TPTP Problem File: SLH0676^1.p

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

% Refs     : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source   : [Des23]
% Names    : FO_Theory_Rewriting/0082_FOR_Check/prob_00070_002436__18875842_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 1645 ( 582 unt; 403 typ;   0 def)
%            Number of atoms       : 3381 (1565 equ;   0 cnn)
%            Maximal formula atoms :   11 (   2 avg)
%            Number of connectives : 11647 ( 592   ~;  83   |; 219   &;9240   @)
%                                         (   0 <=>;1513  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   29 (   7 avg)
%            Number of types       :  105 ( 104 usr)
%            Number of type conns  :  991 ( 991   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  302 ( 299 usr;  13 con; 0-3 aty)
%            Number of variables   : 3943 ( 105   ^;3678   !; 160   ?;3943   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 12:03:01.053
%------------------------------------------------------------------------------
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thf(ty_n_t__String__Ochar,type,
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thf(ty_n_t__Int__Oint,type,
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% Explicit typings (299)
thf(sy_c_Basic__Utils_Oadd__elem__list__lists_001t__FSet__Ofset_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
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thf(sy_c_FSet_Ofset_OFSet_Opred__fset_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
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thf(sy_c_FSet_Ofset_Ofset_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_Mt__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J_J,type,
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thf(sy_c_FSet_Ofset_Ofset_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
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thf(sy_c_FSet_Ofset_Ofset_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
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thf(sy_c_FSet_Ofset_Ofset_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
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thf(sy_c_FSet__Utils_OfId__on_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_Mt__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
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thf(sy_c_FSet__Utils_OfId__on_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
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thf(sy_c_FSet__Utils_OfId__on_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
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thf(sy_c_FSet__Utils_OfImage_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_Mt__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_Mt__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
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thf(sy_c_Fun_Oid_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_Fun_Oid_001t__Nat__Onat,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
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thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
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thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
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thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
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thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint,type,
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thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
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thf(sy_c_Relation_OImage_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_Mt__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
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thf(sy_c_Relation_OImage_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_Mt__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
    image_5835567378187231857rm_a_b: set_Pr8389173773336266149rm_a_b > set_Pr4386577575007340137rm_a_b > set_Pr2972776593051762503rm_a_b ).

thf(sy_c_Relation_OImage_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    image_871640889055226643rm_a_b: set_Pr2972776593051762503rm_a_b > set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b ).

thf(sy_c_Relation_OImage_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    image_1687478359920701578rm_a_b: set_Pr6510905305361987970rm_a_b > set_Pr4386577575007340137rm_a_b > set_term_a_b ).

thf(sy_c_Relation_OImage_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_Mt__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
    image_1883868062718963134rm_a_b: set_Pr1230253970132291838rm_a_b > set_term_a_b > set_Pr2972776593051762503rm_a_b ).

thf(sy_c_Relation_OImage_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    image_9078632467741184736rm_a_b: set_Pr4802440762858034464rm_a_b > set_term_a_b > set_Pr4386577575007340137rm_a_b ).

thf(sy_c_Relation_OImage_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    image_6923787618150785661rm_a_b: set_Pr4386577575007340137rm_a_b > set_term_a_b > set_term_a_b ).

thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Int__Oint,type,
    modulo_modulo_int: int > int > int ).

thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Nat__Onat,type,
    modulo_modulo_nat: nat > nat > nat ).

thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    collec99177395925924084rm_a_b: ( produc357393685978478089rm_a_b > $o ) > set_Pr4386577575007340137rm_a_b ).

thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Int__Oint,type,
    set_or4662586982721622107an_int: int > int > set_int ).

thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat,type,
    set_or4665077453230672383an_nat: nat > nat > set_nat ).

thf(sy_c_String_Ochar_Osize__char,type,
    size_char: char > nat ).

thf(sy_c_Utils_Oconcat__index__split_001t__FSet__Ofset_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
    concat3863192905831170157rm_a_b: product_prod_nat_nat > list_l3960025731926578043rm_a_b > product_prod_nat_nat ).

thf(sy_c_Utils_Otrancl__list_001t__FSet__Ofset_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
    trancl6568930836738952055rm_a_b: set_Pr3641583360614959815rm_a_b > set_Pr2797156446527370311rm_a_b ).

thf(sy_c_Utils_Otrancl__list_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    trancl_list_term_a_b: set_Pr4386577575007340137rm_a_b > set_Pr8564414093027780873rm_a_b ).

thf(sy_c_Utils_Otrancl__listp_001t__FSet__Ofset_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
    trancl5091790982450174325rm_a_b: ( fset_P3577670608877728751rm_a_b > fset_P3577670608877728751rm_a_b > $o ) > list_f7931864845379430901rm_a_b > list_f7931864845379430901rm_a_b > $o ).

thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__FSet__Ofset_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__FSet__Ofset_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J_J_J_J,type,
    accp_P3725179739350892143rm_a_b: ( produc7576762995520877624rm_a_b > produc7576762995520877624rm_a_b > $o ) > produc7576762995520877624rm_a_b > $o ).

thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__Set__Oset_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J_J_J_J,type,
    accp_P8087403281110482147rm_a_b: ( produc770370126490721196rm_a_b > produc770370126490721196rm_a_b > $o ) > produc770370126490721196rm_a_b > $o ).

thf(sy_c_member_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
    member_nat_nat: ( nat > nat ) > set_nat_nat > $o ).

thf(sy_c_member_001t__FOR____Certificate__Oftrs,type,
    member_FOR_ftrs: fOR_ftrs > set_FOR_ftrs > $o ).

thf(sy_c_member_001t__FSet__Ofset_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
    member2349228652077105432rm_a_b: fset_P3577670608877728751rm_a_b > set_fs4190738840245686991rm_a_b > $o ).

thf(sy_c_member_001t__Int__Oint,type,
    member_int: int > set_int > $o ).

thf(sy_c_member_001t__List__Olist_It__FSet__Ofset_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J_J,type,
    member3062333693255306910rm_a_b: list_f7931864845379430901rm_a_b > set_li926548939377968981rm_a_b > $o ).

thf(sy_c_member_001t__List__Olist_It__Set__Oset_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J_J,type,
    member857836065738737176rm_a_b: list_s5629610930313288943rm_a_b > set_li8389183165251233743rm_a_b > $o ).

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

thf(sy_c_member_001t__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
    member7226740684066999833at_nat: produc8199716216217303280at_nat > set_Pr9093778441882193744at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__FSet__Ofset_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J_Mt__FSet__Ofset_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J_J,type,
    member2354150401426528528rm_a_b: produc7740921438345627367rm_a_b > set_Pr3641583360614959815rm_a_b > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__FSet__Ofset_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J_J_Mt__List__Olist_It__FSet__Ofset_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J_J_J,type,
    member6557215607983664784rm_a_b: produc4312625336439893351rm_a_b > set_Pr2797156446527370311rm_a_b > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Set__Oset_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J_J_Mt__List__Olist_It__Set__Oset_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J_J_J,type,
    member4739679974255130896rm_a_b: produc8638778270066257639rm_a_b > set_Pr921323698366220999rm_a_b > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Term__Oterm_Itf__a_Mtf__b_J_J_Mt__List__Olist_It__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
    member4405265420456397394rm_a_b: produc51424535725745577rm_a_b > set_Pr8564414093027780873rm_a_b > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_Mt__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
    member8417600551952982416rm_a_b: produc4523971788519308903rm_a_b > set_Pr2972776593051762503rm_a_b > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    member5869715511025134514rm_a_b: produc357393685978478089rm_a_b > set_Pr4386577575007340137rm_a_b > $o ).

thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
    member8823642877475723282rm_a_b: set_Pr4386577575007340137rm_a_b > set_se1467346398288745161rm_a_b > $o ).

thf(sy_c_member_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    member_term_a_b: term_a_b > set_term_a_b > $o ).

thf(sy_v_Rs,type,
    rs: list_f7931864845379430901rm_a_b ).

thf(sy_v_is,type,
    is: list_FOR_ftrs ).

% Relevant facts (1238)
thf(fact_0_fset__cong,axiom,
    ! [X: fset_term_a_b,Y: fset_term_a_b] :
      ( ( ( fset_term_a_b2 @ X )
        = ( fset_term_a_b2 @ Y ) )
      = ( X = Y ) ) ).

% fset_cong
thf(fact_1_fset__cong,axiom,
    ! [X: fset_P2527920629275104717rm_a_b,Y: fset_P2527920629275104717rm_a_b] :
      ( ( ( fset_P119123501830535382rm_a_b @ X )
        = ( fset_P119123501830535382rm_a_b @ Y ) )
      = ( X = Y ) ) ).

% fset_cong
thf(fact_2_fset__cong,axiom,
    ! [X: fset_P3577670608877728751rm_a_b,Y: fset_P3577670608877728751rm_a_b] :
      ( ( ( fset_P6170650533370837496rm_a_b @ X )
        = ( fset_P6170650533370837496rm_a_b @ Y ) )
      = ( X = Y ) ) ).

% fset_cong
thf(fact_3_assms,axiom,
    ! [X2: fOR_ftrs] :
      ( ( member_FOR_ftrs @ X2 @ ( set_FOR_ftrs2 @ is ) )
     => ( ord_less_nat @ ( fOR_case_ftrs_nat @ id_nat @ id_nat @ X2 ) @ ( size_s1967422219310127201rm_a_b @ rs ) ) ) ).

% assms
thf(fact_4_map__eq__map__tailrec,axiom,
    map_fs2147491833908496126rm_a_b = map_ta2600070521075040840rm_a_b ).

% map_eq_map_tailrec
thf(fact_5_fId__on_Orep__eq,axiom,
    ! [X: fset_P2527920629275104717rm_a_b] :
      ( ( fset_P1685187771707802710rm_a_b @ ( fSet_f5531784347500406511rm_a_b @ X ) )
      = ( id_on_4065744930832257284rm_a_b @ ( fset_P119123501830535382rm_a_b @ X ) ) ) ).

% fId_on.rep_eq
thf(fact_6_fId__on_Orep__eq,axiom,
    ! [X: fset_P3577670608877728751rm_a_b] :
      ( ( fset_P119123501830535382rm_a_b @ ( fSet_f6257059229746172561rm_a_b @ X ) )
      = ( id_on_115818524817812774rm_a_b @ ( fset_P6170650533370837496rm_a_b @ X ) ) ) ).

% fId_on.rep_eq
thf(fact_7_fId__on_Orep__eq,axiom,
    ! [X: fset_term_a_b] :
      ( ( fset_P6170650533370837496rm_a_b @ ( fSet_fId_on_term_a_b @ X ) )
      = ( id_on_term_a_b @ ( fset_term_a_b2 @ X ) ) ) ).

% fId_on.rep_eq
thf(fact_8_fset_Opred__cong,axiom,
    ! [X: fset_term_a_b,Ya: fset_term_a_b,P: term_a_b > $o,Pa: term_a_b > $o] :
      ( ( X = Ya )
     => ( ! [Z: term_a_b] :
            ( ( member_term_a_b @ Z @ ( fset_term_a_b2 @ Ya ) )
           => ( ( P @ Z )
              = ( Pa @ Z ) ) )
       => ( ( pred_fset_term_a_b @ P @ X )
          = ( pred_fset_term_a_b @ Pa @ Ya ) ) ) ) ).

% fset.pred_cong
thf(fact_9_fset_Opred__cong,axiom,
    ! [X: fset_P2527920629275104717rm_a_b,Ya: fset_P2527920629275104717rm_a_b,P: produc4523971788519308903rm_a_b > $o,Pa: produc4523971788519308903rm_a_b > $o] :
      ( ( X = Ya )
     => ( ! [Z: produc4523971788519308903rm_a_b] :
            ( ( member8417600551952982416rm_a_b @ Z @ ( fset_P119123501830535382rm_a_b @ Ya ) )
           => ( ( P @ Z )
              = ( Pa @ Z ) ) )
       => ( ( pred_f8568102129774320521rm_a_b @ P @ X )
          = ( pred_f8568102129774320521rm_a_b @ Pa @ Ya ) ) ) ) ).

% fset.pred_cong
thf(fact_10_fset_Opred__cong,axiom,
    ! [X: fset_P3577670608877728751rm_a_b,Ya: fset_P3577670608877728751rm_a_b,P: produc357393685978478089rm_a_b > $o,Pa: produc357393685978478089rm_a_b > $o] :
      ( ( X = Ya )
     => ( ! [Z: produc357393685978478089rm_a_b] :
            ( ( member5869715511025134514rm_a_b @ Z @ ( fset_P6170650533370837496rm_a_b @ Ya ) )
           => ( ( P @ Z )
              = ( Pa @ Z ) ) )
       => ( ( pred_f8979122959637397547rm_a_b @ P @ X )
          = ( pred_f8979122959637397547rm_a_b @ Pa @ Ya ) ) ) ) ).

% fset.pred_cong
thf(fact_11_fset_Opred__mono__strong,axiom,
    ! [P: term_a_b > $o,X: fset_term_a_b,Pa: term_a_b > $o] :
      ( ( pred_fset_term_a_b @ P @ X )
     => ( ! [Z: term_a_b] :
            ( ( member_term_a_b @ Z @ ( fset_term_a_b2 @ X ) )
           => ( ( P @ Z )
             => ( Pa @ Z ) ) )
       => ( pred_fset_term_a_b @ Pa @ X ) ) ) ).

% fset.pred_mono_strong
thf(fact_12_fset_Opred__mono__strong,axiom,
    ! [P: produc4523971788519308903rm_a_b > $o,X: fset_P2527920629275104717rm_a_b,Pa: produc4523971788519308903rm_a_b > $o] :
      ( ( pred_f8568102129774320521rm_a_b @ P @ X )
     => ( ! [Z: produc4523971788519308903rm_a_b] :
            ( ( member8417600551952982416rm_a_b @ Z @ ( fset_P119123501830535382rm_a_b @ X ) )
           => ( ( P @ Z )
             => ( Pa @ Z ) ) )
       => ( pred_f8568102129774320521rm_a_b @ Pa @ X ) ) ) ).

% fset.pred_mono_strong
thf(fact_13_fset_Opred__mono__strong,axiom,
    ! [P: produc357393685978478089rm_a_b > $o,X: fset_P3577670608877728751rm_a_b,Pa: produc357393685978478089rm_a_b > $o] :
      ( ( pred_f8979122959637397547rm_a_b @ P @ X )
     => ( ! [Z: produc357393685978478089rm_a_b] :
            ( ( member5869715511025134514rm_a_b @ Z @ ( fset_P6170650533370837496rm_a_b @ X ) )
           => ( ( P @ Z )
             => ( Pa @ Z ) ) )
       => ( pred_f8979122959637397547rm_a_b @ Pa @ X ) ) ) ).

% fset.pred_mono_strong
thf(fact_14_fImage_Orep__eq,axiom,
    ! [X: fset_P5492165508529054490rm_a_b,Xa: fset_term_a_b] :
      ( ( fset_P6170650533370837496rm_a_b @ ( fSet_f2752768869892530229rm_a_b @ X @ Xa ) )
      = ( image_9078632467741184736rm_a_b @ ( fset_P2116452259805630523rm_a_b @ X ) @ ( fset_term_a_b2 @ Xa ) ) ) ).

% fImage.rep_eq
thf(fact_15_fImage_Orep__eq,axiom,
    ! [X: fset_P7739237331454329583rm_a_b,Xa: fset_P2527920629275104717rm_a_b] :
      ( ( fset_P6170650533370837496rm_a_b @ ( fSet_f3945364838595517600rm_a_b @ X @ Xa ) )
      = ( image_1022933872408103221rm_a_b @ ( fset_P5488872620772533496rm_a_b @ X ) @ ( fset_P119123501830535382rm_a_b @ Xa ) ) ) ).

% fImage.rep_eq
thf(fact_16_fImage_Orep__eq,axiom,
    ! [X: fset_P7200630051033007996rm_a_b,Xa: fset_P3577670608877728751rm_a_b] :
      ( ( fset_term_a_b2 @ ( fSet_f4584986798926822879rm_a_b @ X @ Xa ) )
      = ( image_1687478359920701578rm_a_b @ ( fset_P1382941437477160349rm_a_b @ X ) @ ( fset_P6170650533370837496rm_a_b @ Xa ) ) ) ).

% fImage.rep_eq
thf(fact_17_fImage_Orep__eq,axiom,
    ! [X: fset_P1968521938398262234rm_a_b,Xa: fset_P2527920629275104717rm_a_b] :
      ( ( fset_term_a_b2 @ ( fSet_f1441844802894473405rm_a_b @ X @ Xa ) )
      = ( image_1384895276050394088rm_a_b @ ( fset_P7858287489005908731rm_a_b @ X ) @ ( fset_P119123501830535382rm_a_b @ Xa ) ) ) ).

% fImage.rep_eq
thf(fact_18_fImage_Orep__eq,axiom,
    ! [X: fset_P6847904874274293035rm_a_b,Xa: fset_P3577670608877728751rm_a_b] :
      ( ( fset_P119123501830535382rm_a_b @ ( fSet_f8757998344374646236rm_a_b @ X @ Xa ) )
      = ( image_5835567378187231857rm_a_b @ ( fset_P8203092514377958708rm_a_b @ X ) @ ( fset_P6170650533370837496rm_a_b @ Xa ) ) ) ).

% fImage.rep_eq
thf(fact_19_fImage_Orep__eq,axiom,
    ! [X: fset_P4267757793467315704rm_a_b,Xa: fset_term_a_b] :
      ( ( fset_P119123501830535382rm_a_b @ ( fSet_f1940817589563042451rm_a_b @ X @ Xa ) )
      = ( image_1883868062718963134rm_a_b @ ( fset_P8824575414863537177rm_a_b @ X ) @ ( fset_term_a_b2 @ Xa ) ) ) ).

