wrt_Osimps_I1_J,axiom,
    ! [Pa: fun_a_fun_a_o] : aa_list_a_o(sorted_wrt_a(Pa),nil_a) ).

% sorted_wrt.simps(1)
tff(fact_3814_sorted__wrt_Osimps_I1_J,axiom,
    ! [Pa: fun_b_fun_b_o] : aa_list_b_o(sorted_wrt_b(Pa),nil_b) ).

% sorted_wrt.simps(1)
tff(fact_3815_sorted__wrt_Osimps_I1_J,axiom,
    ! [Pa: fun_Pr5295903952624206638assn_o] : aa_lis1404944545740551261assn_o(sorted3986126766855596574n_assn(Pa),nil_Pr5671120429643327159n_assn) ).

% sorted_wrt.simps(1)
tff(fact_3816_sorted__wrt_Osimps_I1_J,axiom,
    ! [Pa: fun_nat_fun_nat_o] : aa_list_nat_o(sorted_wrt_nat(Pa),nil_nat) ).

% sorted_wrt.simps(1)
tff(fact_3817_sorted__wrt_Osimps_I1_J,axiom,
    ! [Pa: fun_int_fun_int_o] : aa_list_int_o(sorted_wrt_int(Pa),nil_int) ).

% sorted_wrt.simps(1)
tff(fact_3818_length__induct,axiom,
    ! [Pa: fun_list_nat_o,Xs: list_nat] :
      ( ! [Xs2: list_nat] :
          ( ! [Ys6: list_nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_nat(Ys6)),size_size_list_nat(Xs2))
             => aa_list_nat_o(Pa,Ys6) )
         => aa_list_nat_o(Pa,Xs2) )
     => aa_list_nat_o(Pa,Xs) ) ).

% length_induct
tff(fact_3819_length__induct,axiom,
    ! [Pa: fun_list_int_o,Xs: list_int] :
      ( ! [Xs2: list_int] :
          ( ! [Ys6: list_int] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_int(Ys6)),size_size_list_int(Xs2))
             => aa_list_int_o(Pa,Ys6) )
         => aa_list_int_o(Pa,Xs2) )
     => aa_list_int_o(Pa,Xs) ) ).

% length_induct
tff(fact_3820_finite__maxlen,axiom,
    ! [M: set_list_nat] :
      ( finite8100373058378681591st_nat(M)
     => ? [N7: nat] :
        ! [X4: list_nat] :
          ( member_list_nat(X4,M)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_nat(X4)),N7) ) ) ).

% finite_maxlen
tff(fact_3821_finite__maxlen,axiom,
    ! [M: set_list_int] :
      ( finite3922522038869484883st_int(M)
     => ? [N7: nat] :
        ! [X4: list_int] :
          ( member_list_int(X4,M)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_int(X4)),N7) ) ) ).

% finite_maxlen
tff(fact_3822_map__eq__imp__length__eq,axiom,
    ! [F: fun_Pr3816910165196019820n_assn,Xs: list_P8527749157015355191n_assn,G2: fun_Pr3816910165196019820n_assn,Ys: list_P8527749157015355191n_assn] :
      ( ( aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,F),Xs) = aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,G2),Ys) )
     => ( size_s6829681357464350627n_assn(Xs) = size_s6829681357464350627n_assn(Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_3823_map__eq__imp__length__eq,axiom,
    ! [F: fun_Pr3816910165196019820n_assn,Xs: list_P8527749157015355191n_assn,G2: fun_nat_assn,Ys: list_nat] :
      ( ( aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,F),Xs) = map_nat_assn(G2,Ys) )
     => ( size_s6829681357464350627n_assn(Xs) = size_size_list_nat(Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_3824_map__eq__imp__length__eq,axiom,
    ! [F: fun_Pr3816910165196019820n_assn,Xs: list_P8527749157015355191n_assn,G2: fun_int_assn,Ys: list_int] :
      ( ( aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,F),Xs) = map_int_assn(G2,Ys) )
     => ( size_s6829681357464350627n_assn(Xs) = size_size_list_int(Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_3825_map__eq__imp__length__eq,axiom,
    ! [F: fun_nat_assn,Xs: list_nat,G2: fun_Pr3816910165196019820n_assn,Ys: list_P8527749157015355191n_assn] :
      ( ( map_nat_assn(F,Xs) = aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,G2),Ys) )
     => ( size_size_list_nat(Xs) = size_s6829681357464350627n_assn(Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_3826_map__eq__imp__length__eq,axiom,
    ! [F: fun_nat_nat,Xs: list_nat,G2: fun_nat_nat,Ys: list_nat] :
      ( ( aa_list_nat_list_nat(aa_fun4736978287202700568st_nat(map_nat_nat,F),Xs) = aa_list_nat_list_nat(aa_fun4736978287202700568st_nat(map_nat_nat,G2),Ys) )
     => ( size_size_list_nat(Xs) = size_size_list_nat(Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_3827_map__eq__imp__length__eq,axiom,
    ! [F: fun_nat_nat,Xs: list_nat,G2: fun_int_nat,Ys: list_int] :
      ( ( aa_list_nat_list_nat(aa_fun4736978287202700568st_nat(map_nat_nat,F),Xs) = map_int_nat(G2,Ys) )
     => ( size_size_list_nat(Xs) = size_size_list_int(Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_3828_map__eq__imp__length__eq,axiom,
    ! [F: fun_int_assn,Xs: list_int,G2: fun_Pr3816910165196019820n_assn,Ys: list_P8527749157015355191n_assn] :
      ( ( map_int_assn(F,Xs) = aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,G2),Ys) )
     => ( size_size_list_int(Xs) = size_s6829681357464350627n_assn(Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_3829_map__eq__imp__length__eq,axiom,
    ! [F: fun_int_nat,Xs: list_int,G2: fun_nat_nat,Ys: list_nat] :
      ( ( map_int_nat(F,Xs) = aa_list_nat_list_nat(aa_fun4736978287202700568st_nat(map_nat_nat,G2),Ys) )
     => ( size_size_list_int(Xs) = size_size_list_nat(Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_3830_sorted__rev__iff__nth__Suc,axiom,
    ! [Xs: list_nat] :
      ( aa_list_nat_o(sorted_wrt_nat(ord_less_eq_nat),rev_nat(Xs))
    <=> ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,I2)),size_size_list_nat(Xs))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(nth_nat(Xs),aa_nat_nat(suc,I2))),aa_nat_nat(nth_nat(Xs),I2)) ) ) ).

% sorted_rev_iff_nth_Suc
tff(fact_3831_sorted__rev__iff__nth__Suc,axiom,
    ! [Xs: list_int] :
      ( aa_list_int_o(sorted_wrt_int(ord_less_eq_int),rev_int(Xs))
    <=> ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,I2)),size_size_list_int(Xs))
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(nth_int(Xs),aa_nat_nat(suc,I2))),aa_nat_int(nth_int(Xs),I2)) ) ) ).

% sorted_rev_iff_nth_Suc
tff(fact_3832_sorted__wrt__mergesort__by__rel,axiom,
    ! [R: fun_nat_fun_nat_o,Xs: list_nat] :
      ( ! [X2: nat,Y2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(R,X2),Y2)
          | aa_nat_o(aa_nat_fun_nat_o(R,Y2),X2) )
     => ( ! [X2: nat,Y2: nat,Z3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(R,X2),Y2)
           => ( aa_nat_o(aa_nat_fun_nat_o(R,Y2),Z3)
             => aa_nat_o(aa_nat_fun_nat_o(R,X2),Z3) ) )
       => aa_list_nat_o(sorted_wrt_nat(R),aa_list_nat_list_nat(mergesort_by_rel_nat(R),Xs)) ) ) ).

% sorted_wrt_mergesort_by_rel
tff(fact_3833_sorted__wrt__mergesort__by__rel,axiom,
    ! [R: fun_int_fun_int_o,Xs: list_int] :
      ( ! [X2: int,Y2: int] :
          ( aa_int_o(aa_int_fun_int_o(R,X2),Y2)
          | aa_int_o(aa_int_fun_int_o(R,Y2),X2) )
     => ( ! [X2: int,Y2: int,Z3: int] :
            ( aa_int_o(aa_int_fun_int_o(R,X2),Y2)
           => ( aa_int_o(aa_int_fun_int_o(R,Y2),Z3)
             => aa_int_o(aa_int_fun_int_o(R,X2),Z3) ) )
       => aa_list_int_o(sorted_wrt_int(R),aa_list_int_list_int(mergesort_by_rel_int(R),Xs)) ) ) ).

% sorted_wrt_mergesort_by_rel
tff(fact_3834_sorted__iff__nth__mono__less,axiom,
    ! [Xs: list_nat] :
      ( aa_list_nat_o(sorted_wrt_nat(ord_less_eq_nat),Xs)
    <=> ! [I2: nat,J2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J2)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J2),size_size_list_nat(Xs))
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(nth_nat(Xs),I2)),aa_nat_nat(nth_nat(Xs),J2)) ) ) ) ).

% sorted_iff_nth_mono_less
tff(fact_3835_sorted__iff__nth__mono__less,axiom,
    ! [Xs: list_int] :
      ( aa_list_int_o(sorted_wrt_int(ord_less_eq_int),Xs)
    <=> ! [I2: nat,J2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J2)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J2),size_size_list_int(Xs))
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(nth_int(Xs),I2)),aa_nat_int(nth_int(Xs),J2)) ) ) ) ).

% sorted_iff_nth_mono_less
tff(fact_3836_sorted01,axiom,
    ! [Xs: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_nat(Xs)),one_one_nat)
     => aa_list_nat_o(sorted_wrt_nat(ord_less_eq_nat),Xs) ) ).

% sorted01
tff(fact_3837_sorted01,axiom,
    ! [Xs: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_int(Xs)),one_one_nat)
     => aa_list_int_o(sorted_wrt_int(ord_less_eq_int),Xs) ) ).

% sorted01
tff(fact_3838_list_Osize_I4_J,axiom,
    ! [X21: produc6575502325842934193n_assn,X22: list_P8527749157015355191n_assn] : size_s6829681357464350627n_assn(aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X21),X22)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_s6829681357464350627n_assn(X22)),aa_nat_nat(suc,zero_zero_nat)) ).

% list.size(4)
tff(fact_3839_list_Osize_I4_J,axiom,
    ! [X21: nat,X22: list_nat] : size_size_list_nat(aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X21),X22)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_size_list_nat(X22)),aa_nat_nat(suc,zero_zero_nat)) ).

% list.size(4)
tff(fact_3840_list_Osize_I4_J,axiom,
    ! [X21: int,X22: list_int] : size_size_list_int(aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X21),X22)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_size_list_int(X22)),aa_nat_nat(suc,zero_zero_nat)) ).

% list.size(4)
tff(fact_3841_length__Suc__rev__conv,axiom,
    ! [Xs: list_a,N: nat] :
      ( ( size_size_list_a(Xs) = aa_nat_nat(suc,N) )
    <=> ? [Ys3: list_a,Y3: a] :
          ( ( Xs = aa_list_a_list_a(aa_lis2442066584718795531list_a(append_a,Ys3),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,Y3),nil_a)) )
          & ( size_size_list_a(Ys3) = N ) ) ) ).

% length_Suc_rev_conv
tff(fact_3842_length__Suc__rev__conv,axiom,
    ! [Xs: list_b,N: nat] :
      ( ( size_size_list_b(Xs) = aa_nat_nat(suc,N) )
    <=> ? [Ys3: list_b,Y3: b] :
          ( ( Xs = aa_list_b_list_b(aa_lis927650220358621516list_b(append_b,Ys3),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,Y3),nil_b)) )
          & ( size_size_list_b(Ys3) = N ) ) ) ).

% length_Suc_rev_conv
tff(fact_3843_length__Suc__rev__conv,axiom,
    ! [Xs: list_P8527749157015355191n_assn,N: nat] :
      ( ( size_s6829681357464350627n_assn(Xs) = aa_nat_nat(suc,N) )
    <=> ? [Ys3: list_P8527749157015355191n_assn,Y3: produc6575502325842934193n_assn] :
          ( ( Xs = aa_lis2797772165808192636n_assn(aa_lis4738365772350288766n_assn(append282499809098378956n_assn,Ys3),aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,Y3),nil_Pr5671120429643327159n_assn)) )
          & ( size_s6829681357464350627n_assn(Ys3) = N ) ) ) ).

% length_Suc_rev_conv
tff(fact_3844_length__Suc__rev__conv,axiom,
    ! [Xs: list_nat,N: nat] :
      ( ( size_size_list_nat(Xs) = aa_nat_nat(suc,N) )
    <=> ? [Ys3: list_nat,Y3: nat] :
          ( ( Xs = aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Ys3),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Y3),nil_nat)) )
          & ( size_size_list_nat(Ys3) = N ) ) ) ).

% length_Suc_rev_conv
tff(fact_3845_length__Suc__rev__conv,axiom,
    ! [Xs: list_int,N: nat] :
      ( ( size_size_list_int(Xs) = aa_nat_nat(suc,N) )
    <=> ? [Ys3: list_int,Y3: int] :
          ( ( Xs = aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Ys3),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Y3),nil_int)) )
          & ( size_size_list_int(Ys3) = N ) ) ) ).

% length_Suc_rev_conv
tff(fact_3846_length__Suc__conv__rev,axiom,
    ! [Xs: list_a,N: nat] :
      ( ( size_size_list_a(Xs) = aa_nat_nat(suc,N) )
    <=> ? [Y3: a,Ys3: list_a] :
          ( ( Xs = aa_list_a_list_a(aa_lis2442066584718795531list_a(append_a,Ys3),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,Y3),nil_a)) )
          & ( size_size_list_a(Ys3) = N ) ) ) ).

% length_Suc_conv_rev
tff(fact_3847_length__Suc__conv__rev,axiom,
    ! [Xs: list_b,N: nat] :
      ( ( size_size_list_b(Xs) = aa_nat_nat(suc,N) )
    <=> ? [Y3: b,Ys3: list_b] :
          ( ( Xs = aa_list_b_list_b(aa_lis927650220358621516list_b(append_b,Ys3),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,Y3),nil_b)) )
          & ( size_size_list_b(Ys3) = N ) ) ) ).

% length_Suc_conv_rev
tff(fact_3848_length__Suc__conv__rev,axiom,
    ! [Xs: list_P8527749157015355191n_assn,N: nat] :
      ( ( size_s6829681357464350627n_assn(Xs) = aa_nat_nat(suc,N) )
    <=> ? [Y3: produc6575502325842934193n_assn,Ys3: list_P8527749157015355191n_assn] :
          ( ( Xs = aa_lis2797772165808192636n_assn(aa_lis4738365772350288766n_assn(append282499809098378956n_assn,Ys3),aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,Y3),nil_Pr5671120429643327159n_assn)) )
          & ( size_s6829681357464350627n_assn(Ys3) = N ) ) ) ).

% length_Suc_conv_rev
tff(fact_3849_length__Suc__conv__rev,axiom,
    ! [Xs: list_nat,N: nat] :
      ( ( size_size_list_nat(Xs) = aa_nat_nat(suc,N) )
    <=> ? [Y3: nat,Ys3: list_nat] :
          ( ( Xs = aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Ys3),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Y3),nil_nat)) )
          & ( size_size_list_nat(Ys3) = N ) ) ) ).

% length_Suc_conv_rev
tff(fact_3850_length__Suc__conv__rev,axiom,
    ! [Xs: list_int,N: nat] :
      ( ( size_size_list_int(Xs) = aa_nat_nat(suc,N) )
    <=> ? [Y3: int,Ys3: list_int] :
          ( ( Xs = aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Ys3),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Y3),nil_int)) )
          & ( size_size_list_int(Ys3) = N ) ) ) ).

% length_Suc_conv_rev
tff(fact_3851_length__append__singleton,axiom,
    ! [Xs: list_a,X: a] : size_size_list_a(aa_list_a_list_a(aa_lis2442066584718795531list_a(append_a,Xs),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X),nil_a))) = aa_nat_nat(suc,size_size_list_a(Xs)) ).

% length_append_singleton
tff(fact_3852_length__append__singleton,axiom,
    ! [Xs: list_b,X: b] : size_size_list_b(aa_list_b_list_b(aa_lis927650220358621516list_b(append_b,Xs),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X),nil_b))) = aa_nat_nat(suc,size_size_list_b(Xs)) ).

% length_append_singleton
tff(fact_3853_length__append__singleton,axiom,
    ! [Xs: list_P8527749157015355191n_assn,X: produc6575502325842934193n_assn] : size_s6829681357464350627n_assn(aa_lis2797772165808192636n_assn(aa_lis4738365772350288766n_assn(append282499809098378956n_assn,Xs),aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X),nil_Pr5671120429643327159n_assn))) = aa_nat_nat(suc,size_s6829681357464350627n_assn(Xs)) ).

% length_append_singleton
tff(fact_3854_length__append__singleton,axiom,
    ! [Xs: list_nat,X: nat] : size_size_list_nat(aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Xs),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X),nil_nat))) = aa_nat_nat(suc,size_size_list_nat(Xs)) ).

% length_append_singleton
tff(fact_3855_length__append__singleton,axiom,
    ! [Xs: list_int,X: int] : size_size_list_int(aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,Xs),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X),nil_int))) = aa_nat_nat(suc,size_size_list_int(Xs)) ).

% length_append_singleton
tff(fact_3856_strict__sorted__imp__sorted,axiom,
    ! [Xs: list_nat] :
      ( aa_list_nat_o(sorted_wrt_nat(ord_less_nat),Xs)
     => aa_list_nat_o(sorted_wrt_nat(ord_less_eq_nat),Xs) ) ).

% strict_sorted_imp_sorted
tff(fact_3857_strict__sorted__imp__sorted,axiom,
    ! [Xs: list_int] :
      ( aa_list_int_o(sorted_wrt_int(ord_less_int),Xs)
     => aa_list_int_o(sorted_wrt_int(ord_less_eq_int),Xs) ) ).

% strict_sorted_imp_sorted
tff(fact_3858_sorted__nth__mono,axiom,
    ! [Xs: list_nat,I: nat,J: nat] :
      ( aa_list_nat_o(sorted_wrt_nat(ord_less_eq_nat),Xs)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_size_list_nat(Xs))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(nth_nat(Xs),I)),aa_nat_nat(nth_nat(Xs),J)) ) ) ) ).

% sorted_nth_mono
tff(fact_3859_sorted__nth__mono,axiom,
    ! [Xs: list_int,I: nat,J: nat] :
      ( aa_list_int_o(sorted_wrt_int(ord_less_eq_int),Xs)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_size_list_int(Xs))
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(nth_int(Xs),I)),aa_nat_int(nth_int(Xs),J)) ) ) ) ).

% sorted_nth_mono
tff(fact_3860_sorted__iff__nth__mono,axiom,
    ! [Xs: list_nat] :
      ( aa_list_nat_o(sorted_wrt_nat(ord_less_eq_nat),Xs)
    <=> ! [I2: nat,J2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J2)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J2),size_size_list_nat(Xs))
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(nth_nat(Xs),I2)),aa_nat_nat(nth_nat(Xs),J2)) ) ) ) ).