% fImage.rep_eq
thf(fact_20_fImage_Orep__eq,axiom,
    ! [X: fset_P2477867035931772237rm_a_b,Xa: fset_P2527920629275104717rm_a_b] :
      ( ( fset_P119123501830535382rm_a_b @ ( fSet_f9057368216645834622rm_a_b @ X @ Xa ) )
      = ( image_4755580888768615315rm_a_b @ ( fset_P1685187771707802710rm_a_b @ X ) @ ( fset_P119123501830535382rm_a_b @ Xa ) ) ) ).

% fImage.rep_eq
thf(fact_21_fImage_Orep__eq,axiom,
    ! [X: fset_P3577670608877728751rm_a_b,Xa: fset_term_a_b] :
      ( ( fset_term_a_b2 @ ( fSet_f8210585146631509224rm_a_b @ X @ Xa ) )
      = ( image_6923787618150785661rm_a_b @ ( fset_P6170650533370837496rm_a_b @ X ) @ ( fset_term_a_b2 @ Xa ) ) ) ).

% fImage.rep_eq
thf(fact_22_fImage_Orep__eq,axiom,
    ! [X: fset_P2527920629275104717rm_a_b,Xa: fset_P3577670608877728751rm_a_b] :
      ( ( fset_P6170650533370837496rm_a_b @ ( fSet_f3823212753407577598rm_a_b @ X @ Xa ) )
      = ( image_871640889055226643rm_a_b @ ( fset_P119123501830535382rm_a_b @ X ) @ ( fset_P6170650533370837496rm_a_b @ Xa ) ) ) ).

% fImage.rep_eq
thf(fact_23_rotate1__map,axiom,
    ! [F: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b,Xs: list_s5629610930313288943rm_a_b] :
      ( ( rotate753530590164781500rm_a_b @ ( map_se1470442168586473604rm_a_b @ F @ Xs ) )
      = ( map_se1470442168586473604rm_a_b @ F @ ( rotate753530590164781500rm_a_b @ Xs ) ) ) ).

% rotate1_map
thf(fact_24_rotate1__map,axiom,
    ! [F: set_Pr4386577575007340137rm_a_b > fset_P3577670608877728751rm_a_b,Xs: list_s5629610930313288943rm_a_b] :
      ( ( rotate8515649208237718978rm_a_b @ ( map_se2528531069056196746rm_a_b @ F @ Xs ) )
      = ( map_se2528531069056196746rm_a_b @ F @ ( rotate753530590164781500rm_a_b @ Xs ) ) ) ).

% rotate1_map
thf(fact_25_rotate1__map,axiom,
    ! [F: fset_P3577670608877728751rm_a_b > fset_P3577670608877728751rm_a_b,Xs: list_f7931864845379430901rm_a_b] :
      ( ( rotate8515649208237718978rm_a_b @ ( map_fs9003807414418110980rm_a_b @ F @ Xs ) )
      = ( map_fs9003807414418110980rm_a_b @ F @ ( rotate8515649208237718978rm_a_b @ Xs ) ) ) ).

% rotate1_map
thf(fact_26_rotate1__map,axiom,
    ! [F: fset_P3577670608877728751rm_a_b > set_Pr4386577575007340137rm_a_b,Xs: list_f7931864845379430901rm_a_b] :
      ( ( rotate753530590164781500rm_a_b @ ( map_fs2147491833908496126rm_a_b @ F @ Xs ) )
      = ( map_fs2147491833908496126rm_a_b @ F @ ( rotate8515649208237718978rm_a_b @ Xs ) ) ) ).

% rotate1_map
thf(fact_27_list_Omap__disc__iff,axiom,
    ! [F: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b,A: list_s5629610930313288943rm_a_b] :
      ( ( ( map_se1470442168586473604rm_a_b @ F @ A )
        = nil_se7334864872594423151rm_a_b )
      = ( A = nil_se7334864872594423151rm_a_b ) ) ).

% list.map_disc_iff
thf(fact_28_list_Omap__disc__iff,axiom,
    ! [F: set_Pr4386577575007340137rm_a_b > fset_P3577670608877728751rm_a_b,A: list_s5629610930313288943rm_a_b] :
      ( ( ( map_se2528531069056196746rm_a_b @ F @ A )
        = nil_fs4020363321024022517rm_a_b )
      = ( A = nil_se7334864872594423151rm_a_b ) ) ).

% list.map_disc_iff
thf(fact_29_list_Omap__disc__iff,axiom,
    ! [F: fset_P3577670608877728751rm_a_b > fset_P3577670608877728751rm_a_b,A: list_f7931864845379430901rm_a_b] :
      ( ( ( map_fs9003807414418110980rm_a_b @ F @ A )
        = nil_fs4020363321024022517rm_a_b )
      = ( A = nil_fs4020363321024022517rm_a_b ) ) ).

% list.map_disc_iff
thf(fact_30_list_Omap__disc__iff,axiom,
    ! [F: fset_P3577670608877728751rm_a_b > set_Pr4386577575007340137rm_a_b,A: list_f7931864845379430901rm_a_b] :
      ( ( ( map_fs2147491833908496126rm_a_b @ F @ A )
        = nil_se7334864872594423151rm_a_b )
      = ( A = nil_fs4020363321024022517rm_a_b ) ) ).

% list.map_disc_iff
thf(fact_31_Nil__is__map__conv,axiom,
    ! [F: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b,Xs: list_s5629610930313288943rm_a_b] :
      ( ( nil_se7334864872594423151rm_a_b
        = ( map_se1470442168586473604rm_a_b @ F @ Xs ) )
      = ( Xs = nil_se7334864872594423151rm_a_b ) ) ).

% Nil_is_map_conv
thf(fact_32_Nil__is__map__conv,axiom,
    ! [F: set_Pr4386577575007340137rm_a_b > fset_P3577670608877728751rm_a_b,Xs: list_s5629610930313288943rm_a_b] :
      ( ( nil_fs4020363321024022517rm_a_b
        = ( map_se2528531069056196746rm_a_b @ F @ Xs ) )
      = ( Xs = nil_se7334864872594423151rm_a_b ) ) ).

% Nil_is_map_conv
thf(fact_33_Nil__is__map__conv,axiom,
    ! [F: fset_P3577670608877728751rm_a_b > fset_P3577670608877728751rm_a_b,Xs: list_f7931864845379430901rm_a_b] :
      ( ( nil_fs4020363321024022517rm_a_b
        = ( map_fs9003807414418110980rm_a_b @ F @ Xs ) )
      = ( Xs = nil_fs4020363321024022517rm_a_b ) ) ).

% Nil_is_map_conv
thf(fact_34_Nil__is__map__conv,axiom,
    ! [F: fset_P3577670608877728751rm_a_b > set_Pr4386577575007340137rm_a_b,Xs: list_f7931864845379430901rm_a_b] :
      ( ( nil_se7334864872594423151rm_a_b
        = ( map_fs2147491833908496126rm_a_b @ F @ Xs ) )
      = ( Xs = nil_fs4020363321024022517rm_a_b ) ) ).

% Nil_is_map_conv
thf(fact_35_map__is__Nil__conv,axiom,
    ! [F: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b,Xs: list_s5629610930313288943rm_a_b] :
      ( ( ( map_se1470442168586473604rm_a_b @ F @ Xs )
        = nil_se7334864872594423151rm_a_b )
      = ( Xs = nil_se7334864872594423151rm_a_b ) ) ).

% map_is_Nil_conv
thf(fact_36_map__is__Nil__conv,axiom,
    ! [F: set_Pr4386577575007340137rm_a_b > fset_P3577670608877728751rm_a_b,Xs: list_s5629610930313288943rm_a_b] :
      ( ( ( map_se2528531069056196746rm_a_b @ F @ Xs )
        = nil_fs4020363321024022517rm_a_b )
      = ( Xs = nil_se7334864872594423151rm_a_b ) ) ).

% map_is_Nil_conv
thf(fact_37_map__is__Nil__conv,axiom,
    ! [F: fset_P3577670608877728751rm_a_b > fset_P3577670608877728751rm_a_b,Xs: list_f7931864845379430901rm_a_b] :
      ( ( ( map_fs9003807414418110980rm_a_b @ F @ Xs )
        = nil_fs4020363321024022517rm_a_b )
      = ( Xs = nil_fs4020363321024022517rm_a_b ) ) ).

% map_is_Nil_conv
thf(fact_38_map__is__Nil__conv,axiom,
    ! [F: fset_P3577670608877728751rm_a_b > set_Pr4386577575007340137rm_a_b,Xs: list_f7931864845379430901rm_a_b] :
      ( ( ( map_fs2147491833908496126rm_a_b @ F @ Xs )
        = nil_se7334864872594423151rm_a_b )
      = ( Xs = nil_fs4020363321024022517rm_a_b ) ) ).

% map_is_Nil_conv
thf(fact_39_distinct__adj__mapD,axiom,
    ! [F: fset_P3577670608877728751rm_a_b > set_Pr4386577575007340137rm_a_b,Xs: list_f7931864845379430901rm_a_b] :
      ( ( distin160423016700913479rm_a_b @ ( map_fs2147491833908496126rm_a_b @ F @ Xs ) )
     => ( distin8264758955140711885rm_a_b @ Xs ) ) ).

% distinct_adj_mapD
thf(fact_40_length__map,axiom,
    ! [F: fset_P3577670608877728751rm_a_b > set_Pr4386577575007340137rm_a_b,Xs: list_f7931864845379430901rm_a_b] :
      ( ( size_s3663623692424790363rm_a_b @ ( map_fs2147491833908496126rm_a_b @ F @ Xs ) )
      = ( size_s1967422219310127201rm_a_b @ Xs ) ) ).

% length_map
thf(fact_41_length__map,axiom,
    ! [F: fset_P3577670608877728751rm_a_b > fset_P3577670608877728751rm_a_b,Xs: list_f7931864845379430901rm_a_b] :
      ( ( size_s1967422219310127201rm_a_b @ ( map_fs9003807414418110980rm_a_b @ F @ Xs ) )
      = ( size_s1967422219310127201rm_a_b @ Xs ) ) ).

% length_map
thf(fact_42_map__eq__conv,axiom,
    ! [F: fset_P3577670608877728751rm_a_b > set_Pr4386577575007340137rm_a_b,Xs: list_f7931864845379430901rm_a_b,G: fset_P3577670608877728751rm_a_b > set_Pr4386577575007340137rm_a_b] :
      ( ( ( map_fs2147491833908496126rm_a_b @ F @ Xs )
        = ( map_fs2147491833908496126rm_a_b @ G @ Xs ) )
      = ( ! [X3: fset_P3577670608877728751rm_a_b] :
            ( ( member2349228652077105432rm_a_b @ X3 @ ( set_fs2829528993146950532rm_a_b @ Xs ) )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) ) ) ) ).

% map_eq_conv
thf(fact_43_list_Omap__id0,axiom,
    ( ( map_nat_nat @ id_nat )
    = id_list_nat ) ).

% list.map_id0
thf(fact_44_rotate1__is__Nil__conv,axiom,
    ! [Xs: list_s5629610930313288943rm_a_b] :
      ( ( ( rotate753530590164781500rm_a_b @ Xs )
        = nil_se7334864872594423151rm_a_b )
      = ( Xs = nil_se7334864872594423151rm_a_b ) ) ).

% rotate1_is_Nil_conv
thf(fact_45_rotate1__is__Nil__conv,axiom,
    ! [Xs: list_f7931864845379430901rm_a_b] :
      ( ( ( rotate8515649208237718978rm_a_b @ Xs )
        = nil_fs4020363321024022517rm_a_b )
      = ( Xs = nil_fs4020363321024022517rm_a_b ) ) ).

% rotate1_is_Nil_conv
thf(fact_46_length__rotate1,axiom,
    ! [Xs: list_s5629610930313288943rm_a_b] :
      ( ( size_s3663623692424790363rm_a_b @ ( rotate753530590164781500rm_a_b @ Xs ) )
      = ( size_s3663623692424790363rm_a_b @ Xs ) ) ).

% length_rotate1
thf(fact_47_length__rotate1,axiom,
    ! [Xs: list_f7931864845379430901rm_a_b] :
      ( ( size_s1967422219310127201rm_a_b @ ( rotate8515649208237718978rm_a_b @ Xs ) )
      = ( size_s1967422219310127201rm_a_b @ Xs ) ) ).

% length_rotate1
thf(fact_48_set__rotate1,axiom,
    ! [Xs: list_s5629610930313288943rm_a_b] :
      ( ( set_se5504503741092831358rm_a_b @ ( rotate753530590164781500rm_a_b @ Xs ) )
      = ( set_se5504503741092831358rm_a_b @ Xs ) ) ).

% set_rotate1
thf(fact_49_set__rotate1,axiom,
    ! [Xs: list_f7931864845379430901rm_a_b] :
      ( ( set_fs2829528993146950532rm_a_b @ ( rotate8515649208237718978rm_a_b @ Xs ) )
      = ( set_fs2829528993146950532rm_a_b @ Xs ) ) ).

% set_rotate1
thf(fact_50_set__rotate1,axiom,
    ! [Xs: list_FOR_ftrs] :
      ( ( set_FOR_ftrs2 @ ( rotate1_FOR_ftrs @ Xs ) )
      = ( set_FOR_ftrs2 @ Xs ) ) ).

% set_rotate1
thf(fact_51_rotate1_Osimps_I1_J,axiom,
    ( ( rotate753530590164781500rm_a_b @ nil_se7334864872594423151rm_a_b )
    = nil_se7334864872594423151rm_a_b ) ).

% rotate1.simps(1)
thf(fact_52_rotate1_Osimps_I1_J,axiom,
    ( ( rotate8515649208237718978rm_a_b @ nil_fs4020363321024022517rm_a_b )
    = nil_fs4020363321024022517rm_a_b ) ).

% rotate1.simps(1)
thf(fact_53_length__induct,axiom,
    ! [P: list_f7931864845379430901rm_a_b > $o,Xs: list_f7931864845379430901rm_a_b] :
      ( ! [Xs2: list_f7931864845379430901rm_a_b] :
          ( ! [Ys: list_f7931864845379430901rm_a_b] :
              ( ( ord_less_nat @ ( size_s1967422219310127201rm_a_b @ Ys ) @ ( size_s1967422219310127201rm_a_b @ Xs2 ) )
             => ( P @ Ys ) )
         => ( P @ Xs2 ) )
     => ( P @ Xs ) ) ).

% length_induct
thf(fact_54_distinct__adj__Nil,axiom,
    distin160423016700913479rm_a_b @ nil_se7334864872594423151rm_a_b ).

% distinct_adj_Nil
thf(fact_55_distinct__adj__Nil,axiom,
    distin8264758955140711885rm_a_b @ nil_fs4020363321024022517rm_a_b ).

% distinct_adj_Nil
thf(fact_56_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs2: list_f7931864845379430901rm_a_b] :
      ( ( size_s1967422219310127201rm_a_b @ Xs2 )
      = N ) ).

% Ex_list_of_length
thf(fact_57_neq__if__length__neq,axiom,
    ! [Xs: list_f7931864845379430901rm_a_b,Ys2: list_f7931864845379430901rm_a_b] :
      ( ( ( size_s1967422219310127201rm_a_b @ Xs )
       != ( size_s1967422219310127201rm_a_b @ Ys2 ) )
     => ( Xs != Ys2 ) ) ).

% neq_if_length_neq
thf(fact_58_ex__map__conv,axiom,
    ! [Ys2: list_s5629610930313288943rm_a_b,F: fset_P3577670608877728751rm_a_b > set_Pr4386577575007340137rm_a_b] :
      ( ( ? [Xs3: list_f7931864845379430901rm_a_b] :
            ( Ys2
            = ( map_fs2147491833908496126rm_a_b @ F @ Xs3 ) ) )
      = ( ! [X3: set_Pr4386577575007340137rm_a_b] :
            ( ( member8823642877475723282rm_a_b @ X3 @ ( set_se5504503741092831358rm_a_b @ Ys2 ) )
           => ? [Y2: fset_P3577670608877728751rm_a_b] :
                ( X3
                = ( F @ Y2 ) ) ) ) ) ).

% ex_map_conv
thf(fact_59_map__cong,axiom,
    ! [Xs: list_f7931864845379430901rm_a_b,Ys2: list_f7931864845379430901rm_a_b,F: fset_P3577670608877728751rm_a_b > set_Pr4386577575007340137rm_a_b,G: fset_P3577670608877728751rm_a_b > set_Pr4386577575007340137rm_a_b] :
      ( ( Xs = Ys2 )
     => ( ! [X4: fset_P3577670608877728751rm_a_b] :
            ( ( member2349228652077105432rm_a_b @ X4 @ ( set_fs2829528993146950532rm_a_b @ Ys2 ) )
           => ( ( F @ X4 )
              = ( G @ X4 ) ) )
       => ( ( map_fs2147491833908496126rm_a_b @ F @ Xs )
          = ( map_fs2147491833908496126rm_a_b @ G @ Ys2 ) ) ) ) ).

% map_cong
thf(fact_60_map__idI,axiom,
    ! [Xs: list_P8875379029341186191rm_a_b,F: produc357393685978478089rm_a_b > produc357393685978478089rm_a_b] :
      ( ! [X4: produc357393685978478089rm_a_b] :
          ( ( member5869715511025134514rm_a_b @ X4 @ ( set_Pr2370880052973118494rm_a_b @ Xs ) )
         => ( ( F @ X4 )
            = X4 ) )
     => ( ( map_Pr3281474655358167172rm_a_b @ F @ Xs )
        = Xs ) ) ).

% map_idI
thf(fact_61_map__idI,axiom,
    ! [Xs: list_FOR_ftrs,F: fOR_ftrs > fOR_ftrs] :
      ( ! [X4: fOR_ftrs] :
          ( ( member_FOR_ftrs @ X4 @ ( set_FOR_ftrs2 @ Xs ) )
         => ( ( F @ X4 )
            = X4 ) )
     => ( ( map_FO5875617154011697824R_ftrs @ F @ Xs )
        = Xs ) ) ).

% map_idI
thf(fact_62_map__ext,axiom,
    ! [Xs: list_f7931864845379430901rm_a_b,F: fset_P3577670608877728751rm_a_b > set_Pr4386577575007340137rm_a_b,G: fset_P3577670608877728751rm_a_b > set_Pr4386577575007340137rm_a_b] :
      ( ! [X4: fset_P3577670608877728751rm_a_b] :
          ( ( member2349228652077105432rm_a_b @ X4 @ ( set_fs2829528993146950532rm_a_b @ Xs ) )
         => ( ( F @ X4 )
            = ( G @ X4 ) ) )
     => ( ( map_fs2147491833908496126rm_a_b @ F @ Xs )
        = ( map_fs2147491833908496126rm_a_b @ G @ Xs ) ) ) ).

% map_ext
thf(fact_63_list_Omap__ident__strong,axiom,
    ! [T: list_P8875379029341186191rm_a_b,F: produc357393685978478089rm_a_b > produc357393685978478089rm_a_b] :
      ( ! [Z: produc357393685978478089rm_a_b] :
          ( ( member5869715511025134514rm_a_b @ Z @ ( set_Pr2370880052973118494rm_a_b @ T ) )
         => ( ( F @ Z )
            = Z ) )
     => ( ( map_Pr3281474655358167172rm_a_b @ F @ T )
        = T ) ) ).

% list.map_ident_strong
thf(fact_64_list_Omap__ident__strong,axiom,
    ! [T: list_FOR_ftrs,F: fOR_ftrs > fOR_ftrs] :
      ( ! [Z: fOR_ftrs] :
          ( ( member_FOR_ftrs @ Z @ ( set_FOR_ftrs2 @ T ) )
         => ( ( F @ Z )
            = Z ) )
     => ( ( map_FO5875617154011697824R_ftrs @ F @ T )
        = T ) ) ).

% list.map_ident_strong
thf(fact_65_list_Oinj__map__strong,axiom,
    ! [X: list_f7931864845379430901rm_a_b,Xa: list_f7931864845379430901rm_a_b,F: fset_P3577670608877728751rm_a_b > set_Pr4386577575007340137rm_a_b,Fa: fset_P3577670608877728751rm_a_b > set_Pr4386577575007340137rm_a_b] :
      ( ! [Z: fset_P3577670608877728751rm_a_b,Za: fset_P3577670608877728751rm_a_b] :
          ( ( member2349228652077105432rm_a_b @ Z @ ( set_fs2829528993146950532rm_a_b @ X ) )
         => ( ( member2349228652077105432rm_a_b @ Za @ ( set_fs2829528993146950532rm_a_b @ Xa ) )
           => ( ( ( F @ Z )
                = ( Fa @ Za ) )
             => ( Z = Za ) ) ) )
     => ( ( ( map_fs2147491833908496126rm_a_b @ F @ X )
          = ( map_fs2147491833908496126rm_a_b @ Fa @ Xa ) )
       => ( X = Xa ) ) ) ).

% list.inj_map_strong
thf(fact_66_list_Omap__cong0,axiom,
    ! [X: list_f7931864845379430901rm_a_b,F: fset_P3577670608877728751rm_a_b > set_Pr4386577575007340137rm_a_b,G: fset_P3577670608877728751rm_a_b > set_Pr4386577575007340137rm_a_b] :
      ( ! [Z: fset_P3577670608877728751rm_a_b] :
          ( ( member2349228652077105432rm_a_b @ Z @ ( set_fs2829528993146950532rm_a_b @ X ) )
         => ( ( F @ Z )
            = ( G @ Z ) ) )
     => ( ( map_fs2147491833908496126rm_a_b @ F @ X )
        = ( map_fs2147491833908496126rm_a_b @ G @ X ) ) ) ).

% list.map_cong0
thf(fact_67_list_Omap__cong,axiom,
    ! [X: list_f7931864845379430901rm_a_b,Ya: list_f7931864845379430901rm_a_b,F: fset_P3577670608877728751rm_a_b > set_Pr4386577575007340137rm_a_b,G: fset_P3577670608877728751rm_a_b > set_Pr4386577575007340137rm_a_b] :
      ( ( X = Ya )
     => ( ! [Z: fset_P3577670608877728751rm_a_b] :
            ( ( member2349228652077105432rm_a_b @ Z @ ( set_fs2829528993146950532rm_a_b @ Ya ) )
           => ( ( F @ Z )
              = ( G @ Z ) ) )
       => ( ( map_fs2147491833908496126rm_a_b @ F @ X )
          = ( map_fs2147491833908496126rm_a_b @ G @ Ya ) ) ) ) ).

% list.map_cong
thf(fact_68_map__eq__imp__length__eq,axiom,
    ! [F: fset_P3577670608877728751rm_a_b > set_Pr4386577575007340137rm_a_b,Xs: list_f7931864845379430901rm_a_b,G: fset_P3577670608877728751rm_a_b > set_Pr4386577575007340137rm_a_b,Ys2: list_f7931864845379430901rm_a_b] :
      ( ( ( map_fs2147491833908496126rm_a_b @ F @ Xs )
        = ( map_fs2147491833908496126rm_a_b @ G @ Ys2 ) )
     => ( ( size_s1967422219310127201rm_a_b @ Xs )
        = ( size_s1967422219310127201rm_a_b @ Ys2 ) ) ) ).

% map_eq_imp_length_eq
thf(fact_69_list_Osimps_I8_J,axiom,
    ! [F: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b] :
      ( ( map_se1470442168586473604rm_a_b @ F @ nil_se7334864872594423151rm_a_b )
      = nil_se7334864872594423151rm_a_b ) ).

% list.simps(8)
thf(fact_70_list_Osimps_I8_J,axiom,
    ! [F: set_Pr4386577575007340137rm_a_b > fset_P3577670608877728751rm_a_b] :
      ( ( map_se2528531069056196746rm_a_b @ F @ nil_se7334864872594423151rm_a_b )
      = nil_fs4020363321024022517rm_a_b ) ).

% list.simps(8)
thf(fact_71_list_Osimps_I8_J,axiom,
    ! [F: fset_P3577670608877728751rm_a_b > fset_P3577670608877728751rm_a_b] :
      ( ( map_fs9003807414418110980rm_a_b @ F @ nil_fs4020363321024022517rm_a_b )
      = nil_fs4020363321024022517rm_a_b ) ).

% list.simps(8)
thf(fact_72_list_Osimps_I8_J,axiom,
    ! [F: fset_P3577670608877728751rm_a_b > set_Pr4386577575007340137rm_a_b] :
      ( ( map_fs2147491833908496126rm_a_b @ F @ nil_fs4020363321024022517rm_a_b )
      = nil_se7334864872594423151rm_a_b ) ).

% list.simps(8)
thf(fact_73_map__id,axiom,
    ! [T: list_nat] :
      ( ( map_nat_nat @ id_nat @ T )
      = T ) ).

% map_id
thf(fact_74_id__apply,axiom,
    ( id_nat
    = ( ^ [X3: nat] : X3 ) ) ).

% id_apply
thf(fact_75_in__set__simps_I3_J,axiom,
    ! [X: produc357393685978478089rm_a_b] :
      ~ ( member5869715511025134514rm_a_b @ X @ ( set_Pr2370880052973118494rm_a_b @ nil_Pr6942174756412032271rm_a_b ) ) ).

% in_set_simps(3)
thf(fact_76_in__set__simps_I3_J,axiom,
    ! [X: set_Pr4386577575007340137rm_a_b] :
      ~ ( member8823642877475723282rm_a_b @ X @ ( set_se5504503741092831358rm_a_b @ nil_se7334864872594423151rm_a_b ) ) ).

% in_set_simps(3)
thf(fact_77_in__set__simps_I3_J,axiom,
    ! [X: fset_P3577670608877728751rm_a_b] :
      ~ ( member2349228652077105432rm_a_b @ X @ ( set_fs2829528993146950532rm_a_b @ nil_fs4020363321024022517rm_a_b ) ) ).

% in_set_simps(3)
thf(fact_78_in__set__simps_I3_J,axiom,
    ! [X: fOR_ftrs] :
      ~ ( member_FOR_ftrs @ X @ ( set_FOR_ftrs2 @ nil_FOR_ftrs ) ) ).

% in_set_simps(3)
thf(fact_79_list__ex1__simps_I1_J,axiom,
    ! [P: set_Pr4386577575007340137rm_a_b > $o] :
      ~ ( list_e3628350431938618930rm_a_b @ P @ nil_se7334864872594423151rm_a_b ) ).

% list_ex1_simps(1)
thf(fact_80_list__ex1__simps_I1_J,axiom,
    ! [P: fset_P3577670608877728751rm_a_b > $o] :
      ~ ( list_e747710766979891000rm_a_b @ P @ nil_fs4020363321024022517rm_a_b ) ).

% list_ex1_simps(1)
thf(fact_81_length__greater__0__conv,axiom,
    ! [Xs: list_s5629610930313288943rm_a_b] :
      ( ( ord_less_nat @ zero_zero_nat @ ( size_s3663623692424790363rm_a_b @ Xs ) )
      = ( Xs != nil_se7334864872594423151rm_a_b ) ) ).

% length_greater_0_conv
thf(fact_82_length__greater__0__conv,axiom,
    ! [Xs: list_f7931864845379430901rm_a_b] :
      ( ( ord_less_nat @ zero_zero_nat @ ( size_s1967422219310127201rm_a_b @ Xs ) )
      = ( Xs != nil_fs4020363321024022517rm_a_b ) ) ).

% length_greater_0_conv
thf(fact_83_length__removeAll__less,axiom,
    ! [X: produc357393685978478089rm_a_b,Xs: list_P8875379029341186191rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ X @ ( set_Pr2370880052973118494rm_a_b @ Xs ) )
     => ( ord_less_nat @ ( size_s8600120110137556219rm_a_b @ ( remove405047584712840725rm_a_b @ X @ Xs ) ) @ ( size_s8600120110137556219rm_a_b @ Xs ) ) ) ).

% length_removeAll_less
thf(fact_84_length__removeAll__less,axiom,
    ! [X: fOR_ftrs,Xs: list_FOR_ftrs] :
      ( ( member_FOR_ftrs @ X @ ( set_FOR_ftrs2 @ Xs ) )
     => ( ord_less_nat @ ( size_s7038034906319482769R_ftrs @ ( removeAll_FOR_ftrs @ X @ Xs ) ) @ ( size_s7038034906319482769R_ftrs @ Xs ) ) ) ).

% length_removeAll_less
thf(fact_85_length__removeAll__less,axiom,
    ! [X: fset_P3577670608877728751rm_a_b,Xs: list_f7931864845379430901rm_a_b] :
      ( ( member2349228652077105432rm_a_b @ X @ ( set_fs2829528993146950532rm_a_b @ Xs ) )
     => ( ord_less_nat @ ( size_s1967422219310127201rm_a_b @ ( remove3100474213990342139rm_a_b @ X @ Xs ) ) @ ( size_s1967422219310127201rm_a_b @ Xs ) ) ) ).

% length_removeAll_less
thf(fact_86_nth__map,axiom,
    ! [N: nat,Xs: list_f7931864845379430901rm_a_b,F: fset_P3577670608877728751rm_a_b > set_Pr4386577575007340137rm_a_b] :
      ( ( ord_less_nat @ N @ ( size_s1967422219310127201rm_a_b @ Xs ) )
     => ( ( nth_se1992752789547675248rm_a_b @ ( map_fs2147491833908496126rm_a_b @ F @ Xs ) @ N )
        = ( F @ ( nth_fs3949069584078227574rm_a_b @ Xs @ N ) ) ) ) ).

% nth_map
thf(fact_87_id__def,axiom,
    ( id_nat
    = ( ^ [X3: nat] : X3 ) ) ).

% id_def
thf(fact_88_eq__id__iff,axiom,
    ! [F: nat > nat] :
      ( ( ! [X3: nat] :
            ( ( F @ X3 )
            = X3 ) )
      = ( F = id_nat ) ) ).

% eq_id_iff
thf(fact_89_length__pos__if__in__set,axiom,
    ! [X: produc357393685978478089rm_a_b,Xs: list_P8875379029341186191rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ X @ ( set_Pr2370880052973118494rm_a_b @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s8600120110137556219rm_a_b @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_90_length__pos__if__in__set,axiom,
    ! [X: fOR_ftrs,Xs: list_FOR_ftrs] :
      ( ( member_FOR_ftrs @ X @ ( set_FOR_ftrs2 @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s7038034906319482769R_ftrs @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_91_length__pos__if__in__set,axiom,
    ! [X: fset_P3577670608877728751rm_a_b,Xs: list_f7931864845379430901rm_a_b] :
      ( ( member2349228652077105432rm_a_b @ X @ ( set_fs2829528993146950532rm_a_b @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s1967422219310127201rm_a_b @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_92_mem__Collect__eq,axiom,
    ! [A: produc357393685978478089rm_a_b,P: produc357393685978478089rm_a_b > $o] :
      ( ( member5869715511025134514rm_a_b @ A @ ( collec99177395925924084rm_a_b @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_93_Collect__mem__eq,axiom,
    ! [A2: set_Pr4386577575007340137rm_a_b] :
      ( ( collec99177395925924084rm_a_b
        @ ^ [X3: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X3 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_94_nat__neq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( M != N )
      = ( ( ord_less_nat @ M @ N )
        | ( ord_less_nat @ N @ M ) ) ) ).

% nat_neq_iff
thf(fact_95_less__nat__zero__code,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% less_nat_zero_code
thf(fact_96_neq0__conv,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
      = ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% neq0_conv
thf(fact_97_bot__nat__0_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
      = ( ord_less_nat @ zero_zero_nat @ A ) ) ).

% bot_nat_0.not_eq_extremum
thf(fact_98_removeAll__id,axiom,
    ! [X: produc357393685978478089rm_a_b,Xs: list_P8875379029341186191rm_a_b] :
      ( ~ ( member5869715511025134514rm_a_b @ X @ ( set_Pr2370880052973118494rm_a_b @ Xs ) )
     => ( ( remove405047584712840725rm_a_b @ X @ Xs )
        = Xs ) ) ).

% removeAll_id
thf(fact_99_removeAll__id,axiom,
    ! [X: fOR_ftrs,Xs: list_FOR_ftrs] :
      ( ~ ( member_FOR_ftrs @ X @ ( set_FOR_ftrs2 @ Xs ) )
     => ( ( removeAll_FOR_ftrs @ X @ Xs )
        = Xs ) ) ).

% removeAll_id
thf(fact_100_length__0__conv,axiom,
    ! [Xs: list_s5629610930313288943rm_a_b] :
      ( ( ( size_s3663623692424790363rm_a_b @ Xs )
        = zero_zero_nat )
      = ( Xs = nil_se7334864872594423151rm_a_b ) ) ).

% length_0_conv
thf(fact_101_length__0__conv,axiom,
    ! [Xs: list_f7931864845379430901rm_a_b] :
      ( ( ( size_s1967422219310127201rm_a_b @ Xs )
        = zero_zero_nat )
      = ( Xs = nil_fs4020363321024022517rm_a_b ) ) ).

% length_0_conv
thf(fact_102_infinite__descent0,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N2: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N2 )
           => ( ~ ( P @ N2 )
             => ? [M2: nat] :
                  ( ( ord_less_nat @ M2 @ N2 )
                  & ~ ( P @ M2 ) ) ) )
       => ( P @ N ) ) ) ).

% infinite_descent0
thf(fact_103_gr__implies__not0,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not0
thf(fact_104_less__zeroE,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% less_zeroE
thf(fact_105_not__less0,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% not_less0
thf(fact_106_not__gr0,axiom,
    ! [N: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
      = ( N = zero_zero_nat ) ) ).

% not_gr0
thf(fact_107_gr0I,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% gr0I
thf(fact_108_bot__nat__0_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ zero_zero_nat ) ).

% bot_nat_0.extremum_strict
thf(fact_109_Ex__list__of__length__P,axiom,
    ! [N: nat,P: fset_P3577670608877728751rm_a_b > nat > $o] :
      ( ! [I: nat] :
          ( ( ord_less_nat @ I @ N )
         => ? [X2: fset_P3577670608877728751rm_a_b] : ( P @ X2 @ I ) )
     => ? [Xs2: list_f7931864845379430901rm_a_b] :
          ( ( ( size_s1967422219310127201rm_a_b @ Xs2 )
            = N )
          & ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ N )
             => ( P @ ( nth_fs3949069584078227574rm_a_b @ Xs2 @ I2 ) @ I2 ) ) ) ) ).

% Ex_list_of_length_P
thf(fact_110_removeAll_Osimps_I1_J,axiom,
    ! [X: set_Pr4386577575007340137rm_a_b] :
      ( ( remove4495598804540913269rm_a_b @ X @ nil_se7334864872594423151rm_a_b )
      = nil_se7334864872594423151rm_a_b ) ).

% removeAll.simps(1)
thf(fact_111_removeAll_Osimps_I1_J,axiom,
    ! [X: fset_P3577670608877728751rm_a_b] :
      ( ( remove3100474213990342139rm_a_b @ X @ nil_fs4020363321024022517rm_a_b )
      = nil_fs4020363321024022517rm_a_b ) ).