% sorted_iff_nth_mono
tff(fact_3861_sorted__iff__nth__mono,axiom,
    ! [Xs: list_int] :
      ( aa_list_int_o(sorted_wrt_int(ord_less_eq_int),Xs)
    <=> ! [I2: nat,J2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J2)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J2),size_size_list_int(Xs))
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(nth_int(Xs),I2)),aa_nat_int(nth_int(Xs),J2)) ) ) ) ).

% sorted_iff_nth_mono
tff(fact_3862_sorted2,axiom,
    ! [X: nat,Y: nat,Zs3: list_nat] :
      ( aa_list_nat_o(sorted_wrt_nat(ord_less_eq_nat),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Y),Zs3)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
        & aa_list_nat_o(sorted_wrt_nat(ord_less_eq_nat),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Y),Zs3)) ) ) ).

% sorted2
tff(fact_3863_sorted2,axiom,
    ! [X: int,Y: int,Zs3: list_int] :
      ( aa_list_int_o(sorted_wrt_int(ord_less_eq_int),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Y),Zs3)))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
        & aa_list_int_o(sorted_wrt_int(ord_less_eq_int),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Y),Zs3)) ) ) ).

% sorted2
tff(fact_3864_sorted0,axiom,
    aa_list_nat_o(sorted_wrt_nat(ord_less_eq_nat),nil_nat) ).

% sorted0
tff(fact_3865_sorted0,axiom,
    aa_list_int_o(sorted_wrt_int(ord_less_eq_int),nil_int) ).

% sorted0
tff(fact_3866_strict__sorted__simps_I1_J,axiom,
    aa_list_nat_o(sorted_wrt_nat(ord_less_nat),nil_nat) ).

% strict_sorted_simps(1)
tff(fact_3867_strict__sorted__simps_I1_J,axiom,
    aa_list_int_o(sorted_wrt_int(ord_less_int),nil_int) ).

% strict_sorted_simps(1)
tff(fact_3868_sorted__wrt1,axiom,
    ! [Pa: fun_a_fun_a_o,X: a] : aa_list_a_o(sorted_wrt_a(Pa),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,X),nil_a)) ).

% sorted_wrt1
tff(fact_3869_sorted__wrt1,axiom,
    ! [Pa: fun_b_fun_b_o,X: b] : aa_list_b_o(sorted_wrt_b(Pa),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,X),nil_b)) ).

% sorted_wrt1
tff(fact_3870_sorted__wrt1,axiom,
    ! [Pa: fun_Pr5295903952624206638assn_o,X: produc6575502325842934193n_assn] : aa_lis1404944545740551261assn_o(sorted3986126766855596574n_assn(Pa),aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,X),nil_Pr5671120429643327159n_assn)) ).

% sorted_wrt1
tff(fact_3871_sorted__wrt1,axiom,
    ! [Pa: fun_nat_fun_nat_o,X: nat] : aa_list_nat_o(sorted_wrt_nat(Pa),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X),nil_nat)) ).

% sorted_wrt1
tff(fact_3872_sorted__wrt1,axiom,
    ! [Pa: fun_int_fun_int_o,X: int] : aa_list_int_o(sorted_wrt_int(Pa),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X),nil_int)) ).

% sorted_wrt1
tff(fact_3873_sorted__remove1,axiom,
    ! [Xs: list_nat,A: nat] :
      ( aa_list_nat_o(sorted_wrt_nat(ord_less_eq_nat),Xs)
     => aa_list_nat_o(sorted_wrt_nat(ord_less_eq_nat),remove1_nat(A,Xs)) ) ).

% sorted_remove1
tff(fact_3874_sorted__remove1,axiom,
    ! [Xs: list_int,A: int] :
      ( aa_list_int_o(sorted_wrt_int(ord_less_eq_int),Xs)
     => aa_list_int_o(sorted_wrt_int(ord_less_eq_int),remove1_int(A,Xs)) ) ).

% sorted_remove1
tff(fact_3875_sorted__list__of__set_Osorted__sorted__key__list__of__set,axiom,
    ! [A2: set_int] : aa_list_int_o(sorted_wrt_int(ord_less_eq_int),linord2612477271533052124et_int(A2)) ).

% sorted_list_of_set.sorted_sorted_key_list_of_set
tff(fact_3876_sorted__list__of__set_Osorted__sorted__key__list__of__set,axiom,
    ! [A2: set_nat] : aa_list_nat_o(sorted_wrt_nat(ord_less_eq_nat),linord2614967742042102400et_nat(A2)) ).

% sorted_list_of_set.sorted_sorted_key_list_of_set
tff(fact_3877_sorted__list__of__set_Ostrict__sorted__key__list__of__set,axiom,
    ! [A2: set_int] : aa_list_int_o(sorted_wrt_int(ord_less_int),linord2612477271533052124et_int(A2)) ).

% sorted_list_of_set.strict_sorted_key_list_of_set
tff(fact_3878_sorted__list__of__set_Ostrict__sorted__key__list__of__set,axiom,
    ! [A2: set_nat] : aa_list_nat_o(sorted_wrt_nat(ord_less_nat),linord2614967742042102400et_nat(A2)) ).

% sorted_list_of_set.strict_sorted_key_list_of_set
tff(fact_3879_rev__nth,axiom,
    ! [N: nat,Xs: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),size_size_list_nat(Xs))
     => ( aa_nat_nat(nth_nat(rev_nat(Xs)),N) = aa_nat_nat(nth_nat(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,size_size_list_nat(Xs)),aa_nat_nat(suc,N))) ) ) ).

% rev_nth
tff(fact_3880_rev__nth,axiom,
    ! [N: nat,Xs: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),size_size_list_int(Xs))
     => ( aa_nat_int(nth_int(rev_int(Xs)),N) = aa_nat_int(nth_int(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,size_size_list_int(Xs)),aa_nat_nat(suc,N))) ) ) ).

% rev_nth
tff(fact_3881_sorted__mergesort__by__rel,axiom,
    ! [Xs: list_nat] : aa_list_nat_o(sorted_wrt_nat(ord_less_eq_nat),aa_list_nat_list_nat(mergesort_by_rel_nat(ord_less_eq_nat),Xs)) ).

% sorted_mergesort_by_rel
tff(fact_3882_sorted__mergesort__by__rel,axiom,
    ! [Xs: list_int] : aa_list_int_o(sorted_wrt_int(ord_less_eq_int),aa_list_int_list_int(mergesort_by_rel_int(ord_less_eq_int),Xs)) ).

% sorted_mergesort_by_rel
tff(fact_3883_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_int,Zs3: list_a,Ws2: list_a,Pa: fun_li472312507700329092st_a_o] :
      ( ( size_size_list_nat(Xs) = size_size_list_int(Ys) )
     => ( ( size_size_list_int(Ys) = size_size_list_a(Zs3) )
       => ( ( size_size_list_a(Zs3) = size_size_list_a(Ws2) )
         => ( aa_list_a_o(aa_lis4963054610627903775st_a_o(aa_lis6681097040608381428st_a_o(aa_lis4583567994034305133st_a_o(Pa,nil_nat),nil_int),nil_a),nil_a)
           => ( ! [X2: nat,Xs2: list_nat,Y2: int,Ys2: list_int,Z3: a,Zs2: list_a,W3: a,Ws3: list_a] :
                  ( ( size_size_list_nat(Xs2) = size_size_list_int(Ys2) )
                 => ( ( size_size_list_int(Ys2) = size_size_list_a(Zs2) )
                   => ( ( size_size_list_a(Zs2) = size_size_list_a(Ws3) )
                     => ( aa_list_a_o(aa_lis4963054610627903775st_a_o(aa_lis6681097040608381428st_a_o(aa_lis4583567994034305133st_a_o(Pa,Xs2),Ys2),Zs2),Ws3)
                       => aa_list_a_o(aa_lis4963054610627903775st_a_o(aa_lis6681097040608381428st_a_o(aa_lis4583567994034305133st_a_o(Pa,aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X2),Xs2)),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Y2),Ys2)),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,Z3),Zs2)),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,W3),Ws3)) ) ) ) )
             => aa_list_a_o(aa_lis4963054610627903775st_a_o(aa_lis6681097040608381428st_a_o(aa_lis4583567994034305133st_a_o(Pa,Xs),Ys),Zs3),Ws2) ) ) ) ) ) ).

% list_induct4
tff(fact_3884_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_int,Zs3: list_a,Ws2: list_b,Pa: fun_li19591848630169155st_b_o] :
      ( ( size_size_list_nat(Xs) = size_size_list_int(Ys) )
     => ( ( size_size_list_int(Ys) = size_size_list_a(Zs3) )
       => ( ( size_size_list_a(Zs3) = size_size_list_b(Ws2) )
         => ( aa_list_b_o(aa_lis2175599028655227742st_b_o(aa_lis6045448040003394739st_b_o(aa_lis7234238424532900780st_b_o(Pa,nil_nat),nil_int),nil_a),nil_b)
           => ( ! [X2: nat,Xs2: list_nat,Y2: int,Ys2: list_int,Z3: a,Zs2: list_a,W3: b,Ws3: list_b] :
                  ( ( size_size_list_nat(Xs2) = size_size_list_int(Ys2) )
                 => ( ( size_size_list_int(Ys2) = size_size_list_a(Zs2) )
                   => ( ( size_size_list_a(Zs2) = size_size_list_b(Ws3) )
                     => ( aa_list_b_o(aa_lis2175599028655227742st_b_o(aa_lis6045448040003394739st_b_o(aa_lis7234238424532900780st_b_o(Pa,Xs2),Ys2),Zs2),Ws3)
                       => aa_list_b_o(aa_lis2175599028655227742st_b_o(aa_lis6045448040003394739st_b_o(aa_lis7234238424532900780st_b_o(Pa,aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X2),Xs2)),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Y2),Ys2)),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,Z3),Zs2)),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,W3),Ws3)) ) ) ) )
             => aa_list_b_o(aa_lis2175599028655227742st_b_o(aa_lis6045448040003394739st_b_o(aa_lis7234238424532900780st_b_o(Pa,Xs),Ys),Zs3),Ws2) ) ) ) ) ) ).

% list_induct4
tff(fact_3885_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_int,Zs3: list_b,Ws2: list_a,Pa: fun_li6592160461839019653st_a_o] :
      ( ( size_size_list_nat(Xs) = size_size_list_int(Ys) )
     => ( ( size_size_list_int(Ys) = size_size_list_b(Zs3) )
       => ( ( size_size_list_b(Zs3) = size_size_list_a(Ws2) )
         => ( aa_list_a_o(aa_lis7224773892088682016st_a_o(aa_lis8942816322069159669st_a_o(aa_lis5835330142748433646st_a_o(Pa,nil_nat),nil_int),nil_b),nil_a)
           => ( ! [X2: nat,Xs2: list_nat,Y2: int,Ys2: list_int,Z3: b,Zs2: list_b,W3: a,Ws3: list_a] :
                  ( ( size_size_list_nat(Xs2) = size_size_list_int(Ys2) )
                 => ( ( size_size_list_int(Ys2) = size_size_list_b(Zs2) )
                   => ( ( size_size_list_b(Zs2) = size_size_list_a(Ws3) )
                     => ( aa_list_a_o(aa_lis7224773892088682016st_a_o(aa_lis8942816322069159669st_a_o(aa_lis5835330142748433646st_a_o(Pa,Xs2),Ys2),Zs2),Ws3)
                       => aa_list_a_o(aa_lis7224773892088682016st_a_o(aa_lis8942816322069159669st_a_o(aa_lis5835330142748433646st_a_o(Pa,aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X2),Xs2)),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Y2),Ys2)),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,Z3),Zs2)),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,W3),Ws3)) ) ) ) )
             => aa_list_a_o(aa_lis7224773892088682016st_a_o(aa_lis8942816322069159669st_a_o(aa_lis5835330142748433646st_a_o(Pa,Xs),Ys),Zs3),Ws2) ) ) ) ) ) ).

% list_induct4
tff(fact_3886_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_int,Zs3: list_b,Ws2: list_b,Pa: fun_li6139439802768859716st_b_o] :
      ( ( size_size_list_nat(Xs) = size_size_list_int(Ys) )
     => ( ( size_size_list_int(Ys) = size_size_list_b(Zs3) )
       => ( ( size_size_list_b(Zs3) = size_size_list_b(Ws2) )
         => ( aa_list_b_o(aa_lis4437318310116005983st_b_o(aa_lis8307167321464172980st_b_o(aa_lis8486000573247029293st_b_o(Pa,nil_nat),nil_int),nil_b),nil_b)
           => ( ! [X2: nat,Xs2: list_nat,Y2: int,Ys2: list_int,Z3: b,Zs2: list_b,W3: b,Ws3: list_b] :
                  ( ( size_size_list_nat(Xs2) = size_size_list_int(Ys2) )
                 => ( ( size_size_list_int(Ys2) = size_size_list_b(Zs2) )
                   => ( ( size_size_list_b(Zs2) = size_size_list_b(Ws3) )
                     => ( aa_list_b_o(aa_lis4437318310116005983st_b_o(aa_lis8307167321464172980st_b_o(aa_lis8486000573247029293st_b_o(Pa,Xs2),Ys2),Zs2),Ws3)
                       => aa_list_b_o(aa_lis4437318310116005983st_b_o(aa_lis8307167321464172980st_b_o(aa_lis8486000573247029293st_b_o(Pa,aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X2),Xs2)),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Y2),Ys2)),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,Z3),Zs2)),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,W3),Ws3)) ) ) ) )
             => aa_list_b_o(aa_lis4437318310116005983st_b_o(aa_lis8307167321464172980st_b_o(aa_lis8486000573247029293st_b_o(Pa,Xs),Ys),Zs3),Ws2) ) ) ) ) ) ).

% list_induct4
tff(fact_3887_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_int,Zs3: list_a,Ws2: list_nat,Pa: fun_li2388536625477207910_nat_o] :
      ( ( size_size_list_nat(Xs) = size_size_list_int(Ys) )
     => ( ( size_size_list_int(Ys) = size_size_list_a(Zs3) )
       => ( ( size_size_list_a(Zs3) = size_size_list_nat(Ws2) )
         => ( aa_list_nat_o(aa_lis6662471382306767049_nat_o(aa_lis5536955518701567178_nat_o(aa_lis7321481625703335719_nat_o(Pa,nil_nat),nil_int),nil_a),nil_nat)
           => ( ! [X2: nat,Xs2: list_nat,Y2: int,Ys2: list_int,Z3: a,Zs2: list_a,W3: nat,Ws3: list_nat] :
                  ( ( size_size_list_nat(Xs2) = size_size_list_int(Ys2) )
                 => ( ( size_size_list_int(Ys2) = size_size_list_a(Zs2) )
                   => ( ( size_size_list_a(Zs2) = size_size_list_nat(Ws3) )
                     => ( aa_list_nat_o(aa_lis6662471382306767049_nat_o(aa_lis5536955518701567178_nat_o(aa_lis7321481625703335719_nat_o(Pa,Xs2),Ys2),Zs2),Ws3)
                       => aa_list_nat_o(aa_lis6662471382306767049_nat_o(aa_lis5536955518701567178_nat_o(aa_lis7321481625703335719_nat_o(Pa,aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X2),Xs2)),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Y2),Ys2)),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,Z3),Zs2)),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,W3),Ws3)) ) ) ) )
             => aa_list_nat_o(aa_lis6662471382306767049_nat_o(aa_lis5536955518701567178_nat_o(aa_lis7321481625703335719_nat_o(Pa,Xs),Ys),Zs3),Ws2) ) ) ) ) ) ).

% list_induct4
tff(fact_3888_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_int,Zs3: list_b,Ws2: list_nat,Pa: fun_li1744653883581039909_nat_o] :
      ( ( size_size_list_nat(Xs) = size_size_list_int(Ys) )
     => ( ( size_size_list_int(Ys) = size_size_list_b(Zs3) )
       => ( ( size_size_list_b(Zs3) = size_size_list_nat(Ws2) )
         => ( aa_list_nat_o(aa_lis3653689927639571848_nat_o(aa_lis2528174064034371977_nat_o(aa_lis1953204246618813542_nat_o(Pa,nil_nat),nil_int),nil_b),nil_nat)
           => ( ! [X2: nat,Xs2: list_nat,Y2: int,Ys2: list_int,Z3: b,Zs2: list_b,W3: nat,Ws3: list_nat] :
                  ( ( size_size_list_nat(Xs2) = size_size_list_int(Ys2) )
                 => ( ( size_size_list_int(Ys2) = size_size_list_b(Zs2) )
                   => ( ( size_size_list_b(Zs2) = size_size_list_nat(Ws3) )
                     => ( aa_list_nat_o(aa_lis3653689927639571848_nat_o(aa_lis2528174064034371977_nat_o(aa_lis1953204246618813542_nat_o(Pa,Xs2),Ys2),Zs2),Ws3)
                       => aa_list_nat_o(aa_lis3653689927639571848_nat_o(aa_lis2528174064034371977_nat_o(aa_lis1953204246618813542_nat_o(Pa,aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X2),Xs2)),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Y2),Ys2)),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,Z3),Zs2)),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,W3),Ws3)) ) ) ) )
             => aa_list_nat_o(aa_lis3653689927639571848_nat_o(aa_lis2528174064034371977_nat_o(aa_lis1953204246618813542_nat_o(Pa,Xs),Ys),Zs3),Ws2) ) ) ) ) ) ).

% list_induct4
tff(fact_3889_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_int,Zs3: list_a,Ws2: list_int,Pa: fun_li3628523779957217802_int_o] :
      ( ( size_size_list_nat(Xs) = size_size_list_int(Ys) )
     => ( ( size_size_list_int(Ys) = size_size_list_a(Zs3) )
       => ( ( size_size_list_a(Zs3) = size_size_list_int(Ws2) )
         => ( aa_list_int_o(aa_lis4417326414636321133_int_o(aa_lis5608906010924528494_int_o(aa_lis6456045083995990987_int_o(Pa,nil_nat),nil_int),nil_a),nil_int)
           => ( ! [X2: nat,Xs2: list_nat,Y2: int,Ys2: list_int,Z3: a,Zs2: list_a,W3: int,Ws3: list_int] :
                  ( ( size_size_list_nat(Xs2) = size_size_list_int(Ys2) )
                 => ( ( size_size_list_int(Ys2) = size_size_list_a(Zs2) )
                   => ( ( size_size_list_a(Zs2) = size_size_list_int(Ws3) )
                     => ( aa_list_int_o(aa_lis4417326414636321133_int_o(aa_lis5608906010924528494_int_o(aa_lis6456045083995990987_int_o(Pa,Xs2),Ys2),Zs2),Ws3)
                       => aa_list_int_o(aa_lis4417326414636321133_int_o(aa_lis5608906010924528494_int_o(aa_lis6456045083995990987_int_o(Pa,aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X2),Xs2)),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Y2),Ys2)),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,Z3),Zs2)),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,W3),Ws3)) ) ) ) )
             => aa_list_int_o(aa_lis4417326414636321133_int_o(aa_lis5608906010924528494_int_o(aa_lis6456045083995990987_int_o(Pa,Xs),Ys),Zs3),Ws2) ) ) ) ) ) ).