% removeAll.simps(1)
thf(fact_112_ex__set__conv__ex__nth,axiom,
    ! [Xs: list_FOR_ftrs,P: fOR_ftrs > $o] :
      ( ( ? [X3: fOR_ftrs] :
            ( ( member_FOR_ftrs @ X3 @ ( set_FOR_ftrs2 @ Xs ) )
            & ( P @ X3 ) ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s7038034906319482769R_ftrs @ Xs ) )
            & ( P @ ( nth_FOR_ftrs @ Xs @ I3 ) ) ) ) ) ).

% ex_set_conv_ex_nth
thf(fact_113_ex__set__conv__ex__nth,axiom,
    ! [Xs: list_f7931864845379430901rm_a_b,P: fset_P3577670608877728751rm_a_b > $o] :
      ( ( ? [X3: fset_P3577670608877728751rm_a_b] :
            ( ( member2349228652077105432rm_a_b @ X3 @ ( set_fs2829528993146950532rm_a_b @ Xs ) )
            & ( P @ X3 ) ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s1967422219310127201rm_a_b @ Xs ) )
            & ( P @ ( nth_fs3949069584078227574rm_a_b @ Xs @ I3 ) ) ) ) ) ).

% ex_set_conv_ex_nth
thf(fact_114_map__nth__eq__conv,axiom,
    ! [Xs: list_f7931864845379430901rm_a_b,Ys2: list_s5629610930313288943rm_a_b,F: fset_P3577670608877728751rm_a_b > set_Pr4386577575007340137rm_a_b] :
      ( ( ( size_s1967422219310127201rm_a_b @ Xs )
        = ( size_s3663623692424790363rm_a_b @ Ys2 ) )
     => ( ( ( map_fs2147491833908496126rm_a_b @ F @ Xs )
          = Ys2 )
        = ( ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ ( size_s3663623692424790363rm_a_b @ Ys2 ) )
             => ( ( F @ ( nth_fs3949069584078227574rm_a_b @ Xs @ I3 ) )
                = ( nth_se1992752789547675248rm_a_b @ Ys2 @ I3 ) ) ) ) ) ) ).

% map_nth_eq_conv
thf(fact_115_map__nth__eq__conv,axiom,
    ! [Xs: list_f7931864845379430901rm_a_b,Ys2: list_f7931864845379430901rm_a_b,F: fset_P3577670608877728751rm_a_b > fset_P3577670608877728751rm_a_b] :
      ( ( ( size_s1967422219310127201rm_a_b @ Xs )
        = ( size_s1967422219310127201rm_a_b @ Ys2 ) )
     => ( ( ( map_fs9003807414418110980rm_a_b @ F @ Xs )
          = Ys2 )
        = ( ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ ( size_s1967422219310127201rm_a_b @ Ys2 ) )
             => ( ( F @ ( nth_fs3949069584078227574rm_a_b @ Xs @ I3 ) )
                = ( nth_fs3949069584078227574rm_a_b @ Ys2 @ I3 ) ) ) ) ) ) ).

% map_nth_eq_conv
thf(fact_116_map__nth__conv,axiom,
    ! [F: fset_P3577670608877728751rm_a_b > set_Pr4386577575007340137rm_a_b,Ss: list_f7931864845379430901rm_a_b,G: fset_P3577670608877728751rm_a_b > set_Pr4386577575007340137rm_a_b,Ts: list_f7931864845379430901rm_a_b] :
      ( ( ( map_fs2147491833908496126rm_a_b @ F @ Ss )
        = ( map_fs2147491833908496126rm_a_b @ G @ Ts ) )
     => ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_s1967422219310127201rm_a_b @ Ss ) )
         => ( ( F @ ( nth_fs3949069584078227574rm_a_b @ Ss @ I2 ) )
            = ( G @ ( nth_fs3949069584078227574rm_a_b @ Ts @ I2 ) ) ) ) ) ).

% map_nth_conv
thf(fact_117_map__eq__conv_H,axiom,
    ! [F: fset_P3577670608877728751rm_a_b > set_Pr4386577575007340137rm_a_b,Xs: list_f7931864845379430901rm_a_b,G: fset_P3577670608877728751rm_a_b > set_Pr4386577575007340137rm_a_b,Ys2: list_f7931864845379430901rm_a_b] :
      ( ( ( map_fs2147491833908496126rm_a_b @ F @ Xs )
        = ( map_fs2147491833908496126rm_a_b @ G @ Ys2 ) )
      = ( ( ( size_s1967422219310127201rm_a_b @ Xs )
          = ( size_s1967422219310127201rm_a_b @ Ys2 ) )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s1967422219310127201rm_a_b @ Xs ) )
           => ( ( F @ ( nth_fs3949069584078227574rm_a_b @ Xs @ I3 ) )
              = ( G @ ( nth_fs3949069584078227574rm_a_b @ Ys2 @ I3 ) ) ) ) ) ) ).

% map_eq_conv'
thf(fact_118_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y3: list_f7931864845379430901rm_a_b,Z2: list_f7931864845379430901rm_a_b] : ( Y3 = Z2 ) )
    = ( ^ [Xs3: list_f7931864845379430901rm_a_b,Ys3: list_f7931864845379430901rm_a_b] :
          ( ( ( size_s1967422219310127201rm_a_b @ Xs3 )
            = ( size_s1967422219310127201rm_a_b @ Ys3 ) )
          & ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ ( size_s1967422219310127201rm_a_b @ Xs3 ) )
             => ( ( nth_fs3949069584078227574rm_a_b @ Xs3 @ I3 )
                = ( nth_fs3949069584078227574rm_a_b @ Ys3 @ I3 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_119_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > fset_P3577670608877728751rm_a_b > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ K )
           => ? [X5: fset_P3577670608877728751rm_a_b] : ( P @ I3 @ X5 ) ) )
      = ( ? [Xs3: list_f7931864845379430901rm_a_b] :
            ( ( ( size_s1967422219310127201rm_a_b @ Xs3 )
              = K )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ K )
               => ( P @ I3 @ ( nth_fs3949069584078227574rm_a_b @ Xs3 @ I3 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_120_nth__equalityI,axiom,
    ! [Xs: list_f7931864845379430901rm_a_b,Ys2: list_f7931864845379430901rm_a_b] :
      ( ( ( size_s1967422219310127201rm_a_b @ Xs )
        = ( size_s1967422219310127201rm_a_b @ Ys2 ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s1967422219310127201rm_a_b @ Xs ) )
           => ( ( nth_fs3949069584078227574rm_a_b @ Xs @ I )
              = ( nth_fs3949069584078227574rm_a_b @ Ys2 @ I ) ) )
       => ( Xs = Ys2 ) ) ) ).

% nth_equalityI
thf(fact_121_list__ex1__iff,axiom,
    ( list_e2893281867963234770rm_a_b
    = ( ^ [P2: produc357393685978478089rm_a_b > $o,Xs3: list_P8875379029341186191rm_a_b] :
        ? [X3: produc357393685978478089rm_a_b] :
          ( ( member5869715511025134514rm_a_b @ X3 @ ( set_Pr2370880052973118494rm_a_b @ Xs3 ) )
          & ( P2 @ X3 )
          & ! [Y2: produc357393685978478089rm_a_b] :
              ( ( ( member5869715511025134514rm_a_b @ Y2 @ ( set_Pr2370880052973118494rm_a_b @ Xs3 ) )
                & ( P2 @ Y2 ) )
             => ( Y2 = X3 ) ) ) ) ) ).

% list_ex1_iff
thf(fact_122_list__ex1__iff,axiom,
    ( list_ex1_FOR_ftrs
    = ( ^ [P2: fOR_ftrs > $o,Xs3: list_FOR_ftrs] :
        ? [X3: fOR_ftrs] :
          ( ( member_FOR_ftrs @ X3 @ ( set_FOR_ftrs2 @ Xs3 ) )
          & ( P2 @ X3 )
          & ! [Y2: fOR_ftrs] :
              ( ( ( member_FOR_ftrs @ Y2 @ ( set_FOR_ftrs2 @ Xs3 ) )
                & ( P2 @ Y2 ) )
             => ( Y2 = X3 ) ) ) ) ) ).

% list_ex1_iff
thf(fact_123_all__set__conv__all__nth,axiom,
    ! [Xs: list_FOR_ftrs,P: fOR_ftrs > $o] :
      ( ( ! [X3: fOR_ftrs] :
            ( ( member_FOR_ftrs @ X3 @ ( set_FOR_ftrs2 @ Xs ) )
           => ( P @ X3 ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s7038034906319482769R_ftrs @ Xs ) )
           => ( P @ ( nth_FOR_ftrs @ Xs @ I3 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_124_all__set__conv__all__nth,axiom,
    ! [Xs: list_f7931864845379430901rm_a_b,P: fset_P3577670608877728751rm_a_b > $o] :
      ( ( ! [X3: fset_P3577670608877728751rm_a_b] :
            ( ( member2349228652077105432rm_a_b @ X3 @ ( set_fs2829528993146950532rm_a_b @ Xs ) )
           => ( P @ X3 ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s1967422219310127201rm_a_b @ Xs ) )
           => ( P @ ( nth_fs3949069584078227574rm_a_b @ Xs @ I3 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_125_all__nth__imp__all__set,axiom,
    ! [Xs: list_P8875379029341186191rm_a_b,P: produc357393685978478089rm_a_b > $o,X: produc357393685978478089rm_a_b] :
      ( ! [I: nat] :
          ( ( ord_less_nat @ I @ ( size_s8600120110137556219rm_a_b @ Xs ) )
         => ( P @ ( nth_Pr1098598711982373392rm_a_b @ Xs @ I ) ) )
     => ( ( member5869715511025134514rm_a_b @ X @ ( set_Pr2370880052973118494rm_a_b @ Xs ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_126_all__nth__imp__all__set,axiom,
    ! [Xs: list_FOR_ftrs,P: fOR_ftrs > $o,X: fOR_ftrs] :
      ( ! [I: nat] :
          ( ( ord_less_nat @ I @ ( size_s7038034906319482769R_ftrs @ Xs ) )
         => ( P @ ( nth_FOR_ftrs @ Xs @ I ) ) )
     => ( ( member_FOR_ftrs @ X @ ( set_FOR_ftrs2 @ Xs ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_127_all__nth__imp__all__set,axiom,
    ! [Xs: list_f7931864845379430901rm_a_b,P: fset_P3577670608877728751rm_a_b > $o,X: fset_P3577670608877728751rm_a_b] :
      ( ! [I: nat] :
          ( ( ord_less_nat @ I @ ( size_s1967422219310127201rm_a_b @ Xs ) )
         => ( P @ ( nth_fs3949069584078227574rm_a_b @ Xs @ I ) ) )
     => ( ( member2349228652077105432rm_a_b @ X @ ( set_fs2829528993146950532rm_a_b @ Xs ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_128_in__set__conv__nth,axiom,
    ! [X: produc357393685978478089rm_a_b,Xs: list_P8875379029341186191rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ X @ ( set_Pr2370880052973118494rm_a_b @ Xs ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s8600120110137556219rm_a_b @ Xs ) )
            & ( ( nth_Pr1098598711982373392rm_a_b @ Xs @ I3 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_129_in__set__conv__nth,axiom,
    ! [X: fOR_ftrs,Xs: list_FOR_ftrs] :
      ( ( member_FOR_ftrs @ X @ ( set_FOR_ftrs2 @ Xs ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s7038034906319482769R_ftrs @ Xs ) )
            & ( ( nth_FOR_ftrs @ Xs @ I3 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_130_in__set__conv__nth,axiom,
    ! [X: fset_P3577670608877728751rm_a_b,Xs: list_f7931864845379430901rm_a_b] :
      ( ( member2349228652077105432rm_a_b @ X @ ( set_fs2829528993146950532rm_a_b @ Xs ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s1967422219310127201rm_a_b @ Xs ) )
            & ( ( nth_fs3949069584078227574rm_a_b @ Xs @ I3 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_131_list__ball__nth,axiom,
    ! [N: nat,Xs: list_FOR_ftrs,P: fOR_ftrs > $o] :
      ( ( ord_less_nat @ N @ ( size_s7038034906319482769R_ftrs @ Xs ) )
     => ( ! [X4: fOR_ftrs] :
            ( ( member_FOR_ftrs @ X4 @ ( set_FOR_ftrs2 @ Xs ) )
           => ( P @ X4 ) )
       => ( P @ ( nth_FOR_ftrs @ Xs @ N ) ) ) ) ).

% list_ball_nth
thf(fact_132_list__ball__nth,axiom,
    ! [N: nat,Xs: list_f7931864845379430901rm_a_b,P: fset_P3577670608877728751rm_a_b > $o] :
      ( ( ord_less_nat @ N @ ( size_s1967422219310127201rm_a_b @ Xs ) )
     => ( ! [X4: fset_P3577670608877728751rm_a_b] :
            ( ( member2349228652077105432rm_a_b @ X4 @ ( set_fs2829528993146950532rm_a_b @ Xs ) )
           => ( P @ X4 ) )
       => ( P @ ( nth_fs3949069584078227574rm_a_b @ Xs @ N ) ) ) ) ).

% list_ball_nth
thf(fact_133_nth__mem,axiom,
    ! [N: nat,Xs: list_P8875379029341186191rm_a_b] :
      ( ( ord_less_nat @ N @ ( size_s8600120110137556219rm_a_b @ Xs ) )
     => ( member5869715511025134514rm_a_b @ ( nth_Pr1098598711982373392rm_a_b @ Xs @ N ) @ ( set_Pr2370880052973118494rm_a_b @ Xs ) ) ) ).

% nth_mem
thf(fact_134_nth__mem,axiom,
    ! [N: nat,Xs: list_FOR_ftrs] :
      ( ( ord_less_nat @ N @ ( size_s7038034906319482769R_ftrs @ Xs ) )
     => ( member_FOR_ftrs @ ( nth_FOR_ftrs @ Xs @ N ) @ ( set_FOR_ftrs2 @ Xs ) ) ) ).

% nth_mem
thf(fact_135_nth__mem,axiom,
    ! [N: nat,Xs: list_f7931864845379430901rm_a_b] :
      ( ( ord_less_nat @ N @ ( size_s1967422219310127201rm_a_b @ Xs ) )
     => ( member2349228652077105432rm_a_b @ ( nth_fs3949069584078227574rm_a_b @ Xs @ N ) @ ( set_fs2829528993146950532rm_a_b @ Xs ) ) ) ).

% nth_mem
thf(fact_136_size__neq__size__imp__neq,axiom,
    ! [X: list_f7931864845379430901rm_a_b,Y: list_f7931864845379430901rm_a_b] :
      ( ( ( size_s1967422219310127201rm_a_b @ X )
       != ( size_s1967422219310127201rm_a_b @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_137_size__neq__size__imp__neq,axiom,
    ! [X: char,Y: char] :
      ( ( ( size_size_char @ X )
       != ( size_size_char @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_138_linorder__neqE__nat,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
     => ( ~ ( ord_less_nat @ X @ Y )
       => ( ord_less_nat @ Y @ X ) ) ) ).

% linorder_neqE_nat
thf(fact_139_infinite__descent,axiom,
    ! [P: nat > $o,N: nat] :
      ( ! [N2: nat] :
          ( ~ ( P @ N2 )
         => ? [M2: nat] :
              ( ( ord_less_nat @ M2 @ N2 )
              & ~ ( P @ M2 ) ) )
     => ( P @ N ) ) ).

% infinite_descent
thf(fact_140_nat__less__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ! [N2: nat] :
          ( ! [M2: nat] :
              ( ( ord_less_nat @ M2 @ N2 )
             => ( P @ M2 ) )
         => ( P @ N2 ) )
     => ( P @ N ) ) ).

% nat_less_induct
thf(fact_141_less__irrefl__nat,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ N ) ).

% less_irrefl_nat
thf(fact_142_less__not__refl3,axiom,
    ! [S: nat,T: nat] :
      ( ( ord_less_nat @ S @ T )
     => ( S != T ) ) ).

% less_not_refl3
thf(fact_143_less__not__refl2,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ N @ M )
     => ( M != N ) ) ).

% less_not_refl2
thf(fact_144_less__not__refl,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ N ) ).

% less_not_refl
thf(fact_145_not__gr__zero,axiom,
    ! [N: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
      = ( N = zero_zero_nat ) ) ).

% not_gr_zero
thf(fact_146_map__eq__nth__conv,axiom,
    ! [F: fset_P3577670608877728751rm_a_b > set_Pr4386577575007340137rm_a_b,Xs: list_f7931864845379430901rm_a_b,G: fset_P3577670608877728751rm_a_b > set_Pr4386577575007340137rm_a_b,Ys2: list_f7931864845379430901rm_a_b] :
      ( ( ( map_fs2147491833908496126rm_a_b @ F @ Xs )
        = ( map_fs2147491833908496126rm_a_b @ G @ Ys2 ) )
      = ( ( ( size_s1967422219310127201rm_a_b @ Xs )
          = ( size_s1967422219310127201rm_a_b @ Ys2 ) )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s1967422219310127201rm_a_b @ Ys2 ) )
           => ( ( F @ ( nth_fs3949069584078227574rm_a_b @ Xs @ I3 ) )
              = ( G @ ( nth_fs3949069584078227574rm_a_b @ Ys2 @ I3 ) ) ) ) ) ) ).