% list_induct4
tff(fact_3890_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_int,Zs3: list_b,Ws2: list_int,Pa: fun_li2984641038061049801_int_o] :
      ( ( size_size_list_nat(Xs) = size_size_list_int(Ys) )
     => ( ( size_size_list_int(Ys) = size_size_list_b(Zs3) )
       => ( ( size_size_list_b(Zs3) = size_size_list_int(Ws2) )
         => ( aa_list_int_o(aa_lis1408544959969125932_int_o(aa_lis2600124556257333293_int_o(aa_lis1087767704911468810_int_o(Pa,nil_nat),nil_int),nil_b),nil_int)
           => ( ! [X2: nat,Xs2: list_nat,Y2: int,Ys2: list_int,Z3: b,Zs2: list_b,W3: int,Ws3: list_int] :
                  ( ( size_size_list_nat(Xs2) = size_size_list_int(Ys2) )
                 => ( ( size_size_list_int(Ys2) = size_size_list_b(Zs2) )
                   => ( ( size_size_list_b(Zs2) = size_size_list_int(Ws3) )
                     => ( aa_list_int_o(aa_lis1408544959969125932_int_o(aa_lis2600124556257333293_int_o(aa_lis1087767704911468810_int_o(Pa,Xs2),Ys2),Zs2),Ws3)
                       => aa_list_int_o(aa_lis1408544959969125932_int_o(aa_lis2600124556257333293_int_o(aa_lis1087767704911468810_int_o(Pa,aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X2),Xs2)),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Y2),Ys2)),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,Z3),Zs2)),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,W3),Ws3)) ) ) ) )
             => aa_list_int_o(aa_lis1408544959969125932_int_o(aa_lis2600124556257333293_int_o(aa_lis1087767704911468810_int_o(Pa,Xs),Ys),Zs3),Ws2) ) ) ) ) ) ).

% list_induct4
tff(fact_3891_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_int,Zs3: list_nat,Ws2: list_a,Pa: fun_li5313178076291648842st_a_o] :
      ( ( size_size_list_nat(Xs) = size_size_list_int(Ys) )
     => ( ( size_size_list_int(Ys) = size_size_list_nat(Zs3) )
       => ( ( size_size_list_nat(Zs3) = size_size_list_a(Ws2) )
         => ( aa_list_a_o(aa_lis6242202807773918837st_a_o(aa_lis8168940518702212782st_a_o(aa_lis2954487798061334283st_a_o(Pa,nil_nat),nil_int),nil_nat),nil_a)
           => ( ! [X2: nat,Xs2: list_nat,Y2: int,Ys2: list_int,Z3: nat,Zs2: list_nat,W3: a,Ws3: list_a] :
                  ( ( size_size_list_nat(Xs2) = size_size_list_int(Ys2) )
                 => ( ( size_size_list_int(Ys2) = size_size_list_nat(Zs2) )
                   => ( ( size_size_list_nat(Zs2) = size_size_list_a(Ws3) )
                     => ( aa_list_a_o(aa_lis6242202807773918837st_a_o(aa_lis8168940518702212782st_a_o(aa_lis2954487798061334283st_a_o(Pa,Xs2),Ys2),Zs2),Ws3)
                       => aa_list_a_o(aa_lis6242202807773918837st_a_o(aa_lis8168940518702212782st_a_o(aa_lis2954487798061334283st_a_o(Pa,aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X2),Xs2)),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Y2),Ys2)),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Z3),Zs2)),aa_list_a_list_a(aa_a_f3048006285668266245list_a(cons_a,W3),Ws3)) ) ) ) )
             => aa_list_a_o(aa_lis6242202807773918837st_a_o(aa_lis8168940518702212782st_a_o(aa_lis2954487798061334283st_a_o(Pa,Xs),Ys),Zs3),Ws2) ) ) ) ) ) ).

% list_induct4
tff(fact_3892_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_int,Zs3: list_nat,Ws2: list_b,Pa: fun_li4860457417221488905st_b_o] :
      ( ( size_size_list_nat(Xs) = size_size_list_int(Ys) )
     => ( ( size_size_list_int(Ys) = size_size_list_nat(Zs3) )
       => ( ( size_size_list_nat(Zs3) = size_size_list_b(Ws2) )
         => ( aa_list_b_o(aa_lis3454747225801242804st_b_o(aa_lis7533291518097226093st_b_o(aa_lis5605158228559929930st_b_o(Pa,nil_nat),nil_int),nil_nat),nil_b)
           => ( ! [X2: nat,Xs2: list_nat,Y2: int,Ys2: list_int,Z3: nat,Zs2: list_nat,W3: b,Ws3: list_b] :
                  ( ( size_size_list_nat(Xs2) = size_size_list_int(Ys2) )
                 => ( ( size_size_list_int(Ys2) = size_size_list_nat(Zs2) )
                   => ( ( size_size_list_nat(Zs2) = size_size_list_b(Ws3) )
                     => ( aa_list_b_o(aa_lis3454747225801242804st_b_o(aa_lis7533291518097226093st_b_o(aa_lis5605158228559929930st_b_o(Pa,Xs2),Ys2),Zs2),Ws3)
                       => aa_list_b_o(aa_lis3454747225801242804st_b_o(aa_lis7533291518097226093st_b_o(aa_lis5605158228559929930st_b_o(Pa,aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X2),Xs2)),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,Y2),Ys2)),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,Z3),Zs2)),aa_list_b_list_b(aa_b_f8935218930876307398list_b(cons_b,W3),Ws3)) ) ) ) )
             => aa_list_b_o(aa_lis3454747225801242804st_b_o(aa_lis7533291518097226093st_b_o(aa_lis5605158228559929930st_b_o(Pa,Xs),Ys),Zs3),Ws2) ) ) ) ) ) ).

% list_induct4
tff(fact_3893_nat__compl__induct_H,axiom,
    ! [Pa: fun_nat_o,N: nat] :
      ( aa_nat_o(Pa,zero_zero_nat)
     => ( ! [N7: nat] :
            ( ! [Nn: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nn),N7)
               => aa_nat_o(Pa,Nn) )
           => aa_nat_o(Pa,aa_nat_nat(suc,N7)) )
       => aa_nat_o(Pa,N) ) ) ).

% nat_compl_induct'
tff(fact_3894_nat__compl__induct,axiom,
    ! [Pa: fun_nat_o,N: nat] :
      ( aa_nat_o(Pa,zero_zero_nat)
     => ( ! [N7: nat] :
            ( ! [Nn: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nn),N7)
               => aa_nat_o(Pa,Nn) )
           => aa_nat_o(Pa,aa_nat_nat(suc,N7)) )
       => aa_nat_o(Pa,N) ) ) ).

% nat_compl_induct
tff(fact_3895_nat__in__between__eq_I1_J,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),aa_nat_nat(suc,A)) )
    <=> ( B = aa_nat_nat(suc,A) ) ) ).

% nat_in_between_eq(1)
tff(fact_3896_nat__in__between__eq_I2_J,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),aa_nat_nat(suc,A)) )
    <=> ( B = A ) ) ).

% nat_in_between_eq(2)
tff(fact_3897_Suc__to__right,axiom,
    ! [N: nat,M2: nat] :
      ( ( aa_nat_nat(suc,N) = M2 )
     => ( N = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),aa_nat_nat(suc,zero_zero_nat)) ) ) ).

% Suc_to_right
tff(fact_3898_nz__le__conv__less,axiom,
    ! [K: nat,M2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),M2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),aa_nat_nat(suc,zero_zero_nat))),M2) ) ) ).

% nz_le_conv_less
tff(fact_3899_Suc__n__minus__m__eq,axiom,
    ! [M2: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),M2)
       => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),one_one_nat)) ) ) ) ).

% Suc_n_minus_m_eq
tff(fact_3900_nth__sorted__list__of__set__greaterThanLessThan,axiom,
    ! [N: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),aa_nat_nat(suc,I)))
     => ( aa_nat_nat(nth_nat(linord2614967742042102400et_nat(aa_nat_set_nat(set_or5834768355832116004an_nat(I),J))),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),N)) ) ) ).

% nth_sorted_list_of_set_greaterThanLessThan
tff(fact_3901_sorted__list__of__set__greaterThanLessThan,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,I)),J)
     => ( linord2614967742042102400et_nat(aa_nat_set_nat(set_or5834768355832116004an_nat(I),J)) = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,aa_nat_nat(suc,I)),linord2614967742042102400et_nat(aa_nat_set_nat(set_or5834768355832116004an_nat(aa_nat_nat(suc,I)),J))) ) ) ).

% sorted_list_of_set_greaterThanLessThan
tff(fact_3902_nth__sorted__list__of__set__greaterThanAtMost,axiom,
    ! [N: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I))
     => ( aa_nat_nat(nth_nat(linord2614967742042102400et_nat(aa_nat_set_nat(set_or6659071591806873216st_nat(I),J))),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),N)) ) ) ).

% nth_sorted_list_of_set_greaterThanAtMost
tff(fact_3903_sorted__list__of__set__lessThan__Suc,axiom,
    ! [K: nat] : linord2614967742042102400et_nat(aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,K))) = aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,linord2614967742042102400et_nat(aa_nat_set_nat(set_ord_lessThan_nat,K))),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,K),nil_nat)) ).

% sorted_list_of_set_lessThan_Suc
tff(fact_3904_sorted__list__of__set__greaterThanAtMost,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,I)),J)
     => ( linord2614967742042102400et_nat(aa_nat_set_nat(set_or6659071591806873216st_nat(I),J)) = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,aa_nat_nat(suc,I)),linord2614967742042102400et_nat(aa_nat_set_nat(set_or6659071591806873216st_nat(aa_nat_nat(suc,I)),J))) ) ) ).

% sorted_list_of_set_greaterThanAtMost
tff(fact_3905_minus__assn__def,axiom,
    ! [A: assn,B: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(minus_minus_assn,A),B) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),aa_assn_assn(uminus_uminus_assn,B)) ).

% minus_assn_def
tff(fact_3906_inf__nat__def,axiom,
    inf_inf_nat = ord_min_nat ).

% inf_nat_def
tff(fact_3907_sorted__list__of__set__atMost__Suc,axiom,
    ! [K: nat] : linord2614967742042102400et_nat(aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,K))) = aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,linord2614967742042102400et_nat(aa_nat_set_nat(set_ord_atMost_nat,K))),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,aa_nat_nat(suc,K)),nil_nat)) ).

% sorted_list_of_set_atMost_Suc
tff(fact_3908_upt__rec__numeral,axiom,
    ! [M2: num,N: num] :
      aa_nat_list_nat(upt(numeral_numeral_nat(M2)),numeral_numeral_nat(N)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,numeral_numeral_nat(M2)),numeral_numeral_nat(N)),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,numeral_numeral_nat(M2)),aa_nat_list_nat(upt(aa_nat_nat(suc,numeral_numeral_nat(M2))),numeral_numeral_nat(N))),nil_nat) ).

% upt_rec_numeral
tff(fact_3909_tl__upt,axiom,
    ! [M2: nat,N: nat] : tl_nat(aa_nat_list_nat(upt(M2),N)) = aa_nat_list_nat(upt(aa_nat_nat(suc,M2)),N) ).

% tl_upt
tff(fact_3910_hd__upt,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( hd_nat(aa_nat_list_nat(upt(I),J)) = I ) ) ).

% hd_upt
tff(fact_3911_drop__upt,axiom,
    ! [M2: nat,I: nat,J: nat] : drop_nat(M2,aa_nat_list_nat(upt(I),J)) = aa_nat_list_nat(upt(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),M2)),J) ).

% drop_upt
tff(fact_3912_take__upt,axiom,
    ! [I: nat,M2: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),M2)),N)
     => ( take_nat(M2,aa_nat_list_nat(upt(I),N)) = aa_nat_list_nat(upt(I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),M2)) ) ) ).

% take_upt
tff(fact_3913_upt__0__eq__Nil__conv,axiom,
    ! [J: nat] :
      ( ( aa_nat_list_nat(upt(zero_zero_nat),J) = nil_nat )
    <=> ( J = zero_zero_nat ) ) ).

% upt_0_eq_Nil_conv
tff(fact_3914_upt__conv__Nil,axiom,
    ! [J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I)
     => ( aa_nat_list_nat(upt(I),J) = nil_nat ) ) ).

% upt_conv_Nil
tff(fact_3915_upt__merge,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),K) )
     => ( aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,aa_nat_list_nat(upt(I),J)),aa_nat_list_nat(upt(J),K)) = aa_nat_list_nat(upt(I),K) ) ) ).

% upt_merge
tff(fact_3916_length__upt,axiom,
    ! [I: nat,J: nat] : size_size_list_nat(aa_nat_list_nat(upt(I),J)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I) ).

% length_upt
tff(fact_3917_last__upt,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( last_nat(aa_nat_list_nat(upt(I),J)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),one_one_nat) ) ) ).

% last_upt
tff(fact_3918_upt__eq__Nil__conv,axiom,
    ! [I: nat,J: nat] :
      ( ( aa_nat_list_nat(upt(I),J) = nil_nat )
    <=> ( ( J = zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I) ) ) ).

% upt_eq_Nil_conv
tff(fact_3919_nth__upt,axiom,
    ! [I: nat,K: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),J)
     => ( aa_nat_nat(nth_nat(aa_nat_list_nat(upt(I),J)),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K) ) ) ).

% nth_upt
tff(fact_3920_map__Suc__upt,axiom,
    ! [M2: nat,N: nat] : aa_list_nat_list_nat(aa_fun4736978287202700568st_nat(map_nat_nat,suc),aa_nat_list_nat(upt(M2),N)) = aa_nat_list_nat(upt(aa_nat_nat(suc,M2)),aa_nat_nat(suc,N)) ).

% map_Suc_upt
tff(fact_3921_distinct__upt,axiom,
    ! [I: nat,J: nat] : aa_list_nat_o(distinct_nat,aa_nat_list_nat(upt(I),J)) ).

% distinct_upt
tff(fact_3922_butlast__upt,axiom,
    ! [M2: nat,N: nat] : butlast_nat(aa_nat_list_nat(upt(M2),N)) = aa_nat_list_nat(upt(M2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) ).

% butlast_upt
tff(fact_3923_atLeast__upt,axiom,
    ! [N: nat] : aa_nat_set_nat(set_ord_lessThan_nat,N) = set_nat2(aa_nat_list_nat(upt(zero_zero_nat),N)) ).

% atLeast_upt
tff(fact_3924_upt__conv__Cons__Cons,axiom,
    ! [M2: nat,N: nat,Ns: list_nat,Q2: nat] :
      ( ( aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,M2),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,N),Ns)) = aa_nat_list_nat(upt(M2),Q2) )
    <=> ( aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,N),Ns) = aa_nat_list_nat(upt(aa_nat_nat(suc,M2)),Q2) ) ) ).

% upt_conv_Cons_Cons
tff(fact_3925_upt__0,axiom,
    ! [I: nat] : aa_nat_list_nat(upt(I),zero_zero_nat) = nil_nat ).

% upt_0
tff(fact_3926_sorted__wrt__upt,axiom,
    ! [M2: nat,N: nat] : aa_list_nat_o(sorted_wrt_nat(ord_less_nat),aa_nat_list_nat(upt(M2),N)) ).

% sorted_wrt_upt
tff(fact_3927_sorted__upt,axiom,
    ! [M2: nat,N: nat] : aa_list_nat_o(sorted_wrt_nat(ord_less_eq_nat),aa_nat_list_nat(upt(M2),N)) ).

% sorted_upt
tff(fact_3928_greaterThanAtMost__upt,axiom,
    ! [N: nat,M2: nat] : aa_nat_set_nat(set_or6659071591806873216st_nat(N),M2) = set_nat2(aa_nat_list_nat(upt(aa_nat_nat(suc,N)),aa_nat_nat(suc,M2))) ).

% greaterThanAtMost_upt
tff(fact_3929_upt__eq__append__conv,axiom,
    ! [I: nat,J: nat,Xs: list_nat,Ys: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( ( aa_nat_list_nat(upt(I),J) = aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Xs),Ys) )
      <=> ? [K4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),K4)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K4),J)
            & ( aa_nat_list_nat(upt(I),K4) = Xs )
            & ( aa_nat_list_nat(upt(K4),J) = Ys ) ) ) ) ).

% upt_eq_append_conv
tff(fact_3930_greaterThanLessThan__upt,axiom,
    ! [N: nat,M2: nat] : aa_nat_set_nat(set_or5834768355832116004an_nat(N),M2) = set_nat2(aa_nat_list_nat(upt(aa_nat_nat(suc,N)),M2)) ).

% greaterThanLessThan_upt
tff(fact_3931_atMost__upto,axiom,
    ! [N: nat] : aa_nat_set_nat(set_ord_atMost_nat,N) = set_nat2(aa_nat_list_nat(upt(zero_zero_nat),aa_nat_nat(suc,N))) ).

% atMost_upto
tff(fact_3932_upt__conv__Cons,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( aa_nat_list_nat(upt(I),J) = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,I),aa_nat_list_nat(upt(aa_nat_nat(suc,I)),J)) ) ) ).

% upt_conv_Cons
tff(fact_3933_upt__append,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,aa_nat_list_nat(upt(zero_zero_nat),I)),aa_nat_list_nat(upt(I),J)) = aa_nat_list_nat(upt(zero_zero_nat),J) ) ) ).

% upt_append
tff(fact_3934_upt__add__eq__append,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_list_nat(upt(I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)) = aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,aa_nat_list_nat(upt(I),J)),aa_nat_list_nat(upt(J),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K))) ) ) ).

% upt_add_eq_append
tff(fact_3935_upt__eq__Cons__conv,axiom,
    ! [I: nat,J: nat,X: nat,Xs: list_nat] :
      ( ( aa_nat_list_nat(upt(I),J) = aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,X),Xs) )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
        & ( I = X )
        & ( aa_nat_list_nat(upt(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),one_one_nat)),J) = Xs ) ) ) ).

% upt_eq_Cons_conv
tff(fact_3936_upt__rec,axiom,
    ! [I: nat,J: nat] :
      aa_nat_list_nat(upt(I),J) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,I),aa_nat_list_nat(upt(aa_nat_nat(suc,I)),J)),nil_nat) ).

% upt_rec
tff(fact_3937_upt__eq__lel__conv,axiom,
    ! [La: nat,H: nat,Is1: list_nat,I: nat,Is2: list_nat] :
      ( ( aa_nat_list_nat(upt(La),H) = aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,Is1),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,I),Is2)) )
    <=> ( ( Is1 = aa_nat_list_nat(upt(La),I) )
        & ( Is2 = aa_nat_list_nat(upt(aa_nat_nat(suc,I)),H) )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,La),I)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),H) ) ) ).

% upt_eq_lel_conv
tff(fact_3938_upt__Suc,axiom,
    ! [I: nat,J: nat] :
      aa_nat_list_nat(upt(I),aa_nat_nat(suc,J)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J),aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,aa_nat_list_nat(upt(I),J)),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,J),nil_nat)),nil_nat) ).

% upt_Suc
tff(fact_3939_upt__Suc__append,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_list_nat(upt(I),aa_nat_nat(suc,J)) = aa_list_nat_list_nat(aa_lis3219301671718821851st_nat(append_nat,aa_nat_list_nat(upt(I),J)),aa_list_nat_list_nat(aa_nat3022453877459183435st_nat(cons_nat,J),nil_nat)) ) ) ).

% upt_Suc_append
tff(fact_3940_sorted__list__of__set__range,axiom,
    ! [M2: nat,N: nat] : linord2614967742042102400et_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(M2),N)) = aa_nat_list_nat(upt(M2),N) ).

% sorted_list_of_set_range
tff(fact_3941_atLeastLessThan__upt,axiom,
    ! [I: nat,J: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(I),J) = set_nat2(aa_nat_list_nat(upt(I),J)) ).

% atLeastLessThan_upt
tff(fact_3942_ex__nat__less__eq,axiom,
    ! [N: nat,Pa: fun_nat_o] :
      ( ? [M3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M3),N)
          & aa_nat_o(Pa,M3) )
    <=> ? [X3: nat] :
          ( aa_set_nat_o(member_nat2(X3),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N))
          & aa_nat_o(Pa,X3) ) ) ).

% ex_nat_less_eq
tff(fact_3943_all__nat__less__eq,axiom,
    ! [N: nat,Pa: fun_nat_o] :
      ( ! [M3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M3),N)
         => aa_nat_o(Pa,M3) )
    <=> ! [X3: nat] :
          ( aa_set_nat_o(member_nat2(X3),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N))
         => aa_nat_o(Pa,X3) ) ) ).

% all_nat_less_eq
tff(fact_3944_upto__aux__rec,axiom,
    ! [I: int,J: int,Js: list_int] :
      upto_aux(I,J,Js) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,J),I),Js,upto_aux(I,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,J),Js))) ).

% upto_aux_rec
tff(fact_3945_upto__rec__numeral_I2_J,axiom,
    ! [M2: num,N: num] :
      aa_int_list_int(upto(numeral_numeral_int(M2)),aa_int_int(uminus_uminus_int,numeral_numeral_int(N))) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,numeral_numeral_int(M2)),aa_int_int(uminus_uminus_int,numeral_numeral_int(N))),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,numeral_numeral_int(M2)),aa_int_list_int(upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,numeral_numeral_int(M2)),one_one_int)),aa_int_int(uminus_uminus_int,numeral_numeral_int(N)))),nil_int) ).

% upto_rec_numeral(2)
tff(fact_3946_upto__rec__numeral_I3_J,axiom,
    ! [M2: num,N: num] :
      aa_int_list_int(upto(aa_int_int(uminus_uminus_int,numeral_numeral_int(M2))),numeral_numeral_int(N)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(M2))),numeral_numeral_int(N)),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(M2))),aa_int_list_int(upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(M2))),one_one_int)),numeral_numeral_int(N))),nil_int) ).

% upto_rec_numeral(3)
tff(fact_3947_upto__rec__numeral_I4_J,axiom,
    ! [M2: num,N: num] :
      aa_int_list_int(upto(aa_int_int(uminus_uminus_int,numeral_numeral_int(M2))),aa_int_int(uminus_uminus_int,numeral_numeral_int(N))) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(M2))),aa_int_int(uminus_uminus_int,numeral_numeral_int(N))),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(M2))),aa_int_list_int(upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(M2))),one_one_int)),aa_int_int(uminus_uminus_int,numeral_numeral_int(N)))),nil_int) ).

% upto_rec_numeral(4)
tff(fact_3948_upto__rec__numeral_I1_J,axiom,
    ! [M2: num,N: num] :
      aa_int_list_int(upto(numeral_numeral_int(M2)),numeral_numeral_int(N)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,numeral_numeral_int(M2)),numeral_numeral_int(N)),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,numeral_numeral_int(M2)),aa_int_list_int(upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,numeral_numeral_int(M2)),one_one_int)),numeral_numeral_int(N))),nil_int) ).

% upto_rec_numeral(1)
tff(fact_3949_nth__upto,axiom,
    ! [I: int,K: nat,J: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),aa_nat_int(semiri1314217659103216013at_int,K))),J)
     => ( aa_nat_int(nth_int(aa_int_list_int(upto(I),J)),K) = aa_int_int(aa_int_fun_int_int(plus_plus_int,I),aa_nat_int(semiri1314217659103216013at_int,K)) ) ) ).

% nth_upto
tff(fact_3950_upto__Nil,axiom,
    ! [I: int,J: int] :
      ( ( aa_int_list_int(upto(I),J) = nil_int )
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,J),I) ) ).

% upto_Nil
tff(fact_3951_upto__Nil2,axiom,
    ! [I: int,J: int] :
      ( ( nil_int = aa_int_list_int(upto(I),J) )
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,J),I) ) ).

% upto_Nil2
tff(fact_3952_upto__empty,axiom,
    ! [J: int,I: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,J),I)
     => ( aa_int_list_int(upto(I),J) = nil_int ) ) ).

% upto_empty
tff(fact_3953_upto__single,axiom,
    ! [I: int] : aa_int_list_int(upto(I),I) = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,I),nil_int) ).

% upto_single
tff(fact_3954_distinct__upto,axiom,
    ! [I: int,J: int] : aa_list_int_o(distinct_int,aa_int_list_int(upto(I),J)) ).

% distinct_upto
tff(fact_3955_sorted__upto,axiom,
    ! [M2: int,N: int] : aa_list_int_o(sorted_wrt_int(ord_less_eq_int),aa_int_list_int(upto(M2),N)) ).

% sorted_upto
tff(fact_3956_sorted__wrt__upto,axiom,
    ! [I: int,J: int] : aa_list_int_o(sorted_wrt_int(ord_less_int),aa_int_list_int(upto(I),J)) ).

% sorted_wrt_upto
tff(fact_3957_upto__code,axiom,
    ! [I: int,J: int] : aa_int_list_int(upto(I),J) = upto_aux(I,J,nil_int) ).

% upto_code
tff(fact_3958_upto__aux__def,axiom,
    ! [I: int,J: int,Js: list_int] : upto_aux(I,J,Js) = aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,aa_int_list_int(upto(I),J)),Js) ).

% upto_aux_def
tff(fact_3959_greaterThanAtMost__upto,axiom,
    ! [I: int,J: int] : set_or6656581121297822940st_int(I,J) = set_int2(aa_int_list_int(upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int)),J)) ).

% greaterThanAtMost_upto
tff(fact_3960_atLeastLessThan__upto,axiom,
    ! [I: int,J: int] : set_or4662586982721622107an_int(I,J) = set_int2(aa_int_list_int(upto(I),aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int))) ).

% atLeastLessThan_upto
tff(fact_3961_upto__split2,axiom,
    ! [I: int,J: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J),K)
       => ( aa_int_list_int(upto(I),K) = aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,aa_int_list_int(upto(I),J)),aa_int_list_int(upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,J),one_one_int)),K)) ) ) ) ).

% upto_split2
tff(fact_3962_upto__split1,axiom,
    ! [I: int,J: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J),K)
       => ( aa_int_list_int(upto(I),K) = aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,aa_int_list_int(upto(I),aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int))),aa_int_list_int(upto(J),K)) ) ) ) ).

% upto_split1
tff(fact_3963_greaterThanLessThan__upto,axiom,
    ! [I: int,J: int] : set_or5832277885323065728an_int(I,J) = set_int2(aa_int_list_int(upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int)),aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int))) ).

% greaterThanLessThan_upto
tff(fact_3964_upto_Osimps,axiom,
    ! [I: int,J: int] :
      aa_int_list_int(upto(I),J) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,I),aa_int_list_int(upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int)),J)),nil_int) ).

% upto.simps
tff(fact_3965_upto_Oelims,axiom,
    ! [X: int,Xa: int,Y: list_int] :
      ( ( aa_int_list_int(upto(X),Xa) = Y )
     => ( Y = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Xa),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X),aa_int_list_int(upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,X),one_one_int)),Xa)),nil_int) ) ) ).

% upto.elims
tff(fact_3966_upto__rec1,axiom,
    ! [I: int,J: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
     => ( aa_int_list_int(upto(I),J) = aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,I),aa_int_list_int(upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int)),J)) ) ) ).

% upto_rec1
tff(fact_3967_upto__rec2,axiom,
    ! [I: int,J: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
     => ( aa_int_list_int(upto(I),J) = aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,aa_int_list_int(upto(I),aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int))),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,J),nil_int)) ) ) ).

% upto_rec2
tff(fact_3968_upto__split3,axiom,
    ! [I: int,J: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J),K)
       => ( aa_int_list_int(upto(I),K) = aa_list_int_list_int(aa_lis7051395851445065967st_int(append_int,aa_int_list_int(upto(I),aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int))),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,J),aa_int_list_int(upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,J),one_one_int)),K))) ) ) ) ).

% upto_split3
tff(fact_3969_all__nat__less,axiom,
    ! [N: nat,Pa: fun_nat_o] :
      ( ! [M3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M3),N)
         => aa_nat_o(Pa,M3) )
    <=> ! [X3: nat] :
          ( aa_set_nat_o(member_nat2(X3),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))
         => aa_nat_o(Pa,X3) ) ) ).

% all_nat_less
tff(fact_3970_ex__nat__less,axiom,
    ! [N: nat,Pa: fun_nat_o] :
      ( ? [M3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M3),N)
          & aa_nat_o(Pa,M3) )
    <=> ? [X3: nat] :
          ( aa_set_nat_o(member_nat2(X3),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))
          & aa_nat_o(Pa,X3) ) ) ).

% ex_nat_less
tff(fact_3971_atLeastAtMost__upt,axiom,
    ! [N: nat,M2: nat] : aa_nat_set_nat(set_or1269000886237332187st_nat(N),M2) = set_nat2(aa_nat_list_nat(upt(N),aa_nat_nat(suc,M2))) ).

% atLeastAtMost_upt
tff(fact_3972_map__add__upt_H,axiom,
    ! [Ofs: nat,A: nat,B: nat] : aa_list_nat_list_nat(aa_fun4736978287202700568st_nat(map_nat_nat,aa_nat_fun_nat_nat(aTP_Lamm_ac,Ofs)),aa_nat_list_nat(upt(A),B)) = aa_nat_list_nat(upt(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),Ofs)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),Ofs)) ).

% map_add_upt'
tff(fact_3973_map__add__upt,axiom,
    ! [N: nat,M2: nat] : aa_list_nat_list_nat(aa_fun4736978287202700568st_nat(map_nat_nat,aa_nat_fun_nat_nat(aTP_Lamm_ac,N)),aa_nat_list_nat(upt(zero_zero_nat),M2)) = aa_nat_list_nat(upt(N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N)) ).

% map_add_upt
tff(fact_3974_map__decr__upt,axiom,
    ! [M2: nat,N: nat] : aa_list_nat_list_nat(aa_fun4736978287202700568st_nat(map_nat_nat,aTP_Lamm_ad),aa_nat_list_nat(upt(aa_nat_nat(suc,M2)),aa_nat_nat(suc,N))) = aa_nat_list_nat(upt(M2),N) ).

% map_decr_upt
tff(fact_3975_atLeastAtMost__upto,axiom,
    ! [I: int,J: int] : set_or1266510415728281911st_int(I,J) = set_int2(aa_int_list_int(upto(I),J)) ).

% atLeastAtMost_upto
tff(fact_3976_upt__filter__extend,axiom,
    ! [U: nat,U2: nat,Pa: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,U),U2)
     => ( ! [I3: nat] :
            ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,U),I3)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),U2) )
           => ~ aa_nat_o(Pa,I3) )
       => ( filter_nat(Pa,aa_nat_list_nat(upt(zero_zero_nat),U)) = filter_nat(Pa,aa_nat_list_nat(upt(zero_zero_nat),U2)) ) ) ) ).

% upt_filter_extend
tff(fact_3977_remdups__upt,axiom,
    ! [M2: nat,N: nat] : remdups_nat(aa_nat_list_nat(upt(M2),N)) = aa_nat_list_nat(upt(M2),N) ).

% remdups_upt
tff(fact_3978_sort__upt,axiom,
    ! [M2: nat,N: nat] : linord738340561235409698at_nat(aTP_Lamm_ae,aa_nat_list_nat(upt(M2),N)) = aa_nat_list_nat(upt(M2),N) ).

% sort_upt
tff(fact_3979_sort__upto,axiom,
    ! [I: int,J: int] : linord1735203802627413978nt_int(aTP_Lamm_af,aa_int_list_int(upto(I),J)) = aa_int_list_int(upto(I),J) ).

% sort_upto
tff(fact_3980_merge__true__star,axiom,
    aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,top_top_assn),top_top_assn) = top_top_assn ).

% merge_true_star
tff(fact_3981_assn__basic__inequalities_I1_J,axiom,
    top_top_assn != one_one_assn ).

% assn_basic_inequalities(1)
tff(fact_3982_assn__basic__inequalities_I5_J,axiom,
    top_top_assn != bot_bot_assn ).

% assn_basic_inequalities(5)
tff(fact_3983_norm__assertion__simps_I12_J,axiom,
    ! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),top_top_assn) = top_top_assn ).

% norm_assertion_simps(12)
tff(fact_3984_norm__assertion__simps_I11_J,axiom,
    ! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,top_top_assn),X) = top_top_assn ).

% norm_assertion_simps(11)
tff(fact_3985_norm__assertion__simps_I3_J,axiom,
    ! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,top_top_assn),X) = X ).

% norm_assertion_simps(3)
tff(fact_3986_norm__assertion__simps_I4_J,axiom,
    ! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),top_top_assn) = X ).

% norm_assertion_simps(4)
tff(fact_3987_merge__true__star__ctx,axiom,
    ! [Pa: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,top_top_assn),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,top_top_assn),Pa)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,top_top_assn),Pa) ).

% merge_true_star_ctx
tff(fact_3988_entails__solve__finalize_I1_J,axiom,
    ! [M: list_P8527749157015355191n_assn,Pa: assn] : aa_assn_o(fI_RESULT(M,Pa,one_one_assn),top_top_assn) ).

% entails_solve_finalize(1)
tff(fact_3989_UNIV__bool,axiom,
    top_top_set_o = insert_o($false,insert_o($true,bot_bot_set_o)) ).

% UNIV_bool
tff(fact_3990_upto_Opelims,axiom,
    ! [X: int,Xa: int,Y: list_int] :
      ( ( aa_int_list_int(upto(X),Xa) = Y )
     => ( accp_P1096762738010456898nt_int(upto_rel,product_Pair_int_int(X,Xa))
       => ~ ( ( Y = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Xa),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,X),aa_int_list_int(upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,X),one_one_int)),Xa)),nil_int) )
           => ~ accp_P1096762738010456898nt_int(upto_rel,product_Pair_int_int(X,Xa)) ) ) ) ).

% upto.pelims
tff(fact_3991_upto_Opinduct,axiom,
    ! [A0: int,A1: int,Pa: fun_int_fun_int_o] :
      ( accp_P1096762738010456898nt_int(upto_rel,product_Pair_int_int(A0,A1))
     => ( ! [I3: int,J3: int] :
            ( accp_P1096762738010456898nt_int(upto_rel,product_Pair_int_int(I3,J3))
           => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I3),J3)
               => aa_int_o(aa_int_fun_int_o(Pa,aa_int_int(aa_int_fun_int_int(plus_plus_int,I3),one_one_int)),J3) )
             => aa_int_o(aa_int_fun_int_o(Pa,I3),J3) ) )
       => aa_int_o(aa_int_fun_int_o(Pa,A0),A1) ) ) ).

% upto.pinduct
tff(fact_3992_upto_Opsimps,axiom,
    ! [I: int,J: int] :
      ( accp_P1096762738010456898nt_int(upto_rel,product_Pair_int_int(I,J))
     => ( aa_int_list_int(upto(I),J) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J),aa_list_int_list_int(aa_int854396256184503903st_int(cons_int,I),aa_int_list_int(upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int)),J)),nil_int) ) ) ).

% upto.psimps
tff(fact_3993_length__upto,axiom,
    ! [I: int,J: int] : size_size_list_int(aa_int_list_int(upto(I),J)) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),I)),one_one_int)) ).

% length_upto
tff(fact_3994_list__ex__iff__not__all__inverval__int,axiom,
    ! [Pa: fun_int_o,I: int,J: int] :
      ( aa_list_int_o(list_ex_int(Pa),aa_int_list_int(upto(I),J))
    <=> ~ aa_int_o(aa_int_fun_int_o(all_interval_int(comp_o_o_int(fNot,Pa)),I),J) ) ).

% list_ex_iff_not_all_inverval_int
tff(fact_3995_all__interval__int__def,axiom,
    ! [Pa: fun_int_o,I: int,J: int] :
      ( aa_int_o(aa_int_fun_int_o(all_interval_int(Pa),I),J)
    <=> ! [X3: int] :
          ( aa_set_int_o(member_int2(X3),set_or1266510415728281911st_int(I,J))
         => aa_int_o(Pa,X3) ) ) ).

% all_interval_int_def
tff(fact_3996_list__all__iff__all__interval__int,axiom,
    ! [Pa: fun_int_o,I: int,J: int] :
      ( aa_list_int_o(list_all_int(Pa),aa_int_list_int(upto(I),J))
    <=> aa_int_o(aa_int_fun_int_o(all_interval_int(Pa),I),J) ) ).

% list_all_iff_all_interval_int
tff(fact_3997_list__all__iff__all__interval__nat,axiom,
    ! [Pa: fun_nat_o,I: nat,J: nat] :
      ( aa_list_nat_o(list_all_nat(Pa),aa_nat_list_nat(upt(I),J))
    <=> aa_nat_o(aa_nat_fun_nat_o(all_interval_nat(Pa),I),J) ) ).

% list_all_iff_all_interval_nat
tff(fact_3998_all__interval__nat__def,axiom,
    ! [Pa: fun_nat_o,I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(all_interval_nat(Pa),I),J)
    <=> ! [X3: nat] :
          ( aa_set_nat_o(member_nat2(X3),aa_nat_set_nat(set_or4665077453230672383an_nat(I),J))
         => aa_nat_o(Pa,X3) ) ) ).