% map_eq_nth_conv
thf(fact_147_nth__map__conv,axiom,
    ! [Xs: list_f7931864845379430901rm_a_b,Ys2: list_f7931864845379430901rm_a_b,F: fset_P3577670608877728751rm_a_b > set_Pr4386577575007340137rm_a_b,G: fset_P3577670608877728751rm_a_b > set_Pr4386577575007340137rm_a_b] :
      ( ( ( size_s1967422219310127201rm_a_b @ Xs )
        = ( size_s1967422219310127201rm_a_b @ Ys2 ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s1967422219310127201rm_a_b @ Xs ) )
           => ( ( F @ ( nth_fs3949069584078227574rm_a_b @ Xs @ I ) )
              = ( G @ ( nth_fs3949069584078227574rm_a_b @ Ys2 @ I ) ) ) )
       => ( ( map_fs2147491833908496126rm_a_b @ F @ Xs )
          = ( map_fs2147491833908496126rm_a_b @ G @ Ys2 ) ) ) ) ).

% nth_map_conv
thf(fact_148_in__set__idx,axiom,
    ! [X: produc357393685978478089rm_a_b,Xs: list_P8875379029341186191rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ X @ ( set_Pr2370880052973118494rm_a_b @ Xs ) )
     => ? [I: nat] :
          ( ( ord_less_nat @ I @ ( size_s8600120110137556219rm_a_b @ Xs ) )
          & ( ( nth_Pr1098598711982373392rm_a_b @ Xs @ I )
            = X ) ) ) ).

% in_set_idx
thf(fact_149_in__set__idx,axiom,
    ! [X: fOR_ftrs,Xs: list_FOR_ftrs] :
      ( ( member_FOR_ftrs @ X @ ( set_FOR_ftrs2 @ Xs ) )
     => ? [I: nat] :
          ( ( ord_less_nat @ I @ ( size_s7038034906319482769R_ftrs @ Xs ) )
          & ( ( nth_FOR_ftrs @ Xs @ I )
            = X ) ) ) ).

% in_set_idx
thf(fact_150_in__set__idx,axiom,
    ! [X: fset_P3577670608877728751rm_a_b,Xs: list_f7931864845379430901rm_a_b] :
      ( ( member2349228652077105432rm_a_b @ X @ ( set_fs2829528993146950532rm_a_b @ Xs ) )
     => ? [I: nat] :
          ( ( ord_less_nat @ I @ ( size_s1967422219310127201rm_a_b @ Xs ) )
          & ( ( nth_fs3949069584078227574rm_a_b @ Xs @ I )
            = X ) ) ) ).

% in_set_idx
thf(fact_151_nth__equalityE,axiom,
    ! [Xs: list_f7931864845379430901rm_a_b,Ys2: list_f7931864845379430901rm_a_b] :
      ( ( Xs = Ys2 )
     => ~ ( ( ( size_s1967422219310127201rm_a_b @ Xs )
            = ( size_s1967422219310127201rm_a_b @ Ys2 ) )
         => ~ ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ ( size_s1967422219310127201rm_a_b @ Xs ) )
               => ( ( nth_fs3949069584078227574rm_a_b @ Xs @ I2 )
                  = ( nth_fs3949069584078227574rm_a_b @ Ys2 @ I2 ) ) ) ) ) ).

% nth_equalityE
thf(fact_152_length__nth__simps_I1_J,axiom,
    ( ( size_s3663623692424790363rm_a_b @ nil_se7334864872594423151rm_a_b )
    = zero_zero_nat ) ).

% length_nth_simps(1)
thf(fact_153_length__nth__simps_I1_J,axiom,
    ( ( size_s1967422219310127201rm_a_b @ nil_fs4020363321024022517rm_a_b )
    = zero_zero_nat ) ).

% length_nth_simps(1)
thf(fact_154_permut__sound,axiom,
    ! [I4: nat,As: list_f7931864845379430901rm_a_b,F: nat > nat] :
      ( ( ord_less_nat @ I4 @ ( size_s1967422219310127201rm_a_b @ As ) )
     => ( ( nth_fs3949069584078227574rm_a_b @ ( missin3687330904383079879rm_a_b @ As @ F ) @ I4 )
        = ( nth_fs3949069584078227574rm_a_b @ As @ ( F @ I4 ) ) ) ) ).

% permut_sound
thf(fact_155_forall__finite_I1_J,axiom,
    ! [P: nat > $o,I2: nat] :
      ( ( ord_less_nat @ I2 @ zero_zero_nat )
     => ( P @ I2 ) ) ).

% forall_finite(1)
thf(fact_156_zero__less__iff__neq__zero,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
      = ( N != zero_zero_nat ) ) ).

% zero_less_iff_neq_zero
thf(fact_157_zero__reorient,axiom,
    ! [X: nat] :
      ( ( zero_zero_nat = X )
      = ( X = zero_zero_nat ) ) ).

% zero_reorient
thf(fact_158_zero__reorient,axiom,
    ! [X: int] :
      ( ( zero_zero_int = X )
      = ( X = zero_zero_int ) ) ).

% zero_reorient
thf(fact_159_gr__zeroI,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% gr_zeroI
thf(fact_160_not__less__zero,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% not_less_zero
thf(fact_161_gr__implies__not__zero,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not_zero
thf(fact_162_size__char__eq__0,axiom,
    ( size_size_char
    = ( ^ [C: char] : zero_zero_nat ) ) ).

% size_char_eq_0
thf(fact_163_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).

% less_numeral_extra(3)
thf(fact_164_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).

% less_numeral_extra(3)
thf(fact_165_permut__aux__sound,axiom,
    ! [I4: nat,As: list_f7931864845379430901rm_a_b,F: nat > nat,Bs: list_f7931864845379430901rm_a_b] :
      ( ( ord_less_nat @ I4 @ ( size_s1967422219310127201rm_a_b @ As ) )
     => ( ( nth_fs3949069584078227574rm_a_b @ ( missin4066191743942050865rm_a_b @ As @ F @ Bs ) @ I4 )
        = ( nth_fs3949069584078227574rm_a_b @ Bs @ ( F @ I4 ) ) ) ) ).

% permut_aux_sound
thf(fact_166_length__code,axiom,
    ( size_s1967422219310127201rm_a_b
    = ( gen_le1078766959997619202rm_a_b @ zero_zero_nat ) ) ).

% length_code
thf(fact_167_can__select__set__list__ex1,axiom,
    ! [P: fOR_ftrs > $o,A2: list_FOR_ftrs] :
      ( ( can_select_FOR_ftrs @ P @ ( set_FOR_ftrs2 @ A2 ) )
      = ( list_ex1_FOR_ftrs @ P @ A2 ) ) ).

% can_select_set_list_ex1
thf(fact_168_Utils_Otrancl__listp_Obase,axiom,
    ! [Xs: list_f7931864845379430901rm_a_b,Ys2: list_f7931864845379430901rm_a_b,R: fset_P3577670608877728751rm_a_b > fset_P3577670608877728751rm_a_b > $o] :
      ( ( ( size_s1967422219310127201rm_a_b @ Xs )
        = ( size_s1967422219310127201rm_a_b @ Ys2 ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s1967422219310127201rm_a_b @ Ys2 ) )
           => ( R @ ( nth_fs3949069584078227574rm_a_b @ Xs @ I ) @ ( nth_fs3949069584078227574rm_a_b @ Ys2 @ I ) ) )
       => ( trancl5091790982450174325rm_a_b @ R @ Xs @ Ys2 ) ) ) ).

% Utils.trancl_listp.base
thf(fact_169_remove__nth__P__compat,axiom,
    ! [As: list_f7931864845379430901rm_a_b,Bs: list_f7931864845379430901rm_a_b,P: fset_P3577670608877728751rm_a_b > fset_P3577670608877728751rm_a_b > $o,P3: nat] :
      ( ( ( size_s1967422219310127201rm_a_b @ As )
        = ( size_s1967422219310127201rm_a_b @ Bs ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s1967422219310127201rm_a_b @ As ) )
           => ( P @ ( nth_fs3949069584078227574rm_a_b @ As @ I ) @ ( nth_fs3949069584078227574rm_a_b @ Bs @ I ) ) )
       => ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s1967422219310127201rm_a_b @ ( missin257347171404174568rm_a_b @ P3 @ As ) ) )
           => ( P @ ( nth_fs3949069584078227574rm_a_b @ ( missin257347171404174568rm_a_b @ P3 @ As ) @ I2 ) @ ( nth_fs3949069584078227574rm_a_b @ ( missin257347171404174568rm_a_b @ P3 @ Bs ) @ I2 ) ) ) ) ) ).

% remove_nth_P_compat
thf(fact_170_Ball__at__Least0LessThan__conv,axiom,
    ! [Xs: list_FOR_ftrs,N: nat,P: fOR_ftrs > $o] :
      ( ( ( size_s7038034906319482769R_ftrs @ Xs )
        = N )
     => ( ( ! [X3: nat] :
              ( ( member_nat @ X3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
             => ( P @ ( nth_FOR_ftrs @ Xs @ X3 ) ) ) )
        = ( ! [X3: fOR_ftrs] :
              ( ( member_FOR_ftrs @ X3 @ ( set_FOR_ftrs2 @ Xs ) )
             => ( P @ X3 ) ) ) ) ) ).

% Ball_at_Least0LessThan_conv
thf(fact_171_Ball__at__Least0LessThan__conv,axiom,
    ! [Xs: list_f7931864845379430901rm_a_b,N: nat,P: fset_P3577670608877728751rm_a_b > $o] :
      ( ( ( size_s1967422219310127201rm_a_b @ Xs )
        = N )
     => ( ( ! [X3: nat] :
              ( ( member_nat @ X3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
             => ( P @ ( nth_fs3949069584078227574rm_a_b @ Xs @ X3 ) ) ) )
        = ( ! [X3: fset_P3577670608877728751rm_a_b] :
              ( ( member2349228652077105432rm_a_b @ X3 @ ( set_fs2829528993146950532rm_a_b @ Xs ) )
             => ( P @ X3 ) ) ) ) ) ).

% Ball_at_Least0LessThan_conv
thf(fact_172_of__nat__0__less__iff,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) )
      = ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% of_nat_0_less_iff
thf(fact_173_of__nat__0__less__iff,axiom,
    ! [N: nat] :
      ( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) )
      = ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% of_nat_0_less_iff
thf(fact_174_of__nat__eq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ( semiri1316708129612266289at_nat @ M )
        = ( semiri1316708129612266289at_nat @ N ) )
      = ( M = N ) ) ).

% of_nat_eq_iff
thf(fact_175_of__nat__eq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ( semiri1314217659103216013at_int @ M )
        = ( semiri1314217659103216013at_int @ N ) )
      = ( M = N ) ) ).

% of_nat_eq_iff
thf(fact_176_of__nat__eq__id,axiom,
    semiri1316708129612266289at_nat = id_nat ).

% of_nat_eq_id
thf(fact_177_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( ( semiri1316708129612266289at_nat @ M )
        = zero_zero_nat )
      = ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
thf(fact_178_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( ( semiri1314217659103216013at_int @ M )
        = zero_zero_int )
      = ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
thf(fact_179_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_zero_nat
        = ( semiri1316708129612266289at_nat @ N ) )
      = ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
thf(fact_180_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_zero_int
        = ( semiri1314217659103216013at_int @ N ) )
      = ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
thf(fact_181_of__nat__0,axiom,
    ( ( semiri1316708129612266289at_nat @ zero_zero_nat )
    = zero_zero_nat ) ).

% of_nat_0
thf(fact_182_of__nat__0,axiom,
    ( ( semiri1314217659103216013at_int @ zero_zero_nat )
    = zero_zero_int ) ).

% of_nat_0
thf(fact_183_of__nat__less__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% of_nat_less_iff
thf(fact_184_of__nat__less__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% of_nat_less_iff
thf(fact_185_can__select__def,axiom,
    ( can_se4942192940725369094rm_a_b
    = ( ^ [P2: produc357393685978478089rm_a_b > $o,A3: set_Pr4386577575007340137rm_a_b] :
        ? [X3: produc357393685978478089rm_a_b] :
          ( ( member5869715511025134514rm_a_b @ X3 @ A3 )
          & ( P2 @ X3 )
          & ! [Y2: produc357393685978478089rm_a_b] :
              ( ( ( member5869715511025134514rm_a_b @ Y2 @ A3 )
                & ( P2 @ Y2 ) )
             => ( Y2 = X3 ) ) ) ) ) ).

% can_select_def
thf(fact_186_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).

% of_nat_less_0_iff
thf(fact_187_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).

% of_nat_less_0_iff
thf(fact_188_of__nat__less__imp__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
     => ( ord_less_nat @ M @ N ) ) ).

% of_nat_less_imp_less
thf(fact_189_of__nat__less__imp__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
     => ( ord_less_nat @ M @ N ) ) ).

% of_nat_less_imp_less
thf(fact_190_less__imp__of__nat__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).

% less_imp_of_nat_less
thf(fact_191_less__imp__of__nat__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).

% less_imp_of_nat_less
thf(fact_192_all__nat__less__eq,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ! [M3: nat] :
            ( ( ord_less_nat @ M3 @ N )
           => ( P @ M3 ) ) )
      = ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
           => ( P @ X3 ) ) ) ) ).

% all_nat_less_eq
thf(fact_193_ex__nat__less__eq,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ? [M3: nat] :
            ( ( ord_less_nat @ M3 @ N )
            & ( P @ M3 ) ) )
      = ( ? [X3: nat] :
            ( ( member_nat @ X3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
            & ( P @ X3 ) ) ) ) ).

% ex_nat_less_eq
thf(fact_194_permut__aux_Osimps_I1_J,axiom,
    ! [Uu: nat > nat,Uv: list_s5629610930313288943rm_a_b] :
      ( ( missin781969150674327467rm_a_b @ nil_se7334864872594423151rm_a_b @ Uu @ Uv )
      = nil_se7334864872594423151rm_a_b ) ).

% permut_aux.simps(1)
thf(fact_195_permut__aux_Osimps_I1_J,axiom,
    ! [Uu: nat > nat,Uv: list_f7931864845379430901rm_a_b] :
      ( ( missin4066191743942050865rm_a_b @ nil_fs4020363321024022517rm_a_b @ Uu @ Uv )
      = nil_fs4020363321024022517rm_a_b ) ).

% permut_aux.simps(1)
thf(fact_196_gen__length__code_I1_J,axiom,
    ! [N: nat] :
      ( ( gen_le1402890518825416700rm_a_b @ N @ nil_se7334864872594423151rm_a_b )
      = N ) ).

% gen_length_code(1)
thf(fact_197_gen__length__code_I1_J,axiom,
    ! [N: nat] :
      ( ( gen_le1078766959997619202rm_a_b @ N @ nil_fs4020363321024022517rm_a_b )
      = N ) ).

% gen_length_code(1)
thf(fact_198_size_H__char__eq__0,axiom,
    ( size_char
    = ( ^ [C: char] : zero_zero_nat ) ) ).

% size'_char_eq_0
thf(fact_199_adjust__idx__rev__nth,axiom,
    ! [I4: nat,Xs: list_f7931864845379430901rm_a_b,J: nat] :
      ( ( ord_less_nat @ I4 @ ( size_s1967422219310127201rm_a_b @ Xs ) )
     => ( ( J != I4 )
       => ( ( nth_fs3949069584078227574rm_a_b @ ( missin257347171404174568rm_a_b @ I4 @ Xs ) @ ( missin3815256168798769645dx_rev @ I4 @ J ) )
          = ( nth_fs3949069584078227574rm_a_b @ Xs @ J ) ) ) ) ).

% adjust_idx_rev_nth
thf(fact_200_adjust__idx__nth,axiom,
    ! [I4: nat,Xs: list_f7931864845379430901rm_a_b,J: nat] :
      ( ( ord_less_nat @ I4 @ ( size_s1967422219310127201rm_a_b @ Xs ) )
     => ( ( nth_fs3949069584078227574rm_a_b @ ( missin257347171404174568rm_a_b @ I4 @ Xs ) @ J )
        = ( nth_fs3949069584078227574rm_a_b @ Xs @ ( missing_adjust_idx @ I4 @ J ) ) ) ) ).

% adjust_idx_nth
thf(fact_201_adjust__idx__rev__length,axiom,
    ! [I4: nat,Xs: list_f7931864845379430901rm_a_b,J: nat] :
      ( ( ord_less_nat @ I4 @ ( size_s1967422219310127201rm_a_b @ Xs ) )
     => ( ( ord_less_nat @ J @ ( size_s1967422219310127201rm_a_b @ Xs ) )
       => ( ( J != I4 )
         => ( ord_less_nat @ ( missin3815256168798769645dx_rev @ I4 @ J ) @ ( size_s1967422219310127201rm_a_b @ ( missin257347171404174568rm_a_b @ I4 @ Xs ) ) ) ) ) ) ).

% adjust_idx_rev_length
thf(fact_202_adjust__idx__length,axiom,
    ! [I4: nat,Xs: list_f7931864845379430901rm_a_b,J: nat] :
      ( ( ord_less_nat @ I4 @ ( size_s1967422219310127201rm_a_b @ Xs ) )
     => ( ( ord_less_nat @ J @ ( size_s1967422219310127201rm_a_b @ ( missin257347171404174568rm_a_b @ I4 @ Xs ) ) )
       => ( ord_less_nat @ ( missing_adjust_idx @ I4 @ J ) @ ( size_s1967422219310127201rm_a_b @ Xs ) ) ) ) ).

% adjust_idx_length
thf(fact_203_atLeastLessThan__inj_I2_J,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( ( ( set_or4662586982721622107an_int @ A @ B )
        = ( set_or4662586982721622107an_int @ C2 @ D ) )
     => ( ( ord_less_int @ A @ B )
       => ( ( ord_less_int @ C2 @ D )
         => ( B = D ) ) ) ) ).