% all_interval_nat_def
tff(fact_3999_list__ex__iff__not__all__inverval__nat,axiom,
    ! [Pa: fun_nat_o,I: nat,J: nat] :
      ( aa_list_nat_o(list_ex_nat(Pa),aa_nat_list_nat(upt(I),J))
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(all_interval_nat(comp_o_o_nat(fNot,Pa)),I),J) ) ).

% list_ex_iff_not_all_inverval_nat
tff(fact_4000_FI__RESULT__def,axiom,
    ! [M: list_P8527749157015355191n_assn,UP: assn,UQ: assn,F2: assn] :
      ( aa_assn_o(fI_RESULT(M,UP,UQ),F2)
    <=> ( ! [X3: produc6575502325842934193n_assn] :
            ( member7957490590177025114n_assn(X3,set_Pr1139785259514867910n_assn(M))
           => aa_Pro2491721552408928739assn_o(produc7274209992780475162assn_o(entails),X3) )
       => aa_assn_o(aa_assn_fun_assn_o(entails,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(foldr_assn_assn(times_times_assn,aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,produc9167289414957590229n_assn),M)),one_one_assn)),UP)),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(foldr_assn_assn(times_times_assn,aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,produc2051961928117032727n_assn),M)),one_one_assn)),UQ)),F2)) ) ) ).

% FI_RESULT_def
tff(fact_4001_ent__pure__pre__iff,axiom,
    ! [Pa: assn,B: $o,Q: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Pa),aa_o_assn(pure_assn,(B)))),Q)
    <=> ( (B)
       => aa_assn_o(aa_assn_fun_assn_o(entails,Pa),Q) ) ) ).

% ent_pure_pre_iff
tff(fact_4002_ent__pure__pre__iff__sng,axiom,
    ! [B: $o,Q: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,aa_o_assn(pure_assn,(B))),Q)
    <=> ( (B)
       => aa_assn_o(aa_assn_fun_assn_o(entails,one_one_assn),Q) ) ) ).

% ent_pure_pre_iff_sng
tff(fact_4003_ent__conjI,axiom,
    ! [A2: assn,B2: assn,C3: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,A2),B2)
     => ( aa_assn_o(aa_assn_fun_assn_o(entails,A2),C3)
       => aa_assn_o(aa_assn_fun_assn_o(entails,A2),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,B2),C3)) ) ) ).

% ent_conjI
tff(fact_4004_ent__conjE1,axiom,
    ! [A2: assn,C3: assn,B2: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,A2),C3)
     => aa_assn_o(aa_assn_fun_assn_o(entails,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A2),B2)),C3) ) ).

% ent_conjE1
tff(fact_4005_ent__conjE2,axiom,
    ! [B2: assn,C3: assn,A2: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,B2),C3)
     => aa_assn_o(aa_assn_fun_assn_o(entails,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A2),B2)),C3) ) ).

% ent_conjE2
tff(fact_4006_ent__trans,axiom,
    ! [Pa: assn,Q: assn,R: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,Pa),Q)
     => ( aa_assn_o(aa_assn_fun_assn_o(entails,Q),R)
       => aa_assn_o(aa_assn_fun_assn_o(entails,Pa),R) ) ) ).

% ent_trans
tff(fact_4007_ent__refl,axiom,
    ! [Pa: assn] : aa_assn_o(aa_assn_fun_assn_o(entails,Pa),Pa) ).

% ent_refl
tff(fact_4008_ent__iffI,axiom,
    ! [A2: assn,B2: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,A2),B2)
     => ( aa_assn_o(aa_assn_fun_assn_o(entails,B2),A2)
       => ( A2 = B2 ) ) ) ).

% ent_iffI
tff(fact_4009_is__entails,axiom,
    ! [Pa: assn,Q: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,Pa),Q)
     => aa_assn_o(aa_assn_fun_assn_o(entails,Pa),Q) ) ).

% is_entails
tff(fact_4010_ent__star__mono,axiom,
    ! [Pa: assn,P4: assn,Q: assn,Q3: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,Pa),P4)
     => ( aa_assn_o(aa_assn_fun_assn_o(entails,Q),Q3)
       => aa_assn_o(aa_assn_fun_assn_o(entails,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Pa),Q)),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P4),Q3)) ) ) ).

% ent_star_mono
tff(fact_4011_ent__frame__fwd,axiom,
    ! [Pa: assn,R: assn,Ps2: assn,F2: assn,Q: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,Pa),R)
     => ( aa_assn_o(aa_assn_fun_assn_o(entails,Ps2),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Pa),F2))
       => ( aa_assn_o(aa_assn_fun_assn_o(entails,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,R),F2)),Q)
         => aa_assn_o(aa_assn_fun_assn_o(entails,Ps2),Q) ) ) ) ).

% ent_frame_fwd
tff(fact_4012_fr__rot__rhs,axiom,
    ! [A2: assn,B2: assn,C3: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,A2),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B2),C3))
     => aa_assn_o(aa_assn_fun_assn_o(entails,A2),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,C3),B2)) ) ).

% fr_rot_rhs
tff(fact_4013_fr__refl,axiom,
    ! [A2: assn,B2: assn,C3: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,A2),B2)
     => aa_assn_o(aa_assn_fun_assn_o(entails,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A2),C3)),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B2),C3)) ) ).

% fr_refl
tff(fact_4014_fr__rot,axiom,
    ! [A2: assn,B2: assn,C3: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A2),B2)),C3)
     => aa_assn_o(aa_assn_fun_assn_o(entails,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B2),A2)),C3) ) ).

% fr_rot
tff(fact_4015_ent__true,axiom,
    ! [Pa: assn] : aa_assn_o(aa_assn_fun_assn_o(entails,Pa),top_top_assn) ).

% ent_true
tff(fact_4016_ent__disjI2__direct,axiom,
    ! [B2: assn,A2: assn] : aa_assn_o(aa_assn_fun_assn_o(entails,B2),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A2),B2)) ).

% ent_disjI2_direct
tff(fact_4017_ent__disjI1__direct,axiom,
    ! [A2: assn,B2: assn] : aa_assn_o(aa_assn_fun_assn_o(entails,A2),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A2),B2)) ).

% ent_disjI1_direct
tff(fact_4018_ent__disjI2_H,axiom,
    ! [A2: assn,C3: assn,B2: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,A2),C3)
     => aa_assn_o(aa_assn_fun_assn_o(entails,A2),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,B2),C3)) ) ).

% ent_disjI2'
tff(fact_4019_ent__disjI1_H,axiom,
    ! [A2: assn,B2: assn,C3: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,A2),B2)
     => aa_assn_o(aa_assn_fun_assn_o(entails,A2),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,B2),C3)) ) ).

% ent_disjI1'
tff(fact_4020_ent__disjI2,axiom,
    ! [Pa: assn,Q: assn,R: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Pa),Q)),R)
     => aa_assn_o(aa_assn_fun_assn_o(entails,Q),R) ) ).

% ent_disjI2
tff(fact_4021_ent__disjI1,axiom,
    ! [Pa: assn,Q: assn,R: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Pa),Q)),R)
     => aa_assn_o(aa_assn_fun_assn_o(entails,Pa),R) ) ).

% ent_disjI1
tff(fact_4022_ent__disjE,axiom,
    ! [A2: assn,C3: assn,B2: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,A2),C3)
     => ( aa_assn_o(aa_assn_fun_assn_o(entails,B2),C3)
       => aa_assn_o(aa_assn_fun_assn_o(entails,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A2),B2)),C3) ) ) ).

% ent_disjE
tff(fact_4023_ent__false,axiom,
    ! [Pa: assn] : aa_assn_o(aa_assn_fun_assn_o(entails,bot_bot_assn),Pa) ).

% ent_false
tff(fact_4024_ent__star__mono__true,axiom,
    ! [A2: assn,A10: assn,B2: assn,B8: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,A2),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A10),top_top_assn))
     => ( aa_assn_o(aa_assn_fun_assn_o(entails,B2),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B8),top_top_assn))
       => aa_assn_o(aa_assn_fun_assn_o(entails,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A2),B2)),top_top_assn)),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A10),B8)),top_top_assn)) ) ) ).

% ent_star_mono_true
tff(fact_4025_ent__refl__true,axiom,
    ! [A2: assn] : aa_assn_o(aa_assn_fun_assn_o(entails,A2),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A2),top_top_assn)) ).

% ent_refl_true
tff(fact_4026_ent__true__drop_I1_J,axiom,
    ! [Pa: assn,Q: assn,R: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,Pa),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Q),top_top_assn))
     => aa_assn_o(aa_assn_fun_assn_o(entails,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Pa),R)),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Q),top_top_assn)) ) ).

% ent_true_drop(1)
tff(fact_4027_ent__true__drop_I2_J,axiom,
    ! [Pa: assn,Q: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,Pa),Q)
     => aa_assn_o(aa_assn_fun_assn_o(entails,Pa),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Q),top_top_assn)) ) ).

% ent_true_drop(2)
tff(fact_4028_fi__match__entails,axiom,
    ! [M2: list_P8527749157015355191n_assn] :
      ( ! [X2: produc6575502325842934193n_assn] :
          ( member7957490590177025114n_assn(X2,set_Pr1139785259514867910n_assn(M2))
         => aa_Pro2491721552408928739assn_o(produc7274209992780475162assn_o(entails),X2) )
     => aa_assn_o(aa_assn_fun_assn_o(entails,aa_assn_assn(foldr_assn_assn(times_times_assn,aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,produc9167289414957590229n_assn),M2)),one_one_assn)),aa_assn_assn(foldr_assn_assn(times_times_assn,aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,produc2051961928117032727n_assn),M2)),one_one_assn)) ) ).

% fi_match_entails
tff(fact_4029_FI__QUERY__def,axiom,
    ! [Pa: assn,Q: assn,F2: assn] :
      ( aa_assn_o(fI_QUERY(Pa,Q),F2)
    <=> aa_assn_o(aa_assn_fun_assn_o(entails,Pa),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Q),F2)) ) ).

% FI_QUERY_def
tff(fact_4030_frame__inference__init,axiom,
    ! [Pa: assn,Q: assn,F2: assn] :
      ( aa_assn_o(fI_QUERY(Pa,Q),F2)
     => aa_assn_o(aa_assn_fun_assn_o(entails,Pa),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Q),F2)) ) ).

% frame_inference_init
tff(fact_4031_entails__solve__init_I2_J,axiom,
    ! [Pa: assn,Q: assn] :
      ( aa_assn_o(fI_QUERY(Pa,Q),one_one_assn)
     => aa_assn_o(aa_assn_fun_assn_o(entails,Pa),Q) ) ).

% entails_solve_init(2)
tff(fact_4032_entails__solve__init_I1_J,axiom,
    ! [Pa: assn,Q: assn] :
      ( aa_assn_o(fI_QUERY(Pa,Q),top_top_assn)
     => aa_assn_o(aa_assn_fun_assn_o(entails,Pa),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Q),top_top_assn)) ) ).

% entails_solve_init(1)
tff(fact_4033_FI__match,axiom,
    ! [P3: assn,Q2: assn,M2: list_P8527749157015355191n_assn,Ps: assn,Up: assn,Qs: assn,Uq: assn,F: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,P3),Q2)
     => ( aa_assn_o(fi(aa_lis2797772165808192636n_assn(aa_Pro4357298927425265272n_assn(cons_P2971678138204555879n_assn,produc118845697133431529n_assn(P3,Q2)),M2),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Ps),Up),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Qs),Uq),sln,sln),F)
       => aa_assn_o(fi(M2,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Ps),P3),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Qs),Q2),Up,Uq),F) ) ) ).

% FI_match
tff(fact_4034_FI__def,axiom,
    ! [M2: list_P8527749157015355191n_assn,P3: assn,Q2: assn,Up: assn,Uq: assn,F: assn] :
      ( aa_assn_o(fi(M2,P3,Q2,Up,Uq),F)
    <=> ( ! [X3: produc6575502325842934193n_assn] :
            ( member7957490590177025114n_assn(X3,set_Pr1139785259514867910n_assn(M2))
           => aa_Pro2491721552408928739assn_o(produc7274209992780475162assn_o(entails),X3) )
       => aa_assn_o(aa_assn_fun_assn_o(entails,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(foldr_assn_assn(times_times_assn,aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,produc9167289414957590229n_assn),M2)),one_one_assn)),P3)),Up)),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(foldr_assn_assn(times_times_assn,aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,produc2051961928117032727n_assn),M2)),one_one_assn)),Q2)),Uq)),F)) ) ) ).

% FI_def
tff(fact_4035_ent__wand__frameI,axiom,
    ! [Q: assn,R: assn,F2: assn,S: assn,Pa: assn,X5: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(wand_assn,Q),R)),F2)),S)
     => ( aa_assn_o(aa_assn_fun_assn_o(entails,Pa),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,F2),X5))
       => ( aa_assn_o(aa_assn_fun_assn_o(entails,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Q),X5)),R)
         => aa_assn_o(aa_assn_fun_assn_o(entails,Pa),S) ) ) ) ).

% ent_wand_frameI
tff(fact_4036_ent__wandI,axiom,
    ! [Q: assn,Pa: assn,R: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Q),Pa)),R)
     => aa_assn_o(aa_assn_fun_assn_o(entails,Pa),aa_assn_assn(aa_ass954072293579886259n_assn(wand_assn,Q),R)) ) ).

% ent_wandI
tff(fact_4037_ent__mp,axiom,
    ! [Pa: assn,Q: assn] : aa_assn_o(aa_assn_fun_assn_o(entails,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Pa),aa_assn_assn(aa_ass954072293579886259n_assn(wand_assn,Pa),Q))),Q) ).

% ent_mp
tff(fact_4038_Rep__assn__inject,axiom,
    ! [X: assn,Y: assn] :
      ( ( aa_ass4243947775541419017_nat_o(rep_assn,X) = aa_ass4243947775541419017_nat_o(rep_assn,Y) )
    <=> ( X = Y ) ) ).

% Rep_assn_inject
tff(fact_4039_mod__or__dist,axiom,
    ! [Pa: assn,Q: assn,H: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Pa),Q)),H)
    <=> ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Pa),H)
        | aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Q),H) ) ) ).

% mod_or_dist
tff(fact_4040_mod__h__bot__iff_I5_J,axiom,
    ! [Pa: assn,Q: assn,H: heap_e7401611519738050253t_unit] :
      ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Pa),Q)),produc7507926704131184380et_nat(H,bot_bot_set_nat))
    <=> ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Pa),produc7507926704131184380et_nat(H,bot_bot_set_nat))
        & aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Q),produc7507926704131184380et_nat(H,bot_bot_set_nat)) ) ) ).

% mod_h_bot_iff(5)
tff(fact_4041_mod__pure__star__dist,axiom,
    ! [Pa: assn,B: $o,H: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Pa),aa_o_assn(pure_assn,(B)))),H)
    <=> ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Pa),H)
        & (B) ) ) ).

% mod_pure_star_dist
tff(fact_4042_mod__h__bot__iff_I7_J,axiom,
    ! [Pa: assn,Q: assn,H: heap_e7401611519738050253t_unit] :
      ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Pa),Q)),produc7507926704131184380et_nat(H,bot_bot_set_nat))
    <=> ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Pa),produc7507926704131184380et_nat(H,bot_bot_set_nat))
        | aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Q),produc7507926704131184380et_nat(H,bot_bot_set_nat)) ) ) ).

% mod_h_bot_iff(7)
tff(fact_4043_mod__h__bot__iff_I6_J,axiom,
    ! [Pa: assn,Q: assn,H: heap_e7401611519738050253t_unit] :
      ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Pa),Q)),produc7507926704131184380et_nat(H,bot_bot_set_nat))
    <=> ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Pa),produc7507926704131184380et_nat(H,bot_bot_set_nat))
        & aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Q),produc7507926704131184380et_nat(H,bot_bot_set_nat)) ) ) ).

% mod_h_bot_iff(6)
tff(fact_4044_mod__pure,axiom,
    ! [B: $o,H: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,aa_o_assn(pure_assn,(B))),H)
    <=> ( ( produc8586169260539613262et_nat(H) = bot_bot_set_nat )
        & (B) ) ) ).

% mod_pure
tff(fact_4045_mod__h__bot__iff_I1_J,axiom,
    ! [B: $o,H: heap_e7401611519738050253t_unit] :
      ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,aa_o_assn(pure_assn,(B))),produc7507926704131184380et_nat(H,bot_bot_set_nat))
    <=> (B) ) ).

% mod_h_bot_iff(1)
tff(fact_4046_ent__false__iff,axiom,
    ! [Pa: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,Pa),bot_bot_assn)
    <=> ! [H2: produc3658429121746597890et_nat] : ~ aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Pa),H2) ) ).

% ent_false_iff
tff(fact_4047_ent__pure__post__iff,axiom,
    ! [Pa: assn,Q: assn,B: $o] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,Pa),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Q),aa_o_assn(pure_assn,(B))))
    <=> ( ! [H2: produc3658429121746597890et_nat] :
            ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Pa),H2)
           => (B) )
        & aa_assn_o(aa_assn_fun_assn_o(entails,Pa),Q) ) ) ).

% ent_pure_post_iff
tff(fact_4048_ent__pure__post__iff__sng,axiom,
    ! [Pa: assn,B: $o] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,Pa),aa_o_assn(pure_assn,(B)))
    <=> ( ! [H2: produc3658429121746597890et_nat] :
            ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Pa),H2)
           => (B) )
        & aa_assn_o(aa_assn_fun_assn_o(entails,Pa),one_one_assn) ) ) ).

% ent_pure_post_iff_sng
tff(fact_4049_mod__h__bot__indep,axiom,
    ! [Pa: assn,H: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit] :
      ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Pa),produc7507926704131184380et_nat(H,bot_bot_set_nat))
    <=> aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Pa),produc7507926704131184380et_nat(H3,bot_bot_set_nat)) ) ).

% mod_h_bot_indep
tff(fact_4050_mod__and__dist,axiom,
    ! [Pa: assn,Q: assn,H: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Pa),Q)),H)
    <=> ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Pa),H)
        & aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Q),H) ) ) ).

% mod_and_dist
tff(fact_4051_mod__starD,axiom,
    ! [A2: assn,B2: assn,H: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A2),B2)),H)
     => ? [H1: produc3658429121746597890et_nat,H22: produc3658429121746597890et_nat] :
          ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,A2),H1)
          & aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,B2),H22) ) ) ).

% mod_starD
tff(fact_4052_mod__starE,axiom,
    ! [A: assn,B: assn,H: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),B)),H)
     => ~ ( ? [X_1: produc3658429121746597890et_nat] : aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,A),X_1)
         => ! [H_2: produc3658429121746597890et_nat] : ~ aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,B),H_2) ) ) ).

% mod_starE
tff(fact_4053_mod__false,axiom,
    ! [H: produc3658429121746597890et_nat] : ~ aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,bot_bot_assn),H) ).

% mod_false
tff(fact_4054_ent__fwd,axiom,
    ! [Pa: assn,H: produc3658429121746597890et_nat,Q: assn] :
      ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Pa),H)
     => ( aa_assn_o(aa_assn_fun_assn_o(entails,Pa),Q)
       => aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Q),H) ) ) ).

% ent_fwd
tff(fact_4055_entailsD,axiom,
    ! [Pa: assn,Q: assn,H: produc3658429121746597890et_nat] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,Pa),Q)
     => ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Pa),H)
       => aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Q),H) ) ) ).

% entailsD
tff(fact_4056_entailsI,axiom,
    ! [Pa: assn,Q: assn] :
      ( ! [H4: produc3658429121746597890et_nat] :
          ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Pa),H4)
         => aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Q),H4) )
     => aa_assn_o(aa_assn_fun_assn_o(entails,Pa),Q) ) ).

% entailsI
tff(fact_4057_entails__def,axiom,
    ! [Pa: assn,Q: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,Pa),Q)
    <=> ! [H2: produc3658429121746597890et_nat] :
          ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Pa),H2)
         => aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Q),H2) ) ) ).

% entails_def
tff(fact_4058_mod__frame__fwd,axiom,
    ! [Ps2: assn,H: produc3658429121746597890et_nat,Pa: assn,R: assn,F2: assn] :
      ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Ps2),H)
     => ( aa_assn_o(aa_assn_fun_assn_o(entails,Pa),R)
       => ( aa_assn_o(aa_assn_fun_assn_o(entails,Ps2),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Pa),F2))
         => aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,R),F2)),H) ) ) ) ).

% mod_frame_fwd
tff(fact_4059_star__assnI,axiom,
    ! [Pa: assn,H: heap_e7401611519738050253t_unit,As2: set_nat,Q: assn,As3: set_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Pa),produc7507926704131184380et_nat(H,As2))
     => ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Q),produc7507926704131184380et_nat(H,As3))
       => ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As2),As3) = bot_bot_set_nat )
         => aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Pa),Q)),produc7507926704131184380et_nat(H,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As2),As3))) ) ) ) ).

% star_assnI
tff(fact_4060_mod__star__conv,axiom,
    ! [A2: assn,B2: assn,H: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A2),B2)),H)
    <=> ? [Hr: heap_e7401611519738050253t_unit,As1: set_nat,As22: set_nat] :
          ( ( H = produc7507926704131184380et_nat(Hr,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As1),As22)) )
          & ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As1),As22) = bot_bot_set_nat )
          & aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,A2),produc7507926704131184380et_nat(Hr,As1))
          & aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,B2),produc7507926704131184380et_nat(Hr,As22)) ) ) ).

% mod_star_conv
tff(fact_4061_mod__star__trueE_H,axiom,
    ! [Pa: assn,H: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Pa),top_top_assn)),H)
     => ~ ! [H5: produc3658429121746597890et_nat] :
            ( ( produc1824681642469235216et_nat(H5) = produc1824681642469235216et_nat(H) )
           => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,produc8586169260539613262et_nat(H5)),produc8586169260539613262et_nat(H))
             => ~ aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Pa),H5) ) ) ) ).

% mod_star_trueE'
tff(fact_4062_mod__star__trueI,axiom,
    ! [Pa: assn,H: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Pa),H)
     => aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Pa),top_top_assn)),H) ) ).

% mod_star_trueI
tff(fact_4063_mod__star__trueE,axiom,
    ! [Pa: assn,H: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Pa),top_top_assn)),H)
     => ~ ! [H5: produc3658429121746597890et_nat] : ~ aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Pa),H5) ) ).

% mod_star_trueE
tff(fact_4064_mod__h__bot__iff_I2_J,axiom,
    ! [H: heap_e7401611519738050253t_unit] : aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,top_top_assn),produc7507926704131184380et_nat(H,bot_bot_set_nat)) ).

% mod_h_bot_iff(2)
tff(fact_4065_mod__emp,axiom,
    ! [H: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,one_one_assn),H)
    <=> ( produc8586169260539613262et_nat(H) = bot_bot_set_nat ) ) ).

% mod_emp
tff(fact_4066_mod__emp__simp,axiom,
    ! [H: heap_e7401611519738050253t_unit] : aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,one_one_assn),produc7507926704131184380et_nat(H,bot_bot_set_nat)) ).

% mod_emp_simp
tff(fact_4067_mod__not__dist,axiom,
    ! [Pa: assn,H: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,aa_assn_assn(uminus_uminus_assn,Pa)),H)
    <=> ( aa_Pro653082453696054616_nat_o(in_range,H)
        & ~ aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Pa),H) ) ) ).

% mod_not_dist
tff(fact_4068_in__range__empty,axiom,
    ! [H: heap_e7401611519738050253t_unit] : aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H,bot_bot_set_nat)) ).

% in_range_empty
tff(fact_4069_mod__true,axiom,
    ! [H: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,top_top_assn),H)
    <=> aa_Pro653082453696054616_nat_o(in_range,H) ) ).

% mod_true
tff(fact_4070_in__range__dist__union,axiom,
    ! [H: heap_e7401611519738050253t_unit,As2: set_nat,As3: set_nat] :
      ( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As2),As3)))
    <=> ( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H,As2))
        & aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H,As3)) ) ) ).

% in_range_dist_union
tff(fact_4071_models__in__range,axiom,
    ! [Pa: assn,H: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Pa),H)
     => aa_Pro653082453696054616_nat_o(in_range,H) ) ).

% models_in_range
tff(fact_4072_in__range__subset,axiom,
    ! [As2: set_nat,As3: set_nat,H: heap_e7401611519738050253t_unit] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,As2),As3)
     => ( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H,As3))
       => aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H,As2)) ) ) ).

% in_range_subset
tff(fact_4073_one__assn__raw_Ocases,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ~ ! [H4: heap_e7401611519738050253t_unit,As: set_nat] : X != produc7507926704131184380et_nat(H4,As) ).

% one_assn_raw.cases
tff(fact_4074_times__assn__raw_Ocases,axiom,
    ! [X: produc3112576180687729086et_nat] :
      ~ ! [P: fun_Pr6460944699146269935_nat_o,Q4: fun_Pr6460944699146269935_nat_o,H4: heap_e7401611519738050253t_unit,As: set_nat] : X != produc7291855603632349880et_nat(P,produc6017135503889078682et_nat(Q4,produc7507926704131184380et_nat(H4,As))) ).

% times_assn_raw.cases
tff(fact_4075_wand__assnI,axiom,
    ! [H: heap_e7401611519738050253t_unit,As2: set_nat,Q: assn,R: assn] :
      ( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H,As2))
     => ( ! [H5: heap_e7401611519738050253t_unit,As4: set_nat] :
            ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As2),As4) = bot_bot_set_nat )
           => ( relH(As2,H,H5)
             => ( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H5,As2))
               => ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Q),produc7507926704131184380et_nat(H5,As4))
                 => aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,R),produc7507926704131184380et_nat(H5,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As2),As4))) ) ) ) )
       => aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,aa_assn_assn(aa_ass954072293579886259n_assn(wand_assn,Q),R)),produc7507926704131184380et_nat(H,As2)) ) ) ).

% wand_assnI
tff(fact_4076_times__assn__raw_Osimps,axiom,
    ! [Pa: fun_Pr6460944699146269935_nat_o,Q: fun_Pr6460944699146269935_nat_o,H: heap_e7401611519738050253t_unit,As2: set_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(aa_fun6000889628616002998_nat_o(times_assn_raw,Pa),Q),produc7507926704131184380et_nat(H,As2))
    <=> ? [As1: set_nat,As22: set_nat] :
          ( ( As2 = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As1),As22) )
          & ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As1),As22) = bot_bot_set_nat )
          & aa_Pro653082453696054616_nat_o(Pa,produc7507926704131184380et_nat(H,As1))
          & aa_Pro653082453696054616_nat_o(Q,produc7507926704131184380et_nat(H,As22)) ) ) ).

% times_assn_raw.simps
tff(fact_4077_times__assn__raw_Oelims_I1_J,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,Xa: fun_Pr6460944699146269935_nat_o,Xb: produc3658429121746597890et_nat,Y: $o] :
      ( ( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(aa_fun6000889628616002998_nat_o(times_assn_raw,X),Xa),Xb)
      <=> (Y) )
     => ~ ! [H4: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( Xb = produc7507926704131184380et_nat(H4,As) )
           => ( (Y)
            <=> ~ ? [As1: set_nat,As22: set_nat] :
                    ( ( As = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As1),As22) )
                    & ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As1),As22) = bot_bot_set_nat )
                    & aa_Pro653082453696054616_nat_o(X,produc7507926704131184380et_nat(H4,As1))
                    & aa_Pro653082453696054616_nat_o(Xa,produc7507926704131184380et_nat(H4,As22)) ) ) ) ) ).

% times_assn_raw.elims(1)
tff(fact_4078_relH__dist__union,axiom,
    ! [As2: set_nat,As3: set_nat,H: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit] :
      ( relH(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As2),As3),H,H3)
    <=> ( relH(As2,H,H3)
        & relH(As3,H,H3) ) ) ).

% relH_dist_union
tff(fact_4079_relH__subset,axiom,
    ! [Bs: set_nat,H: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit,As2: set_nat] :
      ( relH(Bs,H,H3)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,As2),Bs)
       => relH(As2,H,H3) ) ) ).

% relH_subset
tff(fact_4080_relH__trans,axiom,
    ! [As2: set_nat,H12: heap_e7401611519738050253t_unit,H23: heap_e7401611519738050253t_unit,H32: heap_e7401611519738050253t_unit] :
      ( relH(As2,H12,H23)
     => ( relH(As2,H23,H32)
       => relH(As2,H12,H32) ) ) ).

% relH_trans
tff(fact_4081_relH__sym,axiom,
    ! [As2: set_nat,H: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit] :
      ( relH(As2,H,H3)
     => relH(As2,H3,H) ) ).

% relH_sym
tff(fact_4082_mod__relH,axiom,
    ! [As2: set_nat,H: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit,Pa: assn] :
      ( relH(As2,H,H3)
     => ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Pa),produc7507926704131184380et_nat(H,As2))
      <=> aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Pa),produc7507926704131184380et_nat(H3,As2)) ) ) ).

% mod_relH
tff(fact_4083_relH__refl,axiom,
    ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
      ( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H,As2))
     => relH(As2,H,H) ) ).

% relH_refl
tff(fact_4084_relH__in__rangeI_I1_J,axiom,
    ! [As2: set_nat,H: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit] :
      ( relH(As2,H,H3)
     => aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H,As2)) ) ).

% relH_in_rangeI(1)
tff(fact_4085_relH__in__rangeI_I2_J,axiom,
    ! [As2: set_nat,H: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit] :
      ( relH(As2,H,H3)
     => aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H3,As2)) ) ).

% relH_in_rangeI(2)
tff(fact_4086_times__assn__raw_Oelims_I3_J,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,Xa: fun_Pr6460944699146269935_nat_o,Xb: produc3658429121746597890et_nat] :
      ( ~ aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(aa_fun6000889628616002998_nat_o(times_assn_raw,X),Xa),Xb)
     => ~ ! [H4: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( Xb = produc7507926704131184380et_nat(H4,As) )
           => ? [As12: set_nat,As23: set_nat] :
                ( ( As = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As12),As23) )
                & ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As12),As23) = bot_bot_set_nat )
                & aa_Pro653082453696054616_nat_o(X,produc7507926704131184380et_nat(H4,As12))
                & aa_Pro653082453696054616_nat_o(Xa,produc7507926704131184380et_nat(H4,As23)) ) ) ) ).

% times_assn_raw.elims(3)
tff(fact_4087_times__assn__raw_Oelims_I2_J,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,Xa: fun_Pr6460944699146269935_nat_o,Xb: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(aa_fun6000889628616002998_nat_o(times_assn_raw,X),Xa),Xb)
     => ~ ! [H4: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( Xb = produc7507926704131184380et_nat(H4,As) )
           => ~ ? [As13: set_nat,As24: set_nat] :
                  ( ( As = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As13),As24) )
                  & ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As13),As24) = bot_bot_set_nat )
                  & aa_Pro653082453696054616_nat_o(X,produc7507926704131184380et_nat(H4,As13))
                  & aa_Pro653082453696054616_nat_o(Xa,produc7507926704131184380et_nat(H4,As24)) ) ) ) ).

% times_assn_raw.elims(2)
tff(fact_4088_times__assn__raw_Opelims_I3_J,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,Xa: fun_Pr6460944699146269935_nat_o,Xb: produc3658429121746597890et_nat] :
      ( ~ aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(aa_fun6000889628616002998_nat_o(times_assn_raw,X),Xa),Xb)
     => ( accp_P1394777952347926645et_nat(times_assn_raw_rel,produc7291855603632349880et_nat(X,produc6017135503889078682et_nat(Xa,Xb)))
       => ~ ! [H4: heap_e7401611519738050253t_unit,As: set_nat] :
              ( ( Xb = produc7507926704131184380et_nat(H4,As) )
             => ( accp_P1394777952347926645et_nat(times_assn_raw_rel,produc7291855603632349880et_nat(X,produc6017135503889078682et_nat(Xa,produc7507926704131184380et_nat(H4,As))))
               => ? [As12: set_nat,As23: set_nat] :
                    ( ( As = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As12),As23) )
                    & ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As12),As23) = bot_bot_set_nat )
                    & aa_Pro653082453696054616_nat_o(X,produc7507926704131184380et_nat(H4,As12))
                    & aa_Pro653082453696054616_nat_o(Xa,produc7507926704131184380et_nat(H4,As23)) ) ) ) ) ) ).

% times_assn_raw.pelims(3)
tff(fact_4089_times__assn__raw_Opelims_I2_J,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,Xa: fun_Pr6460944699146269935_nat_o,Xb: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(aa_fun6000889628616002998_nat_o(times_assn_raw,X),Xa),Xb)
     => ( accp_P1394777952347926645et_nat(times_assn_raw_rel,produc7291855603632349880et_nat(X,produc6017135503889078682et_nat(Xa,Xb)))
       => ~ ! [H4: heap_e7401611519738050253t_unit,As: set_nat] :
              ( ( Xb = produc7507926704131184380et_nat(H4,As) )
             => ( accp_P1394777952347926645et_nat(times_assn_raw_rel,produc7291855603632349880et_nat(X,produc6017135503889078682et_nat(Xa,produc7507926704131184380et_nat(H4,As))))
               => ~ ? [As13: set_nat,As24: set_nat] :
                      ( ( As = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As13),As24) )
                      & ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As13),As24) = bot_bot_set_nat )
                      & aa_Pro653082453696054616_nat_o(X,produc7507926704131184380et_nat(H4,As13))
                      & aa_Pro653082453696054616_nat_o(Xa,produc7507926704131184380et_nat(H4,As24)) ) ) ) ) ) ).

% times_assn_raw.pelims(2)
tff(fact_4090_times__assn__raw_Opelims_I1_J,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,Xa: fun_Pr6460944699146269935_nat_o,Xb: produc3658429121746597890et_nat,Y: $o] :
      ( ( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(aa_fun6000889628616002998_nat_o(times_assn_raw,X),Xa),Xb)
      <=> (Y) )
     => ( accp_P1394777952347926645et_nat(times_assn_raw_rel,produc7291855603632349880et_nat(X,produc6017135503889078682et_nat(Xa,Xb)))
       => ~ ! [H4: heap_e7401611519738050253t_unit,As: set_nat] :
              ( ( Xb = produc7507926704131184380et_nat(H4,As) )
             => ( ( (Y)
                <=> ? [As1: set_nat,As22: set_nat] :
                      ( ( As = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As1),As22) )
                      & ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As1),As22) = bot_bot_set_nat )
                      & aa_Pro653082453696054616_nat_o(X,produc7507926704131184380et_nat(H4,As1))
                      & aa_Pro653082453696054616_nat_o(Xa,produc7507926704131184380et_nat(H4,As22)) ) )
               => ~ accp_P1394777952347926645et_nat(times_assn_raw_rel,produc7291855603632349880et_nat(X,produc6017135503889078682et_nat(Xa,produc7507926704131184380et_nat(H4,As)))) ) ) ) ) ).

% times_assn_raw.pelims(1)
tff(fact_4091_wand__raw_Oelims_I3_J,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,Xa: fun_Pr6460944699146269935_nat_o,Xb: produc3658429121746597890et_nat] :
      ( ~ aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(aa_fun6000889628616002998_nat_o(wand_raw,X),Xa),Xb)
     => ~ ! [H4: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( Xb = produc7507926704131184380et_nat(H4,As) )
           => ( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H4,As))
              & ! [H5: heap_e7401611519738050253t_unit,As4: set_nat] :
                  ( ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As),As4) = bot_bot_set_nat )
                    & relH(As,H4,H5)
                    & aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H5,As))
                    & aa_Pro653082453696054616_nat_o(X,produc7507926704131184380et_nat(H5,As4)) )
                 => aa_Pro653082453696054616_nat_o(Xa,produc7507926704131184380et_nat(H5,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As),As4))) ) ) ) ) ).

% wand_raw.elims(3)
tff(fact_4092_wand__raw_Oelims_I2_J,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,Xa: fun_Pr6460944699146269935_nat_o,Xb: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(aa_fun6000889628616002998_nat_o(wand_raw,X),Xa),Xb)
     => ~ ! [H4: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( Xb = produc7507926704131184380et_nat(H4,As) )
           => ~ ( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H4,As))
                & ! [H6: heap_e7401611519738050253t_unit,As5: set_nat] :
                    ( ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As),As5) = bot_bot_set_nat )
                      & relH(As,H4,H6)
                      & aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H6,As))
                      & aa_Pro653082453696054616_nat_o(X,produc7507926704131184380et_nat(H6,As5)) )
                   => aa_Pro653082453696054616_nat_o(Xa,produc7507926704131184380et_nat(H6,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As),As5))) ) ) ) ) ).

% wand_raw.elims(2)
tff(fact_4093_wand__raw_Oelims_I1_J,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,Xa: fun_Pr6460944699146269935_nat_o,Xb: produc3658429121746597890et_nat,Y: $o] :
      ( ( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(aa_fun6000889628616002998_nat_o(wand_raw,X),Xa),Xb)
      <=> (Y) )
     => ~ ! [H4: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( Xb = produc7507926704131184380et_nat(H4,As) )
           => ( (Y)
            <=> ~ ( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H4,As))
                  & ! [H7: heap_e7401611519738050253t_unit,As6: set_nat] :
                      ( ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As),As6) = bot_bot_set_nat )
                        & relH(As,H4,H7)
                        & aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H7,As))
                        & aa_Pro653082453696054616_nat_o(X,produc7507926704131184380et_nat(H7,As6)) )
                     => aa_Pro653082453696054616_nat_o(Xa,produc7507926704131184380et_nat(H7,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As),As6))) ) ) ) ) ) ).