% atLeastLessThan_inj(2)
thf(fact_204_atLeastLessThan__inj_I2_J,axiom,
    ! [A: nat,B: nat,C2: nat,D: nat] :
      ( ( ( set_or4665077453230672383an_nat @ A @ B )
        = ( set_or4665077453230672383an_nat @ C2 @ D ) )
     => ( ( ord_less_nat @ A @ B )
       => ( ( ord_less_nat @ C2 @ D )
         => ( B = D ) ) ) ) ).

% atLeastLessThan_inj(2)
thf(fact_205_atLeastLessThan__inj_I1_J,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( ( ( set_or4662586982721622107an_int @ A @ B )
        = ( set_or4662586982721622107an_int @ C2 @ D ) )
     => ( ( ord_less_int @ A @ B )
       => ( ( ord_less_int @ C2 @ D )
         => ( A = C2 ) ) ) ) ).

% atLeastLessThan_inj(1)
thf(fact_206_atLeastLessThan__inj_I1_J,axiom,
    ! [A: nat,B: nat,C2: nat,D: nat] :
      ( ( ( set_or4665077453230672383an_nat @ A @ B )
        = ( set_or4665077453230672383an_nat @ C2 @ D ) )
     => ( ( ord_less_nat @ A @ B )
       => ( ( ord_less_nat @ C2 @ D )
         => ( A = C2 ) ) ) ) ).

% atLeastLessThan_inj(1)
thf(fact_207_Ico__eq__Ico,axiom,
    ! [L: int,H: int,L2: int,H2: int] :
      ( ( ( set_or4662586982721622107an_int @ L @ H )
        = ( set_or4662586982721622107an_int @ L2 @ H2 ) )
      = ( ( ( L = L2 )
          & ( H = H2 ) )
        | ( ~ ( ord_less_int @ L @ H )
          & ~ ( ord_less_int @ L2 @ H2 ) ) ) ) ).

% Ico_eq_Ico
thf(fact_208_Ico__eq__Ico,axiom,
    ! [L: nat,H: nat,L2: nat,H2: nat] :
      ( ( ( set_or4665077453230672383an_nat @ L @ H )
        = ( set_or4665077453230672383an_nat @ L2 @ H2 ) )
      = ( ( ( L = L2 )
          & ( H = H2 ) )
        | ( ~ ( ord_less_nat @ L @ H )
          & ~ ( ord_less_nat @ L2 @ H2 ) ) ) ) ).

% Ico_eq_Ico
thf(fact_209_atLeastLessThan__eq__iff,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ C2 @ D )
       => ( ( ( set_or4662586982721622107an_int @ A @ B )
            = ( set_or4662586982721622107an_int @ C2 @ D ) )
          = ( ( A = C2 )
            & ( B = D ) ) ) ) ) ).

% atLeastLessThan_eq_iff
thf(fact_210_atLeastLessThan__eq__iff,axiom,
    ! [A: nat,B: nat,C2: nat,D: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ C2 @ D )
       => ( ( ( set_or4665077453230672383an_nat @ A @ B )
            = ( set_or4665077453230672383an_nat @ C2 @ D ) )
          = ( ( A = C2 )
            & ( B = D ) ) ) ) ) ).

% atLeastLessThan_eq_iff
thf(fact_211_of__nat__id,axiom,
    ( semiri1316708129612266289at_nat
    = ( ^ [N3: nat] : N3 ) ) ).

% of_nat_id
thf(fact_212_adjust__idx__rev2,axiom,
    ! [J: nat,I4: nat] :
      ( ( J != I4 )
     => ( ( missing_adjust_idx @ I4 @ ( missin3815256168798769645dx_rev @ I4 @ J ) )
        = J ) ) ).

% adjust_idx_rev2
thf(fact_213_adjust__idx__rev1,axiom,
    ! [I4: nat,J: nat] :
      ( ( missin3815256168798769645dx_rev @ I4 @ ( missing_adjust_idx @ I4 @ J ) )
      = J ) ).

% adjust_idx_rev1
thf(fact_214_adjust__idx__i,axiom,
    ! [I4: nat,J: nat] :
      ( ( missing_adjust_idx @ I4 @ J )
     != I4 ) ).

% adjust_idx_i
thf(fact_215_pos__int__cases,axiom,
    ! [K: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ~ ! [N2: nat] :
            ( ( K
              = ( semiri1314217659103216013at_int @ N2 ) )
           => ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% pos_int_cases
thf(fact_216_zero__less__imp__eq__int,axiom,
    ! [K: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ? [N2: nat] :
          ( ( ord_less_nat @ zero_zero_nat @ N2 )
          & ( K
            = ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).

% zero_less_imp_eq_int
thf(fact_217_nth__remove__nth__conv,axiom,
    ! [I4: nat,N: nat,Xs: list_f7931864845379430901rm_a_b] :
      ( ( ord_less_nat @ I4 @ ( size_s1967422219310127201rm_a_b @ ( missin257347171404174568rm_a_b @ N @ Xs ) ) )
     => ( ( nth_fs3949069584078227574rm_a_b @ ( missin257347171404174568rm_a_b @ N @ Xs ) @ I4 )
        = ( nth_fs3949069584078227574rm_a_b @ Xs @ ( if_nat @ ( ord_less_nat @ I4 @ N ) @ I4 @ ( suc @ I4 ) ) ) ) ) ).

% nth_remove_nth_conv
thf(fact_218_set__swap,axiom,
    ! [I4: nat,Xs: list_FOR_ftrs,J: nat] :
      ( ( ord_less_nat @ I4 @ ( size_s7038034906319482769R_ftrs @ Xs ) )
     => ( ( ord_less_nat @ J @ ( size_s7038034906319482769R_ftrs @ Xs ) )
       => ( ( set_FOR_ftrs2 @ ( list_update_FOR_ftrs @ ( list_update_FOR_ftrs @ Xs @ I4 @ ( nth_FOR_ftrs @ Xs @ J ) ) @ J @ ( nth_FOR_ftrs @ Xs @ I4 ) ) )
          = ( set_FOR_ftrs2 @ Xs ) ) ) ) ).

% set_swap
thf(fact_219_set__swap,axiom,
    ! [I4: nat,Xs: list_f7931864845379430901rm_a_b,J: nat] :
      ( ( ord_less_nat @ I4 @ ( size_s1967422219310127201rm_a_b @ Xs ) )
     => ( ( ord_less_nat @ J @ ( size_s1967422219310127201rm_a_b @ Xs ) )
       => ( ( set_fs2829528993146950532rm_a_b @ ( list_u7212055091542668893rm_a_b @ ( list_u7212055091542668893rm_a_b @ Xs @ I4 @ ( nth_fs3949069584078227574rm_a_b @ Xs @ J ) ) @ J @ ( nth_fs3949069584078227574rm_a_b @ Xs @ I4 ) ) )
          = ( set_fs2829528993146950532rm_a_b @ Xs ) ) ) ) ).

% set_swap
thf(fact_220_subseteq__set__conv__nth,axiom,
    ! [Ss: list_P8875379029341186191rm_a_b,T2: set_Pr4386577575007340137rm_a_b] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s8600120110137556219rm_a_b @ Ss ) )
           => ( member5869715511025134514rm_a_b @ ( nth_Pr1098598711982373392rm_a_b @ Ss @ I3 ) @ T2 ) ) )
      = ( ord_le118470702582115849rm_a_b @ ( set_Pr2370880052973118494rm_a_b @ Ss ) @ T2 ) ) ).

% subseteq_set_conv_nth
thf(fact_221_subseteq__set__conv__nth,axiom,
    ! [Ss: list_FOR_ftrs,T2: set_FOR_ftrs] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s7038034906319482769R_ftrs @ Ss ) )
           => ( member_FOR_ftrs @ ( nth_FOR_ftrs @ Ss @ I3 ) @ T2 ) ) )
      = ( ord_le1454117246873690627R_ftrs @ ( set_FOR_ftrs2 @ Ss ) @ T2 ) ) ).

% subseteq_set_conv_nth
thf(fact_222_subseteq__set__conv__nth,axiom,
    ! [Ss: list_f7931864845379430901rm_a_b,T2: set_fs4190738840245686991rm_a_b] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s1967422219310127201rm_a_b @ Ss ) )
           => ( member2349228652077105432rm_a_b @ ( nth_fs3949069584078227574rm_a_b @ Ss @ I3 ) @ T2 ) ) )
      = ( ord_le8400580047904139887rm_a_b @ ( set_fs2829528993146950532rm_a_b @ Ss ) @ T2 ) ) ).

% subseteq_set_conv_nth
thf(fact_223_set__list__subset__eq__nth__conv,axiom,
    ! [Xs: list_P8875379029341186191rm_a_b,A2: set_Pr4386577575007340137rm_a_b] :
      ( ( ord_le118470702582115849rm_a_b @ ( set_Pr2370880052973118494rm_a_b @ Xs ) @ A2 )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s8600120110137556219rm_a_b @ Xs ) )
           => ( member5869715511025134514rm_a_b @ ( nth_Pr1098598711982373392rm_a_b @ Xs @ I3 ) @ A2 ) ) ) ) ).

% set_list_subset_eq_nth_conv
thf(fact_224_set__list__subset__eq__nth__conv,axiom,
    ! [Xs: list_FOR_ftrs,A2: set_FOR_ftrs] :
      ( ( ord_le1454117246873690627R_ftrs @ ( set_FOR_ftrs2 @ Xs ) @ A2 )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s7038034906319482769R_ftrs @ Xs ) )
           => ( member_FOR_ftrs @ ( nth_FOR_ftrs @ Xs @ I3 ) @ A2 ) ) ) ) ).

% set_list_subset_eq_nth_conv
thf(fact_225_set__list__subset__eq__nth__conv,axiom,
    ! [Xs: list_f7931864845379430901rm_a_b,A2: set_fs4190738840245686991rm_a_b] :
      ( ( ord_le8400580047904139887rm_a_b @ ( set_fs2829528993146950532rm_a_b @ Xs ) @ A2 )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s1967422219310127201rm_a_b @ Xs ) )
           => ( member2349228652077105432rm_a_b @ ( nth_fs3949069584078227574rm_a_b @ Xs @ I3 ) @ A2 ) ) ) ) ).

% set_list_subset_eq_nth_conv
thf(fact_226_Utils_Otrancl__listp_Ocases,axiom,
    ! [R: fset_P3577670608877728751rm_a_b > fset_P3577670608877728751rm_a_b > $o,A1: list_f7931864845379430901rm_a_b,A22: list_f7931864845379430901rm_a_b] :
      ( ( trancl5091790982450174325rm_a_b @ R @ A1 @ A22 )
     => ( ( ( ( size_s1967422219310127201rm_a_b @ A1 )
            = ( size_s1967422219310127201rm_a_b @ A22 ) )
         => ~ ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ ( size_s1967422219310127201rm_a_b @ A22 ) )
               => ( R @ ( nth_fs3949069584078227574rm_a_b @ A1 @ I2 ) @ ( nth_fs3949069584078227574rm_a_b @ A22 @ I2 ) ) ) )
       => ~ ! [Ys4: list_f7931864845379430901rm_a_b,I: nat,Z: fset_P3577670608877728751rm_a_b] :
              ( ( A22
                = ( list_u7212055091542668893rm_a_b @ Ys4 @ I @ Z ) )
             => ( ( trancl5091790982450174325rm_a_b @ R @ A1 @ Ys4 )
               => ( ( ord_less_nat @ I @ ( size_s1967422219310127201rm_a_b @ Ys4 ) )
                 => ~ ( R @ ( nth_fs3949069584078227574rm_a_b @ Ys4 @ I ) @ Z ) ) ) ) ) ) ).

% Utils.trancl_listp.cases
thf(fact_227_Utils_Otrancl__listp_Osimps,axiom,
    ( trancl5091790982450174325rm_a_b
    = ( ^ [R2: fset_P3577670608877728751rm_a_b > fset_P3577670608877728751rm_a_b > $o,A12: list_f7931864845379430901rm_a_b,A23: list_f7931864845379430901rm_a_b] :
          ( ? [Xs3: list_f7931864845379430901rm_a_b,Ys3: list_f7931864845379430901rm_a_b] :
              ( ( A12 = Xs3 )
              & ( A23 = Ys3 )
              & ( ( size_s1967422219310127201rm_a_b @ Xs3 )
                = ( size_s1967422219310127201rm_a_b @ Ys3 ) )
              & ! [I3: nat] :
                  ( ( ord_less_nat @ I3 @ ( size_s1967422219310127201rm_a_b @ Ys3 ) )
                 => ( R2 @ ( nth_fs3949069584078227574rm_a_b @ Xs3 @ I3 ) @ ( nth_fs3949069584078227574rm_a_b @ Ys3 @ I3 ) ) ) )
          | ? [Xs3: list_f7931864845379430901rm_a_b,Ys3: list_f7931864845379430901rm_a_b,I3: nat,Z3: fset_P3577670608877728751rm_a_b] :
              ( ( A12 = Xs3 )
              & ( A23
                = ( list_u7212055091542668893rm_a_b @ Ys3 @ I3 @ Z3 ) )
              & ( trancl5091790982450174325rm_a_b @ R2 @ Xs3 @ Ys3 )
              & ( ord_less_nat @ I3 @ ( size_s1967422219310127201rm_a_b @ Ys3 ) )
              & ( R2 @ ( nth_fs3949069584078227574rm_a_b @ Ys3 @ I3 ) @ Z3 ) ) ) ) ) ).

% Utils.trancl_listp.simps
thf(fact_228_nat_Oinject,axiom,
    ! [X22: nat,Y22: nat] :
      ( ( ( suc @ X22 )
        = ( suc @ Y22 ) )
      = ( X22 = Y22 ) ) ).

% nat.inject
thf(fact_229_old_Onat_Oinject,axiom,
    ! [Nat: nat,Nat2: nat] :
      ( ( ( suc @ Nat )
        = ( suc @ Nat2 ) )
      = ( Nat = Nat2 ) ) ).

% old.nat.inject
thf(fact_230_le__zero__eq,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ N @ zero_zero_nat )
      = ( N = zero_zero_nat ) ) ).

% le_zero_eq
thf(fact_231_of__nat__le__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% of_nat_le_iff
thf(fact_232_of__nat__le__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% of_nat_le_iff
thf(fact_233_Suc__less__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% Suc_less_eq
thf(fact_234_Suc__mono,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).

% Suc_mono
thf(fact_235_lessI,axiom,
    ! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).

% lessI
thf(fact_236_list__update__nonempty,axiom,
    ! [Xs: list_s5629610930313288943rm_a_b,K: nat,X: set_Pr4386577575007340137rm_a_b] :
      ( ( ( list_u6367744683147468759rm_a_b @ Xs @ K @ X )
        = nil_se7334864872594423151rm_a_b )
      = ( Xs = nil_se7334864872594423151rm_a_b ) ) ).

% list_update_nonempty
thf(fact_237_list__update__nonempty,axiom,
    ! [Xs: list_f7931864845379430901rm_a_b,K: nat,X: fset_P3577670608877728751rm_a_b] :
      ( ( ( list_u7212055091542668893rm_a_b @ Xs @ K @ X )
        = nil_fs4020363321024022517rm_a_b )
      = ( Xs = nil_fs4020363321024022517rm_a_b ) ) ).

% list_update_nonempty
thf(fact_238_length__list__update,axiom,
    ! [Xs: list_f7931864845379430901rm_a_b,I4: nat,X: fset_P3577670608877728751rm_a_b] :
      ( ( size_s1967422219310127201rm_a_b @ ( list_u7212055091542668893rm_a_b @ Xs @ I4 @ X ) )
      = ( size_s1967422219310127201rm_a_b @ Xs ) ) ).

% length_list_update
thf(fact_239_less__Suc0,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
      = ( N = zero_zero_nat ) ) ).

% less_Suc0
thf(fact_240_zero__less__Suc,axiom,
    ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).

% zero_less_Suc
thf(fact_241_atLeastLessThan__iff,axiom,
    ! [I4: int,L: int,U: int] :
      ( ( member_int @ I4 @ ( set_or4662586982721622107an_int @ L @ U ) )
      = ( ( ord_less_eq_int @ L @ I4 )
        & ( ord_less_int @ I4 @ U ) ) ) ).

% atLeastLessThan_iff
thf(fact_242_atLeastLessThan__iff,axiom,
    ! [I4: nat,L: nat,U: nat] :
      ( ( member_nat @ I4 @ ( set_or4665077453230672383an_nat @ L @ U ) )
      = ( ( ord_less_eq_nat @ L @ I4 )
        & ( ord_less_nat @ I4 @ U ) ) ) ).

% atLeastLessThan_iff
thf(fact_243_ivl__subset,axiom,
    ! [I4: int,J: int,M: int,N: int] :
      ( ( ord_less_eq_set_int @ ( set_or4662586982721622107an_int @ I4 @ J ) @ ( set_or4662586982721622107an_int @ M @ N ) )
      = ( ( ord_less_eq_int @ J @ I4 )
        | ( ( ord_less_eq_int @ M @ I4 )
          & ( ord_less_eq_int @ J @ N ) ) ) ) ).

% ivl_subset
thf(fact_244_ivl__subset,axiom,
    ! [I4: nat,J: nat,M: nat,N: nat] :
      ( ( ord_less_eq_set_nat @ ( set_or4665077453230672383an_nat @ I4 @ J ) @ ( set_or4665077453230672383an_nat @ M @ N ) )
      = ( ( ord_less_eq_nat @ J @ I4 )
        | ( ( ord_less_eq_nat @ M @ I4 )
          & ( ord_less_eq_nat @ J @ N ) ) ) ) ).

% ivl_subset
thf(fact_245_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat )
      = ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
thf(fact_246_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int )
      = ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
thf(fact_247_nth__list__update__eq,axiom,
    ! [I4: nat,Xs: list_f7931864845379430901rm_a_b,X: fset_P3577670608877728751rm_a_b] :
      ( ( ord_less_nat @ I4 @ ( size_s1967422219310127201rm_a_b @ Xs ) )
     => ( ( nth_fs3949069584078227574rm_a_b @ ( list_u7212055091542668893rm_a_b @ Xs @ I4 @ X ) @ I4 )
        = X ) ) ).

% nth_list_update_eq
thf(fact_248_int__int__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ( semiri1314217659103216013at_int @ M )
        = ( semiri1314217659103216013at_int @ N ) )
      = ( M = N ) ) ).

% int_int_eq
thf(fact_249_atLeastLessThan__subset__iff,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( ( ord_less_eq_set_int @ ( set_or4662586982721622107an_int @ A @ B ) @ ( set_or4662586982721622107an_int @ C2 @ D ) )
     => ( ( ord_less_eq_int @ B @ A )
        | ( ( ord_less_eq_int @ C2 @ A )
          & ( ord_less_eq_int @ B @ D ) ) ) ) ).

% atLeastLessThan_subset_iff
thf(fact_250_atLeastLessThan__subset__iff,axiom,
    ! [A: nat,B: nat,C2: nat,D: nat] :
      ( ( ord_less_eq_set_nat @ ( set_or4665077453230672383an_nat @ A @ B ) @ ( set_or4665077453230672383an_nat @ C2 @ D ) )
     => ( ( ord_less_eq_nat @ B @ A )
        | ( ( ord_less_eq_nat @ C2 @ A )
          & ( ord_less_eq_nat @ B @ D ) ) ) ) ).

% atLeastLessThan_subset_iff
thf(fact_251_of__nat__mono,axiom,
    ! [I4: nat,J: nat] :
      ( ( ord_less_eq_nat @ I4 @ J )
     => ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I4 ) @ ( semiri1316708129612266289at_nat @ J ) ) ) ).

% of_nat_mono
thf(fact_252_of__nat__mono,axiom,
    ! [I4: nat,J: nat] :
      ( ( ord_less_eq_nat @ I4 @ J )
     => ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I4 ) @ ( semiri1314217659103216013at_int @ J ) ) ) ).

% of_nat_mono
thf(fact_253_exists__least__lemma,axiom,
    ! [P: nat > $o] :
      ( ~ ( P @ zero_zero_nat )
     => ( ? [X_1: nat] : ( P @ X_1 )
       => ? [N2: nat] :
            ( ~ ( P @ N2 )
            & ( P @ ( suc @ N2 ) ) ) ) ) ).

% exists_least_lemma
thf(fact_254_set__update__subsetI,axiom,
    ! [Xs: list_P8875379029341186191rm_a_b,A2: set_Pr4386577575007340137rm_a_b,X: produc357393685978478089rm_a_b,I4: nat] :
      ( ( ord_le118470702582115849rm_a_b @ ( set_Pr2370880052973118494rm_a_b @ Xs ) @ A2 )
     => ( ( member5869715511025134514rm_a_b @ X @ A2 )
       => ( ord_le118470702582115849rm_a_b @ ( set_Pr2370880052973118494rm_a_b @ ( list_u4211539095652206455rm_a_b @ Xs @ I4 @ X ) ) @ A2 ) ) ) ).

% set_update_subsetI
thf(fact_255_set__update__subsetI,axiom,
    ! [Xs: list_FOR_ftrs,A2: set_FOR_ftrs,X: fOR_ftrs,I4: nat] :
      ( ( ord_le1454117246873690627R_ftrs @ ( set_FOR_ftrs2 @ Xs ) @ A2 )
     => ( ( member_FOR_ftrs @ X @ A2 )
       => ( ord_le1454117246873690627R_ftrs @ ( set_FOR_ftrs2 @ ( list_update_FOR_ftrs @ Xs @ I4 @ X ) ) @ A2 ) ) ) ).

% set_update_subsetI
thf(fact_256_lift__Suc__mono__le,axiom,
    ! [F: nat > nat,N: nat,N4: nat] :
      ( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_nat @ N @ N4 )
       => ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_257_lift__Suc__mono__le,axiom,
    ! [F: nat > int,N: nat,N4: nat] :
      ( ! [N2: nat] : ( ord_less_eq_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_nat @ N @ N4 )
       => ( ord_less_eq_int @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_258_lift__Suc__antimono__le,axiom,
    ! [F: nat > nat,N: nat,N4: nat] :
      ( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
     => ( ( ord_less_eq_nat @ N @ N4 )
       => ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_259_lift__Suc__antimono__le,axiom,
    ! [F: nat > int,N: nat,N4: nat] :
      ( ! [N2: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
     => ( ( ord_less_eq_nat @ N @ N4 )
       => ( ord_less_eq_int @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_260_Suc__inject,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( suc @ X )
        = ( suc @ Y ) )
     => ( X = Y ) ) ).

% Suc_inject
thf(fact_261_n__not__Suc__n,axiom,
    ! [N: nat] :
      ( N
     != ( suc @ N ) ) ).

% n_not_Suc_n
thf(fact_262_list__update_Osimps_I1_J,axiom,
    ! [I4: nat,V: set_Pr4386577575007340137rm_a_b] :
      ( ( list_u6367744683147468759rm_a_b @ nil_se7334864872594423151rm_a_b @ I4 @ V )
      = nil_se7334864872594423151rm_a_b ) ).

% list_update.simps(1)
thf(fact_263_list__update_Osimps_I1_J,axiom,
    ! [I4: nat,V: fset_P3577670608877728751rm_a_b] :
      ( ( list_u7212055091542668893rm_a_b @ nil_fs4020363321024022517rm_a_b @ I4 @ V )
      = nil_fs4020363321024022517rm_a_b ) ).

% list_update.simps(1)
thf(fact_264_list__update__code_I1_J,axiom,
    ! [I4: nat,Y: set_Pr4386577575007340137rm_a_b] :
      ( ( list_u6367744683147468759rm_a_b @ nil_se7334864872594423151rm_a_b @ I4 @ Y )
      = nil_se7334864872594423151rm_a_b ) ).

% list_update_code(1)
thf(fact_265_list__update__code_I1_J,axiom,
    ! [I4: nat,Y: fset_P3577670608877728751rm_a_b] :
      ( ( list_u7212055091542668893rm_a_b @ nil_fs4020363321024022517rm_a_b @ I4 @ Y )
      = nil_fs4020363321024022517rm_a_b ) ).

% list_update_code(1)
thf(fact_266_map__update,axiom,
    ! [F: fset_P3577670608877728751rm_a_b > set_Pr4386577575007340137rm_a_b,Xs: list_f7931864845379430901rm_a_b,K: nat,Y: fset_P3577670608877728751rm_a_b] :
      ( ( map_fs2147491833908496126rm_a_b @ F @ ( list_u7212055091542668893rm_a_b @ Xs @ K @ Y ) )
      = ( list_u6367744683147468759rm_a_b @ ( map_fs2147491833908496126rm_a_b @ F @ Xs ) @ K @ ( F @ Y ) ) ) ).

% map_update
thf(fact_267_le__numeral__extra_I3_J,axiom,
    ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).

% le_numeral_extra(3)
thf(fact_268_le__numeral__extra_I3_J,axiom,
    ord_less_eq_int @ zero_zero_int @ zero_zero_int ).

% le_numeral_extra(3)
thf(fact_269_zero__le,axiom,
    ! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).

% zero_le
thf(fact_270_not0__implies__Suc,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ? [M4: nat] :
          ( N
          = ( suc @ M4 ) ) ) ).

% not0_implies_Suc
thf(fact_271_Zero__not__Suc,axiom,
    ! [M: nat] :
      ( zero_zero_nat
     != ( suc @ M ) ) ).

% Zero_not_Suc
thf(fact_272_Zero__neq__Suc,axiom,
    ! [M: nat] :
      ( zero_zero_nat
     != ( suc @ M ) ) ).

% Zero_neq_Suc
thf(fact_273_Suc__neq__Zero,axiom,
    ! [M: nat] :
      ( ( suc @ M )
     != zero_zero_nat ) ).

% Suc_neq_Zero
thf(fact_274_zero__induct,axiom,
    ! [P: nat > $o,K: nat] :
      ( ( P @ K )
     => ( ! [N2: nat] :
            ( ( P @ ( suc @ N2 ) )
           => ( P @ N2 ) )
       => ( P @ zero_zero_nat ) ) ) ).

% zero_induct
thf(fact_275_diff__induct,axiom,
    ! [P: nat > nat > $o,M: nat,N: nat] :
      ( ! [X4: nat] : ( P @ X4 @ zero_zero_nat )
     => ( ! [Y4: nat] : ( P @ zero_zero_nat @ ( suc @ Y4 ) )
       => ( ! [X4: nat,Y4: nat] :
              ( ( P @ X4 @ Y4 )
             => ( P @ ( suc @ X4 ) @ ( suc @ Y4 ) ) )
         => ( P @ M @ N ) ) ) ) ).

% diff_induct
thf(fact_276_nat__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N2: nat] :
            ( ( P @ N2 )
           => ( P @ ( suc @ N2 ) ) )
       => ( P @ N ) ) ) ).

% nat_induct
thf(fact_277_old_Onat_Oexhaust,axiom,
    ! [Y: nat] :
      ( ( Y != zero_zero_nat )
     => ~ ! [Nat3: nat] :
            ( Y
           != ( suc @ Nat3 ) ) ) ).

% old.nat.exhaust
thf(fact_278_nat_OdiscI,axiom,
    ! [Nat: nat,X22: nat] :
      ( ( Nat
        = ( suc @ X22 ) )
     => ( Nat != zero_zero_nat ) ) ).

% nat.discI
thf(fact_279_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] :
      ( zero_zero_nat
     != ( suc @ Nat2 ) ) ).

% old.nat.distinct(1)
thf(fact_280_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] :
      ( ( suc @ Nat2 )
     != zero_zero_nat ) ).

% old.nat.distinct(2)
thf(fact_281_nat_Odistinct_I1_J,axiom,
    ! [X22: nat] :
      ( zero_zero_nat
     != ( suc @ X22 ) ) ).

% nat.distinct(1)
thf(fact_282_not__less__less__Suc__eq,axiom,
    ! [N: nat,M: nat] :
      ( ~ ( ord_less_nat @ N @ M )
     => ( ( ord_less_nat @ N @ ( suc @ M ) )
        = ( N = M ) ) ) ).

% not_less_less_Suc_eq
thf(fact_283_strict__inc__induct,axiom,
    ! [I4: nat,J: nat,P: nat > $o] :
      ( ( ord_less_nat @ I4 @ J )
     => ( ! [I: nat] :
            ( ( J
              = ( suc @ I ) )
           => ( P @ I ) )
       => ( ! [I: nat] :
              ( ( ord_less_nat @ I @ J )
             => ( ( P @ ( suc @ I ) )
               => ( P @ I ) ) )
         => ( P @ I4 ) ) ) ) ).

% strict_inc_induct
thf(fact_284_less__Suc__induct,axiom,
    ! [I4: nat,J: nat,P: nat > nat > $o] :
      ( ( ord_less_nat @ I4 @ J )
     => ( ! [I: nat] : ( P @ I @ ( suc @ I ) )
       => ( ! [I: nat,J2: nat,K2: nat] :
              ( ( ord_less_nat @ I @ J2 )
             => ( ( ord_less_nat @ J2 @ K2 )
               => ( ( P @ I @ J2 )
                 => ( ( P @ J2 @ K2 )
                   => ( P @ I @ K2 ) ) ) ) )
         => ( P @ I4 @ J ) ) ) ) ).

% less_Suc_induct
thf(fact_285_less__trans__Suc,axiom,
    ! [I4: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I4 @ J )
     => ( ( ord_less_nat @ J @ K )
       => ( ord_less_nat @ ( suc @ I4 ) @ K ) ) ) ).

% less_trans_Suc
thf(fact_286_Suc__less__SucD,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
     => ( ord_less_nat @ M @ N ) ) ).

% Suc_less_SucD
thf(fact_287_less__antisym,axiom,
    ! [N: nat,M: nat] :
      ( ~ ( ord_less_nat @ N @ M )
     => ( ( ord_less_nat @ N @ ( suc @ M ) )
       => ( M = N ) ) ) ).

% less_antisym
thf(fact_288_Suc__less__eq2,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ ( suc @ N ) @ M )
      = ( ? [M5: nat] :
            ( ( M
              = ( suc @ M5 ) )
            & ( ord_less_nat @ N @ M5 ) ) ) ) ).

% Suc_less_eq2
thf(fact_289_Nat_OAll__less__Suc,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( suc @ N ) )
           => ( P @ I3 ) ) )
      = ( ( P @ N )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ N )
           => ( P @ I3 ) ) ) ) ).

% Nat.All_less_Suc
thf(fact_290_not__less__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ~ ( ord_less_nat @ M @ N ) )
      = ( ord_less_nat @ N @ ( suc @ M ) ) ) ).

% not_less_eq
thf(fact_291_less__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N ) )
      = ( ( ord_less_nat @ M @ N )
        | ( M = N ) ) ) ).

% less_Suc_eq
thf(fact_292_Ex__less__Suc,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( suc @ N ) )
            & ( P @ I3 ) ) )
      = ( ( P @ N )
        | ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ N )
            & ( P @ I3 ) ) ) ) ).

% Ex_less_Suc
thf(fact_293_less__SucI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).

% less_SucI
thf(fact_294_less__SucE,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N ) )
     => ( ~ ( ord_less_nat @ M @ N )
       => ( M = N ) ) ) ).

% less_SucE
thf(fact_295_Suc__lessI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ( ( suc @ M )
         != N )
       => ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).

% Suc_lessI
thf(fact_296_Suc__lessE,axiom,
    ! [I4: nat,K: nat] :
      ( ( ord_less_nat @ ( suc @ I4 ) @ K )
     => ~ ! [J2: nat] :
            ( ( ord_less_nat @ I4 @ J2 )
           => ( K
             != ( suc @ J2 ) ) ) ) ).

% Suc_lessE
thf(fact_297_Suc__lessD,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ M ) @ N )
     => ( ord_less_nat @ M @ N ) ) ).

% Suc_lessD
thf(fact_298_Nat_OlessE,axiom,
    ! [I4: nat,K: nat] :
      ( ( ord_less_nat @ I4 @ K )
     => ( ( K
         != ( suc @ I4 ) )
       => ~ ! [J2: nat] :
              ( ( ord_less_nat @ I4 @ J2 )
             => ( K
               != ( suc @ J2 ) ) ) ) ) ).

% Nat.lessE
thf(fact_299_subset__code_I1_J,axiom,
    ! [Xs: list_P8875379029341186191rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
      ( ( ord_le118470702582115849rm_a_b @ ( set_Pr2370880052973118494rm_a_b @ Xs ) @ B2 )
      = ( ! [X3: produc357393685978478089rm_a_b] :
            ( ( member5869715511025134514rm_a_b @ X3 @ ( set_Pr2370880052973118494rm_a_b @ Xs ) )
           => ( member5869715511025134514rm_a_b @ X3 @ B2 ) ) ) ) ).

% subset_code(1)
thf(fact_300_subset__code_I1_J,axiom,
    ! [Xs: list_FOR_ftrs,B2: set_FOR_ftrs] :
      ( ( ord_le1454117246873690627R_ftrs @ ( set_FOR_ftrs2 @ Xs ) @ B2 )
      = ( ! [X3: fOR_ftrs] :
            ( ( member_FOR_ftrs @ X3 @ ( set_FOR_ftrs2 @ Xs ) )
           => ( member_FOR_ftrs @ X3 @ B2 ) ) ) ) ).

% subset_code(1)
thf(fact_301_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > nat,N: nat,M: nat] :
      ( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
        = ( ord_less_nat @ N @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_302_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > int,N: nat,M: nat] :
      ( ! [N2: nat] : ( ord_less_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_int @ ( F @ N ) @ ( F @ M ) )
        = ( ord_less_nat @ N @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_303_lift__Suc__mono__less,axiom,
    ! [F: nat > nat,N: nat,N4: nat] :
      ( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_nat @ N @ N4 )
       => ( ord_less_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_304_lift__Suc__mono__less,axiom,
    ! [F: nat > int,N: nat,N4: nat] :
      ( ! [N2: nat] : ( ord_less_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_nat @ N @ N4 )
       => ( ord_less_int @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_305_of__nat__neq__0,axiom,
    ! [N: nat] :
      ( ( semiri1316708129612266289at_nat @ ( suc @ N ) )
     != zero_zero_nat ) ).

% of_nat_neq_0
thf(fact_306_of__nat__neq__0,axiom,
    ! [N: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ N ) )
     != zero_zero_int ) ).

% of_nat_neq_0
thf(fact_307_gr0__conv__Suc,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
      = ( ? [M3: nat] :
            ( N
            = ( suc @ M3 ) ) ) ) ).

% gr0_conv_Suc
thf(fact_308_gr0__implies__Suc,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ? [M4: nat] :
          ( N
          = ( suc @ M4 ) ) ) ).

% gr0_implies_Suc
thf(fact_309_less__Suc__eq__0__disj,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N ) )
      = ( ( M = zero_zero_nat )
        | ? [J3: nat] :
            ( ( M
              = ( suc @ J3 ) )
            & ( ord_less_nat @ J3 @ N ) ) ) ) ).