% wand_raw.elims(1)
tff(fact_4094_wand__raw_Osimps,axiom,
    ! [Pa: fun_Pr6460944699146269935_nat_o,Q: fun_Pr6460944699146269935_nat_o,H: heap_e7401611519738050253t_unit,As2: set_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(aa_fun6000889628616002998_nat_o(wand_raw,Pa),Q),produc7507926704131184380et_nat(H,As2))
    <=> ( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H,As2))
        & ! [H7: heap_e7401611519738050253t_unit,As6: set_nat] :
            ( ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As2),As6) = bot_bot_set_nat )
              & relH(As2,H,H7)
              & aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H7,As2))
              & aa_Pro653082453696054616_nat_o(Pa,produc7507926704131184380et_nat(H7,As6)) )
           => aa_Pro653082453696054616_nat_o(Q,produc7507926704131184380et_nat(H7,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As2),As6))) ) ) ) ).

% wand_raw.simps
tff(fact_4095_wand__raw_Opelims_I3_J,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,Xa: fun_Pr6460944699146269935_nat_o,Xb: produc3658429121746597890et_nat] :
      ( ~ aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(aa_fun6000889628616002998_nat_o(wand_raw,X),Xa),Xb)
     => ( accp_P1394777952347926645et_nat(wand_raw_rel,produc7291855603632349880et_nat(X,produc6017135503889078682et_nat(Xa,Xb)))
       => ~ ! [H4: heap_e7401611519738050253t_unit,As: set_nat] :
              ( ( Xb = produc7507926704131184380et_nat(H4,As) )
             => ( accp_P1394777952347926645et_nat(wand_raw_rel,produc7291855603632349880et_nat(X,produc6017135503889078682et_nat(Xa,produc7507926704131184380et_nat(H4,As))))
               => ( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H4,As))
                  & ! [H5: heap_e7401611519738050253t_unit,As4: set_nat] :
                      ( ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As),As4) = bot_bot_set_nat )
                        & relH(As,H4,H5)
                        & aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H5,As))
                        & aa_Pro653082453696054616_nat_o(X,produc7507926704131184380et_nat(H5,As4)) )
                     => aa_Pro653082453696054616_nat_o(Xa,produc7507926704131184380et_nat(H5,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As),As4))) ) ) ) ) ) ) ).

% wand_raw.pelims(3)
tff(fact_4096_wand__raw_Opelims_I2_J,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,Xa: fun_Pr6460944699146269935_nat_o,Xb: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(aa_fun6000889628616002998_nat_o(wand_raw,X),Xa),Xb)
     => ( accp_P1394777952347926645et_nat(wand_raw_rel,produc7291855603632349880et_nat(X,produc6017135503889078682et_nat(Xa,Xb)))
       => ~ ! [H4: heap_e7401611519738050253t_unit,As: set_nat] :
              ( ( Xb = produc7507926704131184380et_nat(H4,As) )
             => ( accp_P1394777952347926645et_nat(wand_raw_rel,produc7291855603632349880et_nat(X,produc6017135503889078682et_nat(Xa,produc7507926704131184380et_nat(H4,As))))
               => ~ ( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H4,As))
                    & ! [H6: heap_e7401611519738050253t_unit,As5: set_nat] :
                        ( ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As),As5) = bot_bot_set_nat )
                          & relH(As,H4,H6)
                          & aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H6,As))
                          & aa_Pro653082453696054616_nat_o(X,produc7507926704131184380et_nat(H6,As5)) )
                       => aa_Pro653082453696054616_nat_o(Xa,produc7507926704131184380et_nat(H6,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As),As5))) ) ) ) ) ) ) ).

% wand_raw.pelims(2)
tff(fact_4097_wand__raw_Opelims_I1_J,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,Xa: fun_Pr6460944699146269935_nat_o,Xb: produc3658429121746597890et_nat,Y: $o] :
      ( ( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(aa_fun6000889628616002998_nat_o(wand_raw,X),Xa),Xb)
      <=> (Y) )
     => ( accp_P1394777952347926645et_nat(wand_raw_rel,produc7291855603632349880et_nat(X,produc6017135503889078682et_nat(Xa,Xb)))
       => ~ ! [H4: heap_e7401611519738050253t_unit,As: set_nat] :
              ( ( Xb = produc7507926704131184380et_nat(H4,As) )
             => ( ( (Y)
                <=> ( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H4,As))
                    & ! [H7: heap_e7401611519738050253t_unit,As6: set_nat] :
                        ( ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As),As6) = bot_bot_set_nat )
                          & relH(As,H4,H7)
                          & aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H7,As))
                          & aa_Pro653082453696054616_nat_o(X,produc7507926704131184380et_nat(H7,As6)) )
                       => aa_Pro653082453696054616_nat_o(Xa,produc7507926704131184380et_nat(H7,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As),As6))) ) ) )
               => ~ accp_P1394777952347926645et_nat(wand_raw_rel,produc7291855603632349880et_nat(X,produc6017135503889078682et_nat(Xa,produc7507926704131184380et_nat(H4,As)))) ) ) ) ) ).

% wand_raw.pelims(1)
tff(fact_4098_uminus__assn__def,axiom,
    ! [Pa: assn] : aa_assn_assn(uminus_uminus_assn,Pa) = aa_fun5526454687433275355o_assn(abs_assn,aTP_Lamm_ag(Pa)) ).

% uminus_assn_def
tff(fact_4099_Rep__assn__inverse,axiom,
    ! [X: assn] : aa_fun5526454687433275355o_assn(abs_assn,aa_ass4243947775541419017_nat_o(rep_assn,X)) = X ).

% Rep_assn_inverse
tff(fact_4100_pure__assn__def,axiom,
    ! [B: $o] : aa_o_assn(pure_assn,(B)) = aa_fun5526454687433275355o_assn(abs_assn,pure_a825153325127701367it_nat((B))) ).

% pure_assn_def
tff(fact_4101_Abs__assn__eqI_I2_J,axiom,
    ! [Pa: fun_Pr6460944699146269935_nat_o,Pr: assn] :
      ( ! [H4: produc3658429121746597890et_nat] :
          ( aa_Pro653082453696054616_nat_o(Pa,H4)
        <=> aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Pr),H4) )
     => ( Pr = aa_fun5526454687433275355o_assn(abs_assn,Pa) ) ) ).

% Abs_assn_eqI(2)
tff(fact_4102_Abs__assn__eqI_I1_J,axiom,
    ! [Pa: fun_Pr6460944699146269935_nat_o,Pr: assn] :
      ( ! [H4: produc3658429121746597890et_nat] :
          ( aa_Pro653082453696054616_nat_o(Pa,H4)
        <=> aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Pr),H4) )
     => ( aa_fun5526454687433275355o_assn(abs_assn,Pa) = Pr ) ) ).

% Abs_assn_eqI(1)
tff(fact_4103_bot__assn__def,axiom,
    bot_bot_assn = aa_fun5526454687433275355o_assn(abs_assn,aTP_Lamm_ah) ).

% bot_assn_def
tff(fact_4104_top__assn__def,axiom,
    top_top_assn = aa_fun5526454687433275355o_assn(abs_assn,in_range) ).

% top_assn_def
tff(fact_4105_sup__assn__def,axiom,
    ! [Pa: assn,Q: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Pa),Q) = aa_fun5526454687433275355o_assn(abs_assn,aa_ass4243947775541419017_nat_o(aTP_Lamm_ai(Pa),Q)) ).

% sup_assn_def
tff(fact_4106_inf__assn__def,axiom,
    ! [Pa: assn,Q: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Pa),Q) = aa_fun5526454687433275355o_assn(abs_assn,aa_ass4243947775541419017_nat_o(aTP_Lamm_aj(Pa),Q)) ).

% inf_assn_def
tff(fact_4107_times__assn__def,axiom,
    ! [Pa: assn,Q: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Pa),Q) = aa_fun5526454687433275355o_assn(abs_assn,aa_fun2638059832795814268_nat_o(aa_fun6000889628616002998_nat_o(times_assn_raw,aa_ass4243947775541419017_nat_o(rep_assn,Pa)),aa_ass4243947775541419017_nat_o(rep_assn,Q))) ).

% times_assn_def
tff(fact_4108_wand__assn__def,axiom,
    ! [Pa: assn,Q: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(wand_assn,Pa),Q) = aa_fun5526454687433275355o_assn(abs_assn,aa_fun2638059832795814268_nat_o(aa_fun6000889628616002998_nat_o(wand_raw,aa_ass4243947775541419017_nat_o(rep_assn,Pa)),aa_ass4243947775541419017_nat_o(rep_assn,Q))) ).

% wand_assn_def
tff(fact_4109_one__assn__def,axiom,
    one_one_assn = aa_fun5526454687433275355o_assn(abs_assn,one_assn_raw) ).

% one_assn_def
tff(fact_4110_one__assn__raw_Osimps,axiom,
    ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
      ( aa_Pro653082453696054616_nat_o(one_assn_raw,produc7507926704131184380et_nat(H,As2))
    <=> ( As2 = bot_bot_set_nat ) ) ).

% one_assn_raw.simps
tff(fact_4111_one__assn__raw_Oelims_I1_J,axiom,
    ! [X: produc3658429121746597890et_nat,Y: $o] :
      ( ( aa_Pro653082453696054616_nat_o(one_assn_raw,X)
      <=> (Y) )
     => ~ ! [H4: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( X = produc7507926704131184380et_nat(H4,As) )
           => ( (Y)
            <=> ( As != bot_bot_set_nat ) ) ) ) ).

% one_assn_raw.elims(1)
tff(fact_4112_one__assn__raw_Oelims_I3_J,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ( ~ aa_Pro653082453696054616_nat_o(one_assn_raw,X)
     => ~ ! [H4: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( X = produc7507926704131184380et_nat(H4,As) )
           => ( As = bot_bot_set_nat ) ) ) ).

% one_assn_raw.elims(3)
tff(fact_4113_one__assn__raw_Oelims_I2_J,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(one_assn_raw,X)
     => ~ ! [H4: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( X = produc7507926704131184380et_nat(H4,As) )
           => ( As != bot_bot_set_nat ) ) ) ).

% one_assn_raw.elims(2)
tff(fact_4114_properI,axiom,
    ! [Pa: fun_Pr6460944699146269935_nat_o] :
      ( ! [As: set_nat,H4: heap_e7401611519738050253t_unit] :
          ( aa_Pro653082453696054616_nat_o(Pa,produc7507926704131184380et_nat(H4,As))
         => aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H4,As)) )
     => ( ! [As: set_nat,H4: heap_e7401611519738050253t_unit,H5: heap_e7401611519738050253t_unit] :
            ( aa_Pro653082453696054616_nat_o(Pa,produc7507926704131184380et_nat(H4,As))
           => ( relH(As,H4,H5)
             => ( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H5,As))
               => aa_Pro653082453696054616_nat_o(Pa,produc7507926704131184380et_nat(H5,As)) ) ) )
       => aa_fun1675921005893230629at_o_o(proper,Pa) ) ) ).

% properI
tff(fact_4115_properD2,axiom,
    ! [Pa: fun_Pr6460944699146269935_nat_o,H: heap_e7401611519738050253t_unit,As2: set_nat,H3: heap_e7401611519738050253t_unit] :
      ( aa_fun1675921005893230629at_o_o(proper,Pa)
     => ( aa_Pro653082453696054616_nat_o(Pa,produc7507926704131184380et_nat(H,As2))
       => ( relH(As2,H,H3)
         => ( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H3,As2))
           => aa_Pro653082453696054616_nat_o(Pa,produc7507926704131184380et_nat(H3,As2)) ) ) ) ) ).

% properD2
tff(fact_4116_proper__def,axiom,
    ! [Pa: fun_Pr6460944699146269935_nat_o] :
      ( aa_fun1675921005893230629at_o_o(proper,Pa)
    <=> ! [H2: heap_e7401611519738050253t_unit,H7: heap_e7401611519738050253t_unit,As7: set_nat] :
          ( ( aa_Pro653082453696054616_nat_o(Pa,produc7507926704131184380et_nat(H2,As7))
           => aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H2,As7)) )
          & ( ( aa_Pro653082453696054616_nat_o(Pa,produc7507926704131184380et_nat(H2,As7))
              & relH(As7,H2,H7)
              & aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H7,As7)) )
           => aa_Pro653082453696054616_nat_o(Pa,produc7507926704131184380et_nat(H7,As7)) ) ) ) ).

% proper_def
tff(fact_4117_bool__assn__proper_I4_J,axiom,
    ! [Pa: fun_Pr6460944699146269935_nat_o,Q: fun_Pr6460944699146269935_nat_o] :
      ( aa_fun1675921005893230629at_o_o(proper,Pa)
     => ( aa_fun1675921005893230629at_o_o(proper,Q)
       => aa_fun1675921005893230629at_o_o(proper,aa_fun2638059832795814268_nat_o(aa_fun6000889628616002998_nat_o(aTP_Lamm_ak,Pa),Q)) ) ) ).

% bool_assn_proper(4)
tff(fact_4118_bool__assn__proper_I3_J,axiom,
    ! [Pa: fun_Pr6460944699146269935_nat_o,Q: fun_Pr6460944699146269935_nat_o] :
      ( aa_fun1675921005893230629at_o_o(proper,Pa)
     => ( aa_fun1675921005893230629at_o_o(proper,Q)
       => aa_fun1675921005893230629at_o_o(proper,aa_fun2638059832795814268_nat_o(aa_fun6000889628616002998_nat_o(aTP_Lamm_al,Pa),Q)) ) ) ).

% bool_assn_proper(3)
tff(fact_4119_bool__assn__proper_I2_J,axiom,
    aa_fun1675921005893230629at_o_o(proper,aTP_Lamm_ah) ).

% bool_assn_proper(2)
tff(fact_4120_bool__assn__proper_I1_J,axiom,
    aa_fun1675921005893230629at_o_o(proper,in_range) ).

% bool_assn_proper(1)
tff(fact_4121_times__assn__proper,axiom,
    ! [Pa: fun_Pr6460944699146269935_nat_o,Q: fun_Pr6460944699146269935_nat_o] :
      ( aa_fun1675921005893230629at_o_o(proper,Pa)
     => ( aa_fun1675921005893230629at_o_o(proper,Q)
       => aa_fun1675921005893230629at_o_o(proper,aa_fun2638059832795814268_nat_o(aa_fun6000889628616002998_nat_o(times_assn_raw,Pa),Q)) ) ) ).

% times_assn_proper
tff(fact_4122_wand__proper,axiom,
    ! [Pa: fun_Pr6460944699146269935_nat_o,Q: fun_Pr6460944699146269935_nat_o] : aa_fun1675921005893230629at_o_o(proper,aa_fun2638059832795814268_nat_o(aa_fun6000889628616002998_nat_o(wand_raw,Pa),Q)) ).

% wand_proper
tff(fact_4123_one__assn__proper,axiom,
    aa_fun1675921005893230629at_o_o(proper,one_assn_raw) ).

% one_assn_proper
tff(fact_4124_bool__assn__proper_I5_J,axiom,
    ! [Pa: fun_Pr6460944699146269935_nat_o] :
      ( aa_fun1675921005893230629at_o_o(proper,Pa)
     => aa_fun1675921005893230629at_o_o(proper,aTP_Lamm_am(Pa)) ) ).

% bool_assn_proper(5)
tff(fact_4125_pure__assn__proper,axiom,
    ! [B: $o] : aa_fun1675921005893230629at_o_o(proper,pure_a825153325127701367it_nat((B))) ).

% pure_assn_proper
tff(fact_4126_Abs__assn__inject,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,Y: fun_Pr6460944699146269935_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),collec6407654713642994138_nat_o(proper))
     => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Y),collec6407654713642994138_nat_o(proper))
       => ( ( aa_fun5526454687433275355o_assn(abs_assn,X) = aa_fun5526454687433275355o_assn(abs_assn,Y) )
        <=> ( X = Y ) ) ) ) ).

% Abs_assn_inject
tff(fact_4127_Abs__assn__induct,axiom,
    ! [Pa: fun_assn_o,X: assn] :
      ( ! [Y2: fun_Pr6460944699146269935_nat_o] :
          ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Y2),collec6407654713642994138_nat_o(proper))
         => aa_assn_o(Pa,aa_fun5526454687433275355o_assn(abs_assn,Y2)) )
     => aa_assn_o(Pa,X) ) ).

% Abs_assn_induct
tff(fact_4128_Abs__assn__cases,axiom,
    ! [X: assn] :
      ~ ! [Y2: fun_Pr6460944699146269935_nat_o] :
          ( ( X = aa_fun5526454687433275355o_assn(abs_assn,Y2) )
         => ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Y2),collec6407654713642994138_nat_o(proper)) ) ).

% Abs_assn_cases
tff(fact_4129_Rep__assn__induct,axiom,
    ! [Y: fun_Pr6460944699146269935_nat_o,Pa: fun_fu5135969500543783852at_o_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Y),collec6407654713642994138_nat_o(proper))
     => ( ! [X2: assn] : aa_fun1675921005893230629at_o_o(Pa,aa_ass4243947775541419017_nat_o(rep_assn,X2))
       => aa_fun1675921005893230629at_o_o(Pa,Y) ) ) ).

% Rep_assn_induct
tff(fact_4130_Rep__assn__cases,axiom,
    ! [Y: fun_Pr6460944699146269935_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Y),collec6407654713642994138_nat_o(proper))
     => ~ ! [X2: assn] : Y != aa_ass4243947775541419017_nat_o(rep_assn,X2) ) ).

% Rep_assn_cases
tff(fact_4131_Rep__assn,axiom,
    ! [X: assn] : aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,X)),collec6407654713642994138_nat_o(proper)) ).

% Rep_assn
tff(fact_4132_properD1,axiom,
    ! [Pa: fun_Pr6460944699146269935_nat_o,H: heap_e7401611519738050253t_unit,As2: set_nat] :
      ( aa_fun1675921005893230629at_o_o(proper,Pa)
     => ( aa_Pro653082453696054616_nat_o(Pa,produc7507926704131184380et_nat(H,As2))
       => aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H,As2)) ) ) ).

% properD1
tff(fact_4133_Abs__assn__inverse,axiom,
    ! [Y: fun_Pr6460944699146269935_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Y),collec6407654713642994138_nat_o(proper))
     => ( aa_ass4243947775541419017_nat_o(rep_assn,aa_fun5526454687433275355o_assn(abs_assn,Y)) = Y ) ) ).

% Abs_assn_inverse
tff(fact_4134_proper__iff,axiom,
    ! [Pa: fun_Pr6460944699146269935_nat_o,As2: set_nat,H: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit] :
      ( aa_fun1675921005893230629at_o_o(proper,Pa)
     => ( relH(As2,H,H3)
       => ( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H3,As2))
         => ( aa_Pro653082453696054616_nat_o(Pa,produc7507926704131184380et_nat(H,As2))
          <=> aa_Pro653082453696054616_nat_o(Pa,produc7507926704131184380et_nat(H3,As2)) ) ) ) ) ).

% proper_iff
tff(fact_4135_entt__def__true,axiom,
    ! [Pa: assn,Q: assn] :
      ( aa_assn_o(entailst(Pa),Q)
    <=> aa_assn_o(aa_assn_fun_assn_o(entails,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Pa),top_top_assn)),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Q),top_top_assn)) ) ).

% entt_def_true
tff(fact_4136_entailst__def,axiom,
    ! [A2: assn,B2: assn] :
      ( aa_assn_o(entailst(A2),B2)
    <=> aa_assn_o(aa_assn_fun_assn_o(entails,A2),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B2),top_top_assn)) ) ).

% entailst_def
tff(fact_4137_enttI__true,axiom,
    ! [Pa: assn,Q: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Pa),top_top_assn)),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Q),top_top_assn))
     => aa_assn_o(entailst(Pa),Q) ) ).

% enttI_true
tff(fact_4138_entt__refl,axiom,
    ! [A2: assn] : aa_assn_o(entailst(A2),A2) ).

% entt_refl
tff(fact_4139_entt__true,axiom,
    ! [A2: assn] : aa_assn_o(entailst(A2),top_top_assn) ).

% entt_true
tff(fact_4140_entt__emp,axiom,
    ! [A2: assn] : aa_assn_o(entailst(A2),one_one_assn) ).

% entt_emp
tff(fact_4141_entt__star__true__simp_I2_J,axiom,
    ! [A2: assn,B2: assn] :
      ( aa_assn_o(entailst(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A2),top_top_assn)),B2)
    <=> aa_assn_o(entailst(A2),B2) ) ).

% entt_star_true_simp(2)
tff(fact_4142_entt__star__true__simp_I1_J,axiom,
    ! [A2: assn,B2: assn] :
      ( aa_assn_o(entailst(A2),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B2),top_top_assn))
    <=> aa_assn_o(entailst(A2),B2) ) ).

% entt_star_true_simp(1)
tff(fact_4143_entt__disjI2__direct,axiom,
    ! [B2: assn,A2: assn] : aa_assn_o(entailst(B2),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A2),B2)) ).

% entt_disjI2_direct
tff(fact_4144_entt__disjI1__direct,axiom,
    ! [A2: assn,B2: assn] : aa_assn_o(entailst(A2),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A2),B2)) ).

% entt_disjI1_direct
tff(fact_4145_entt__disjI2_H,axiom,
    ! [A2: assn,C3: assn,B2: assn] :
      ( aa_assn_o(entailst(A2),C3)
     => aa_assn_o(entailst(A2),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,B2),C3)) ) ).

% entt_disjI2'
tff(fact_4146_entt__disjI1_H,axiom,
    ! [A2: assn,B2: assn,C3: assn] :
      ( aa_assn_o(entailst(A2),B2)
     => aa_assn_o(entailst(A2),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,B2),C3)) ) ).

% entt_disjI1'
tff(fact_4147_entt__disjD2,axiom,
    ! [A2: assn,B2: assn,C3: assn] :
      ( aa_assn_o(entailst(aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A2),B2)),C3)
     => aa_assn_o(entailst(B2),C3) ) ).

% entt_disjD2
tff(fact_4148_entt__disjD1,axiom,
    ! [A2: assn,B2: assn,C3: assn] :
      ( aa_assn_o(entailst(aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A2),B2)),C3)
     => aa_assn_o(entailst(A2),C3) ) ).

% entt_disjD1
tff(fact_4149_entt__disjE,axiom,
    ! [A2: assn,M: assn,B2: assn] :
      ( aa_assn_o(entailst(A2),M)
     => ( aa_assn_o(entailst(B2),M)
       => aa_assn_o(entailst(aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A2),B2)),M) ) ) ).

% entt_disjE
tff(fact_4150_entt__frame__fwd,axiom,
    ! [Pa: assn,Q: assn,A2: assn,F2: assn,B2: assn] :
      ( aa_assn_o(entailst(Pa),Q)
     => ( aa_assn_o(entailst(A2),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Pa),F2))
       => ( aa_assn_o(entailst(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Q),F2)),B2)
         => aa_assn_o(entailst(A2),B2) ) ) ) ).

% entt_frame_fwd
tff(fact_4151_entt__star__mono,axiom,
    ! [A2: assn,B2: assn,C3: assn,D: assn] :
      ( aa_assn_o(entailst(A2),B2)
     => ( aa_assn_o(entailst(C3),D)
       => aa_assn_o(entailst(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A2),C3)),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B2),D)) ) ) ).

% entt_star_mono
tff(fact_4152_entt__fr__refl,axiom,
    ! [F2: assn,F4: assn,A2: assn] :
      ( aa_assn_o(entailst(F2),F4)
     => aa_assn_o(entailst(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,F2),A2)),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,F4),A2)) ) ).

% entt_fr_refl
tff(fact_4153_entt__fr__drop,axiom,
    ! [F2: assn,F4: assn,A2: assn] :
      ( aa_assn_o(entailst(F2),F4)
     => aa_assn_o(entailst(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,F2),A2)),F4) ) ).

% entt_fr_drop
tff(fact_4154_entt__trans,axiom,
    ! [A2: assn,B2: assn,C3: assn] :
      ( aa_assn_o(entailst(A2),B2)
     => ( aa_assn_o(entailst(B2),C3)
       => aa_assn_o(entailst(A2),C3) ) ) ).

% entt_trans
tff(fact_4155_ent__imp__entt,axiom,
    ! [Pa: assn,Q: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,Pa),Q)
     => aa_assn_o(entailst(Pa),Q) ) ).

% ent_imp_entt
tff(fact_4156_enttD,axiom,
    ! [A2: assn,B2: assn] :
      ( aa_assn_o(entailst(A2),B2)
     => aa_assn_o(aa_assn_fun_assn_o(entails,A2),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B2),top_top_assn)) ) ).

% enttD
tff(fact_4157_enttI,axiom,
    ! [A2: assn,B2: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,A2),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B2),top_top_assn))
     => aa_assn_o(entailst(A2),B2) ) ).

% enttI
tff(fact_4158_in__range_Oelims_I3_J,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ( ~ aa_Pro653082453696054616_nat_o(in_range,X)
     => ~ ! [H4: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( X = produc7507926704131184380et_nat(H4,As) )
           => ! [X2: nat] :
                ( aa_set_nat_o(member_nat2(X2),As)
               => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),lim_Product_unit(H4)) ) ) ) ).

% in_range.elims(3)
tff(fact_4159_in__range_Osimps,axiom,
    ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
      ( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H,As2))
    <=> ! [X3: nat] :
          ( aa_set_nat_o(member_nat2(X3),As2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),lim_Product_unit(H)) ) ) ).

% in_range.simps
tff(fact_4160_in__range_Oelims_I1_J,axiom,
    ! [X: produc3658429121746597890et_nat,Y: $o] :
      ( ( aa_Pro653082453696054616_nat_o(in_range,X)
      <=> (Y) )
     => ~ ! [H4: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( X = produc7507926704131184380et_nat(H4,As) )
           => ( (Y)
            <=> ~ ! [X3: nat] :
                    ( aa_set_nat_o(member_nat2(X3),As)
                   => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),lim_Product_unit(H4)) ) ) ) ) ).

% in_range.elims(1)
tff(fact_4161_in__range_Oelims_I2_J,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(in_range,X)
     => ~ ! [H4: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( X = produc7507926704131184380et_nat(H4,As) )
           => ~ ! [X4: nat] :
                  ( aa_set_nat_o(member_nat2(X4),As)
                 => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),lim_Product_unit(H4)) ) ) ) ).

% in_range.elims(2)
tff(fact_4162_type__definition__assn,axiom,
    aa_set5091322549519198405at_o_o(type_d5492019815722432281_nat_o(rep_assn,abs_assn),collec6407654713642994138_nat_o(proper)) ).

% type_definition_assn
tff(fact_4163_wait__rule,axiom,
    ! [N: nat] : hoare_8945653483474564448t_unit(one_one_assn,heap_Time_wait(N),aTP_Lamm_an) ).

% wait_rule
tff(fact_4164_inf__unit__def,axiom,
    ! [Uu: product_unit,Uv: product_unit] : aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,Uu),Uv) = product_Unity ).

% inf_unit_def
tff(fact_4165_relH__def,axiom,
    ! [As2: set_nat,H: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit] :
      ( relH(As2,H,H3)
    <=> ( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H,As2))
        & aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H3,As2))
        & ! [T2: typerep,X3: nat] :
            ( aa_set_nat_o(member_nat2(X3),As2)
           => ( ( aa_nat_nat(refs_Product_unit(H,T2),X3) = aa_nat_nat(refs_Product_unit(H3,T2),X3) )
              & ( aa_nat_list_nat(arrays_Product_unit(H,T2),X3) = aa_nat_list_nat(arrays_Product_unit(H3,T2),X3) ) ) ) ) ) ).

% relH_def
tff(fact_4166_Restr__natLeq,axiom,
    ! [N: nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,bNF_Ca8665028551170535155natLeq),produc457027306803732586at_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ao,N)),aTP_Lamm_ap(N))) = collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_aq(N))) ).

% Restr_natLeq
tff(fact_4167_Restr__natLeq2,axiom,
    ! [N: nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,bNF_Ca8665028551170535155natLeq),produc457027306803732586at_nat(aa_nat_set_nat(order_underS_nat(bNF_Ca8665028551170535155natLeq),N),aTP_Lamm_ar(N))) = collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_aq(N))) ).

% Restr_natLeq2
tff(fact_4168_in__range_Opelims_I1_J,axiom,
    ! [X: produc3658429121746597890et_nat,Y: $o] :
      ( ( aa_Pro653082453696054616_nat_o(in_range,X)
      <=> (Y) )
     => ( aa_Pro653082453696054616_nat_o(accp_P5801069581201407417et_nat(in_range_rel),X)
       => ~ ! [H4: heap_e7401611519738050253t_unit,As: set_nat] :
              ( ( X = produc7507926704131184380et_nat(H4,As) )
             => ( ( (Y)
                <=> ! [X3: nat] :
                      ( aa_set_nat_o(member_nat2(X3),As)
                     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),lim_Product_unit(H4)) ) )
               => ~ aa_Pro653082453696054616_nat_o(accp_P5801069581201407417et_nat(in_range_rel),produc7507926704131184380et_nat(H4,As)) ) ) ) ) ).

% in_range.pelims(1)
tff(fact_4169_in__range_Opelims_I2_J,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(in_range,X)
     => ( aa_Pro653082453696054616_nat_o(accp_P5801069581201407417et_nat(in_range_rel),X)
       => ~ ! [H4: heap_e7401611519738050253t_unit,As: set_nat] :
              ( ( X = produc7507926704131184380et_nat(H4,As) )
             => ( aa_Pro653082453696054616_nat_o(accp_P5801069581201407417et_nat(in_range_rel),produc7507926704131184380et_nat(H4,As))
               => ~ ! [X4: nat] :
                      ( aa_set_nat_o(member_nat2(X4),As)
                     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),lim_Product_unit(H4)) ) ) ) ) ) ).

% in_range.pelims(2)
tff(fact_4170_in__range_Opelims_I3_J,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ( ~ aa_Pro653082453696054616_nat_o(in_range,X)
     => ( aa_Pro653082453696054616_nat_o(accp_P5801069581201407417et_nat(in_range_rel),X)
       => ~ ! [H4: heap_e7401611519738050253t_unit,As: set_nat] :
              ( ( X = produc7507926704131184380et_nat(H4,As) )
             => ( aa_Pro653082453696054616_nat_o(accp_P5801069581201407417et_nat(in_range_rel),produc7507926704131184380et_nat(H4,As))
               => ! [X2: nat] :
                    ( aa_set_nat_o(member_nat2(X2),As)
                   => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),lim_Product_unit(H4)) ) ) ) ) ) ).

% in_range.pelims(3)
tff(fact_4171_one__assn__raw_Opelims_I3_J,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ( ~ aa_Pro653082453696054616_nat_o(one_assn_raw,X)
     => ( aa_Pro653082453696054616_nat_o(accp_P5801069581201407417et_nat(one_assn_raw_rel),X)
       => ~ ! [H4: heap_e7401611519738050253t_unit,As: set_nat] :
              ( ( X = produc7507926704131184380et_nat(H4,As) )
             => ( aa_Pro653082453696054616_nat_o(accp_P5801069581201407417et_nat(one_assn_raw_rel),produc7507926704131184380et_nat(H4,As))
               => ( As = bot_bot_set_nat ) ) ) ) ) ).

% one_assn_raw.pelims(3)
tff(fact_4172_one__assn__raw_Opelims_I2_J,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(one_assn_raw,X)
     => ( aa_Pro653082453696054616_nat_o(accp_P5801069581201407417et_nat(one_assn_raw_rel),X)
       => ~ ! [H4: heap_e7401611519738050253t_unit,As: set_nat] :
              ( ( X = produc7507926704131184380et_nat(H4,As) )
             => ( aa_Pro653082453696054616_nat_o(accp_P5801069581201407417et_nat(one_assn_raw_rel),produc7507926704131184380et_nat(H4,As))
               => ( As != bot_bot_set_nat ) ) ) ) ) ).

% one_assn_raw.pelims(2)
tff(fact_4173_one__assn__raw_Opelims_I1_J,axiom,
    ! [X: produc3658429121746597890et_nat,Y: $o] :
      ( ( aa_Pro653082453696054616_nat_o(one_assn_raw,X)
      <=> (Y) )
     => ( aa_Pro653082453696054616_nat_o(accp_P5801069581201407417et_nat(one_assn_raw_rel),X)
       => ~ ! [H4: heap_e7401611519738050253t_unit,As: set_nat] :
              ( ( X = produc7507926704131184380et_nat(H4,As) )
             => ( ( (Y)
                <=> ( As = bot_bot_set_nat ) )
               => ~ aa_Pro653082453696054616_nat_o(accp_P5801069581201407417et_nat(one_assn_raw_rel),produc7507926704131184380et_nat(H4,As)) ) ) ) ) ).

% one_assn_raw.pelims(1)
tff(fact_4174_ATP_Olambda__1,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_ad,Uu) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),aa_nat_nat(suc,zero_zero_nat)) ).

% ATP.lambda_1
tff(fact_4175_ATP_Olambda__2,axiom,
    ! [Uu: assn,Uua: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aTP_Lamm_ag(Uu),Uua)
    <=> ( aa_Pro653082453696054616_nat_o(in_range,Uua)
        & ~ aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Uu),Uua) ) ) ).

% ATP.lambda_2
tff(fact_4176_ATP_Olambda__3,axiom,
    ! [Uu: fun_Pr6460944699146269935_nat_o,Uua: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aTP_Lamm_am(Uu),Uua)
    <=> ( aa_Pro653082453696054616_nat_o(in_range,Uua)
        & ~ aa_Pro653082453696054616_nat_o(Uu,Uua) ) ) ).

% ATP.lambda_3
tff(fact_4177_ATP_Olambda__4,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_ao,Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uua),Uu) ) ).

% ATP.lambda_4
tff(fact_4178_ATP_Olambda__5,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ac,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uu) ).

% ATP.lambda_5
tff(fact_4179_ATP_Olambda__6,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aTP_Lamm_ab(Uu),Uua)
    <=> member8440522571783428010at_nat(Uua,Uu) ) ).

% ATP.lambda_6
tff(fact_4180_ATP_Olambda__7,axiom,
    ! [Uu: set_nat,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_aa(Uu),Uua)
    <=> aa_set_nat_o(member_nat2(Uua),Uu) ) ).

% ATP.lambda_7
tff(fact_4181_ATP_Olambda__8,axiom,
    ! [Uu: set_fu3555266933746557007_nat_o,Uua: fun_Pr6460944699146269935_nat_o] :
      ( aa_fun1675921005893230629at_o_o(aTP_Lamm_a(Uu),Uua)
    <=> aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Uua),Uu) ) ).

% ATP.lambda_8
tff(fact_4182_ATP_Olambda__9,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_set_nat(aTP_Lamm_ar(Uu),Uua) = aa_nat_set_nat(order_underS_nat(bNF_Ca8665028551170535155natLeq),Uu) ).

% ATP.lambda_9
tff(fact_4183_ATP_Olambda__10,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_set_nat(aTP_Lamm_ap(Uu),Uua) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ao,Uu)) ).

% ATP.lambda_10
tff(fact_4184_ATP_Olambda__11,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_aq(Uu),Uua),Uub)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uua),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uua),Uub) ) ) ).

% ATP.lambda_11
tff(fact_4185_ATP_Olambda__12,axiom,
    ! [Uu: assn,Uua: assn,Uub: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(aTP_Lamm_ai(Uu),Uua),Uub)
    <=> ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Uu),Uub)
        | aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Uua),Uub) ) ) ).

% ATP.lambda_12
tff(fact_4186_ATP_Olambda__13,axiom,
    ! [Uu: assn,Uua: assn,Uub: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(aTP_Lamm_aj(Uu),Uua),Uub)
    <=> ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Uu),Uub)
        & aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Uua),Uub) ) ) ).

% ATP.lambda_13
tff(fact_4187_ATP_Olambda__14,axiom,
    ! [Uu: fun_Pr6460944699146269935_nat_o,Uua: fun_Pr6460944699146269935_nat_o,Uub: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(aa_fun6000889628616002998_nat_o(aTP_Lamm_al,Uu),Uua),Uub)
    <=> ( aa_Pro653082453696054616_nat_o(Uu,Uub)
        | aa_Pro653082453696054616_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_14
tff(fact_4188_ATP_Olambda__15,axiom,
    ! [Uu: fun_Pr6460944699146269935_nat_o,Uua: fun_Pr6460944699146269935_nat_o,Uub: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(aa_fun6000889628616002998_nat_o(aTP_Lamm_ak,Uu),Uua),Uub)
    <=> ( aa_Pro653082453696054616_nat_o(Uu,Uub)
        & aa_Pro653082453696054616_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_15
tff(fact_4189_ATP_Olambda__16,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_ae,Uu) = Uu ).

% ATP.lambda_16
tff(fact_4190_ATP_Olambda__17,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_af,Uu) = Uu ).

% ATP.lambda_17
tff(fact_4191_ATP_Olambda__18,axiom,
    ! [Uu: product_unit] : aa_Product_unit_assn(aTP_Lamm_an,Uu) = one_one_assn ).

% ATP.lambda_18
tff(fact_4192_ATP_Olambda__19,axiom,
    ! [Uu: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aTP_Lamm_ah,Uu)
    <=> $false ) ).

% ATP.lambda_19

% Helper facts (2)
tff(help_fNot_2_1_U,axiom,
    ! [P5: $o] :
      ( (P5)
      | aa_o_o(fNot,(P5)) ) ).

tff(help_fNot_1_1_U,axiom,
    ! [P5: $o] :
      ( ~ aa_o_o(fNot,(P5))
      | ~ (P5) ) ).

% Conjectures (1)
tff(conj_0,conjecture,
    vEBT_L4319891404334229444sn_a_b(p,nil_a,l) = aa_o_assn(pure_assn,l = nil_b) ).

%------------------------------------------------------------------------------