% less_Suc_eq_0_disj
thf(fact_310_forall__finite_I3_J,axiom,
    ! [X: nat,P: nat > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( suc @ ( suc @ X ) ) )
           => ( P @ I3 ) ) )
      = ( ( P @ zero_zero_nat )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( suc @ X ) )
           => ( P @ ( suc @ I3 ) ) ) ) ) ).

% forall_finite(3)
thf(fact_311_forall__finite_I2_J,axiom,
    ! [P: nat > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( suc @ zero_zero_nat ) )
           => ( P @ I3 ) ) )
      = ( P @ zero_zero_nat ) ) ).

% forall_finite(2)
thf(fact_312_ex__Suc__conv,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( suc @ N ) )
            & ( P @ I3 ) ) )
      = ( ( P @ zero_zero_nat )
        | ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ N )
            & ( P @ ( suc @ I3 ) ) ) ) ) ).

% ex_Suc_conv
thf(fact_313_all__Suc__conv,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( suc @ N ) )
           => ( P @ I3 ) ) )
      = ( ( P @ zero_zero_nat )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ N )
           => ( P @ ( suc @ I3 ) ) ) ) ) ).

% all_Suc_conv
thf(fact_314_all__less__two,axiom,
    ! [P: nat > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( suc @ ( suc @ zero_zero_nat ) ) )
           => ( P @ I3 ) ) )
      = ( ( P @ zero_zero_nat )
        & ( P @ ( suc @ zero_zero_nat ) ) ) ) ).

% all_less_two
thf(fact_315_Comparator__Generator_OAll__less__Suc,axiom,
    ! [X: nat,P: nat > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( suc @ X ) )
           => ( P @ I3 ) ) )
      = ( ( P @ zero_zero_nat )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ X )
           => ( P @ ( suc @ I3 ) ) ) ) ) ).

% Comparator_Generator.All_less_Suc
thf(fact_316_of__nat__0__le__iff,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) ) ).

% of_nat_0_le_iff
thf(fact_317_of__nat__0__le__iff,axiom,
    ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) ) ).

% of_nat_0_le_iff
thf(fact_318_set__removeAll__subset,axiom,
    ! [X: fOR_ftrs,Xs: list_FOR_ftrs] : ( ord_le1454117246873690627R_ftrs @ ( set_FOR_ftrs2 @ ( removeAll_FOR_ftrs @ X @ Xs ) ) @ ( set_FOR_ftrs2 @ Xs ) ) ).

% set_removeAll_subset
thf(fact_319_adjust__idx__def,axiom,
    ( missing_adjust_idx
    = ( ^ [I3: nat,J3: nat] : ( if_nat @ ( ord_less_nat @ J3 @ I3 ) @ J3 @ ( suc @ J3 ) ) ) ) ).

% adjust_idx_def
thf(fact_320_set__update__memI,axiom,
    ! [N: nat,Xs: list_P8875379029341186191rm_a_b,X: produc357393685978478089rm_a_b] :
      ( ( ord_less_nat @ N @ ( size_s8600120110137556219rm_a_b @ Xs ) )
     => ( member5869715511025134514rm_a_b @ X @ ( set_Pr2370880052973118494rm_a_b @ ( list_u4211539095652206455rm_a_b @ Xs @ N @ X ) ) ) ) ).

% set_update_memI
thf(fact_321_set__update__memI,axiom,
    ! [N: nat,Xs: list_FOR_ftrs,X: fOR_ftrs] :
      ( ( ord_less_nat @ N @ ( size_s7038034906319482769R_ftrs @ Xs ) )
     => ( member_FOR_ftrs @ X @ ( set_FOR_ftrs2 @ ( list_update_FOR_ftrs @ Xs @ N @ X ) ) ) ) ).

% set_update_memI
thf(fact_322_set__update__memI,axiom,
    ! [N: nat,Xs: list_f7931864845379430901rm_a_b,X: fset_P3577670608877728751rm_a_b] :
      ( ( ord_less_nat @ N @ ( size_s1967422219310127201rm_a_b @ Xs ) )
     => ( member2349228652077105432rm_a_b @ X @ ( set_fs2829528993146950532rm_a_b @ ( list_u7212055091542668893rm_a_b @ Xs @ N @ X ) ) ) ) ).

% set_update_memI
thf(fact_323_parallel__list__update,axiom,
    ! [N: nat,R3: fset_P3577670608877728751rm_a_b > fset_P3577670608877728751rm_a_b > $o,P3: list_f7931864845379430901rm_a_b > $o,Xs: list_f7931864845379430901rm_a_b,Ys2: list_f7931864845379430901rm_a_b] :
      ( ! [Xs2: list_f7931864845379430901rm_a_b,I: nat,Y4: fset_P3577670608877728751rm_a_b] :
          ( ( ( size_s1967422219310127201rm_a_b @ Xs2 )
            = N )
         => ( ( ord_less_nat @ I @ N )
           => ( ( R3 @ ( nth_fs3949069584078227574rm_a_b @ Xs2 @ I ) @ Y4 )
             => ( ( P3 @ Xs2 )
               => ( P3 @ ( list_u7212055091542668893rm_a_b @ Xs2 @ I @ Y4 ) ) ) ) ) )
     => ( ( ( size_s1967422219310127201rm_a_b @ Xs )
          = N )
       => ( ( P3 @ Xs )
         => ( ( ( size_s1967422219310127201rm_a_b @ Ys2 )
              = N )
           => ( ! [I: nat] :
                  ( ( ord_less_nat @ I @ N )
                 => ( R3 @ ( nth_fs3949069584078227574rm_a_b @ Xs @ I ) @ ( nth_fs3949069584078227574rm_a_b @ Ys2 @ I ) ) )
             => ( P3 @ Ys2 ) ) ) ) ) ) ).

% parallel_list_update
thf(fact_324_list__update__same__conv,axiom,
    ! [I4: nat,Xs: list_f7931864845379430901rm_a_b,X: fset_P3577670608877728751rm_a_b] :
      ( ( ord_less_nat @ I4 @ ( size_s1967422219310127201rm_a_b @ Xs ) )
     => ( ( ( list_u7212055091542668893rm_a_b @ Xs @ I4 @ X )
          = Xs )
        = ( ( nth_fs3949069584078227574rm_a_b @ Xs @ I4 )
          = X ) ) ) ).

% list_update_same_conv
thf(fact_325_nth__list__update,axiom,
    ! [I4: nat,Xs: list_f7931864845379430901rm_a_b,J: nat,X: fset_P3577670608877728751rm_a_b] :
      ( ( ord_less_nat @ I4 @ ( size_s1967422219310127201rm_a_b @ Xs ) )
     => ( ( ( I4 = J )
         => ( ( nth_fs3949069584078227574rm_a_b @ ( list_u7212055091542668893rm_a_b @ Xs @ I4 @ X ) @ J )
            = X ) )
        & ( ( I4 != J )
         => ( ( nth_fs3949069584078227574rm_a_b @ ( list_u7212055091542668893rm_a_b @ Xs @ I4 @ X ) @ J )
            = ( nth_fs3949069584078227574rm_a_b @ Xs @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_326_remove__nth__len,axiom,
    ! [I4: nat,Xs: list_f7931864845379430901rm_a_b] :
      ( ( ord_less_nat @ I4 @ ( size_s1967422219310127201rm_a_b @ Xs ) )
     => ( ( size_s1967422219310127201rm_a_b @ Xs )
        = ( suc @ ( size_s1967422219310127201rm_a_b @ ( missin257347171404174568rm_a_b @ I4 @ Xs ) ) ) ) ) ).

% remove_nth_len
thf(fact_327_Utils_Otrancl__listp_Olist__trancl,axiom,
    ! [R: fset_P3577670608877728751rm_a_b > fset_P3577670608877728751rm_a_b > $o,Xs: list_f7931864845379430901rm_a_b,Ys2: list_f7931864845379430901rm_a_b,I4: nat,Z4: fset_P3577670608877728751rm_a_b] :
      ( ( trancl5091790982450174325rm_a_b @ R @ Xs @ Ys2 )
     => ( ( ord_less_nat @ I4 @ ( size_s1967422219310127201rm_a_b @ Ys2 ) )
       => ( ( R @ ( nth_fs3949069584078227574rm_a_b @ Ys2 @ I4 ) @ Z4 )
         => ( trancl5091790982450174325rm_a_b @ R @ Xs @ ( list_u7212055091542668893rm_a_b @ Ys2 @ I4 @ Z4 ) ) ) ) ) ).

% Utils.trancl_listp.list_trancl
thf(fact_328_set__list__subset__nth__conv,axiom,
    ! [Xs: list_P8875379029341186191rm_a_b,A2: set_Pr4386577575007340137rm_a_b,I4: nat] :
      ( ( ord_le118470702582115849rm_a_b @ ( set_Pr2370880052973118494rm_a_b @ Xs ) @ A2 )
     => ( ( ord_less_nat @ I4 @ ( size_s8600120110137556219rm_a_b @ Xs ) )
       => ( member5869715511025134514rm_a_b @ ( nth_Pr1098598711982373392rm_a_b @ Xs @ I4 ) @ A2 ) ) ) ).

% set_list_subset_nth_conv
thf(fact_329_set__list__subset__nth__conv,axiom,
    ! [Xs: list_FOR_ftrs,A2: set_FOR_ftrs,I4: nat] :
      ( ( ord_le1454117246873690627R_ftrs @ ( set_FOR_ftrs2 @ Xs ) @ A2 )
     => ( ( ord_less_nat @ I4 @ ( size_s7038034906319482769R_ftrs @ Xs ) )
       => ( member_FOR_ftrs @ ( nth_FOR_ftrs @ Xs @ I4 ) @ A2 ) ) ) ).

% set_list_subset_nth_conv
thf(fact_330_set__list__subset__nth__conv,axiom,
    ! [Xs: list_f7931864845379430901rm_a_b,A2: set_fs4190738840245686991rm_a_b,I4: nat] :
      ( ( ord_le8400580047904139887rm_a_b @ ( set_fs2829528993146950532rm_a_b @ Xs ) @ A2 )
     => ( ( ord_less_nat @ I4 @ ( size_s1967422219310127201rm_a_b @ Xs ) )
       => ( member2349228652077105432rm_a_b @ ( nth_fs3949069584078227574rm_a_b @ Xs @ I4 ) @ A2 ) ) ) ).

% set_list_subset_nth_conv
thf(fact_331_inv__to__set,axiom,
    ! [Ss: list_P8875379029341186191rm_a_b,S2: set_Pr4386577575007340137rm_a_b] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s8600120110137556219rm_a_b @ Ss ) )
           => ( member5869715511025134514rm_a_b @ ( nth_Pr1098598711982373392rm_a_b @ Ss @ I3 ) @ S2 ) ) )
      = ( ord_le118470702582115849rm_a_b @ ( set_Pr2370880052973118494rm_a_b @ Ss ) @ S2 ) ) ).

% inv_to_set
thf(fact_332_inv__to__set,axiom,
    ! [Ss: list_FOR_ftrs,S2: set_FOR_ftrs] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s7038034906319482769R_ftrs @ Ss ) )
           => ( member_FOR_ftrs @ ( nth_FOR_ftrs @ Ss @ I3 ) @ S2 ) ) )
      = ( ord_le1454117246873690627R_ftrs @ ( set_FOR_ftrs2 @ Ss ) @ S2 ) ) ).

% inv_to_set
thf(fact_333_inv__to__set,axiom,
    ! [Ss: list_f7931864845379430901rm_a_b,S2: set_fs4190738840245686991rm_a_b] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s1967422219310127201rm_a_b @ Ss ) )
           => ( member2349228652077105432rm_a_b @ ( nth_fs3949069584078227574rm_a_b @ Ss @ I3 ) @ S2 ) ) )
      = ( ord_le8400580047904139887rm_a_b @ ( set_fs2829528993146950532rm_a_b @ Ss ) @ S2 ) ) ).

% inv_to_set
thf(fact_334_nat__int__comparison_I2_J,axiom,
    ( ord_less_nat
    = ( ^ [A4: nat,B3: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).

% nat_int_comparison(2)
thf(fact_335_int__ops_I1_J,axiom,
    ( ( semiri1314217659103216013at_int @ zero_zero_nat )
    = zero_zero_int ) ).

% int_ops(1)
thf(fact_336_Image__mono,axiom,
    ! [R4: set_Pr2972776593051762503rm_a_b,R3: set_Pr2972776593051762503rm_a_b,A5: set_Pr4386577575007340137rm_a_b,A2: set_Pr4386577575007340137rm_a_b] :
      ( ( ord_le2593958535671617255rm_a_b @ R4 @ R3 )
     => ( ( ord_le118470702582115849rm_a_b @ A5 @ A2 )
       => ( ord_le118470702582115849rm_a_b @ ( image_871640889055226643rm_a_b @ R4 @ A5 ) @ ( image_871640889055226643rm_a_b @ R3 @ A2 ) ) ) ) ).

% Image_mono
thf(fact_337_Image__mono,axiom,
    ! [R4: set_Pr4386577575007340137rm_a_b,R3: set_Pr4386577575007340137rm_a_b,A5: set_term_a_b,A2: set_term_a_b] :
      ( ( ord_le118470702582115849rm_a_b @ R4 @ R3 )
     => ( ( ord_le2705286416250468010rm_a_b @ A5 @ A2 )
       => ( ord_le2705286416250468010rm_a_b @ ( image_6923787618150785661rm_a_b @ R4 @ A5 ) @ ( image_6923787618150785661rm_a_b @ R3 @ A2 ) ) ) ) ).

% Image_mono
thf(fact_338_Image__subsetI,axiom,
    ! [R5: set_Pr2972776593051762503rm_a_b,R6: set_Pr2972776593051762503rm_a_b,X6: set_Pr4386577575007340137rm_a_b] :
      ( ( ord_le2593958535671617255rm_a_b @ R5 @ R6 )
     => ( ord_le118470702582115849rm_a_b @ ( image_871640889055226643rm_a_b @ R5 @ X6 ) @ ( image_871640889055226643rm_a_b @ R6 @ X6 ) ) ) ).

% Image_subsetI
thf(fact_339_Image__subsetI,axiom,
    ! [R5: set_Pr4386577575007340137rm_a_b,R6: set_Pr4386577575007340137rm_a_b,X6: set_term_a_b] :
      ( ( ord_le118470702582115849rm_a_b @ R5 @ R6 )
     => ( ord_le2705286416250468010rm_a_b @ ( image_6923787618150785661rm_a_b @ R5 @ X6 ) @ ( image_6923787618150785661rm_a_b @ R6 @ X6 ) ) ) ).

% Image_subsetI
thf(fact_340_list__decode_Ocases,axiom,
    ! [X: nat] :
      ( ( X != zero_zero_nat )
     => ~ ! [N2: nat] :
            ( X
           != ( suc @ N2 ) ) ) ).

% list_decode.cases
thf(fact_341_bot__nat__0_Oextremum,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).

% bot_nat_0.extremum
thf(fact_342_le0,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).

% le0
thf(fact_343_Suc__le__mono,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
      = ( ord_less_eq_nat @ N @ M ) ) ).

% Suc_le_mono
thf(fact_344_list__update__beyond,axiom,
    ! [Xs: list_f7931864845379430901rm_a_b,I4: nat,X: fset_P3577670608877728751rm_a_b] :
      ( ( ord_less_eq_nat @ ( size_s1967422219310127201rm_a_b @ Xs ) @ I4 )
     => ( ( list_u7212055091542668893rm_a_b @ Xs @ I4 @ X )
        = Xs ) ) ).

% list_update_beyond
thf(fact_345_zle__int,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% zle_int
thf(fact_346_nat__int__comparison_I3_J,axiom,
    ( ord_less_eq_nat
    = ( ^ [A4: nat,B3: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).

% nat_int_comparison(3)
thf(fact_347_bounded__Max__nat,axiom,
    ! [P: nat > $o,X: nat,M6: nat] :
      ( ( P @ X )
     => ( ! [X4: nat] :
            ( ( P @ X4 )
           => ( ord_less_eq_nat @ X4 @ M6 ) )
       => ~ ! [M4: nat] :
              ( ( P @ M4 )
             => ~ ! [X2: nat] :
                    ( ( P @ X2 )
                   => ( ord_less_eq_nat @ X2 @ M4 ) ) ) ) ) ).

% bounded_Max_nat
thf(fact_348_le__refl,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).

% le_refl
thf(fact_349_le__trans,axiom,
    ! [I4: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I4 @ J )
     => ( ( ord_less_eq_nat @ J @ K )
       => ( ord_less_eq_nat @ I4 @ K ) ) ) ).

% le_trans
thf(fact_350_eq__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( M = N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% eq_imp_le
thf(fact_351_le__antisym,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( M = N ) ) ) ).

% le_antisym
thf(fact_352_nat__le__linear,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
      | ( ord_less_eq_nat @ N @ M ) ) ).

% nat_le_linear
thf(fact_353_Nat_Oex__has__greatest__nat,axiom,
    ! [P: nat > $o,K: nat,B: nat] :
      ( ( P @ K )
     => ( ! [Y4: nat] :
            ( ( P @ Y4 )
           => ( ord_less_eq_nat @ Y4 @ B ) )
       => ? [X4: nat] :
            ( ( P @ X4 )
            & ! [Y5: nat] :
                ( ( P @ Y5 )
               => ( ord_less_eq_nat @ Y5 @ X4 ) ) ) ) ) ).

% Nat.ex_has_greatest_nat
thf(fact_354_fset_Opred__mono,axiom,
    ! [P: produc357393685978478089rm_a_b > $o,Pa: produc357393685978478089rm_a_b > $o] :
      ( ( ord_le6645574462093738516_a_b_o @ P @ Pa )
     => ( ord_le8000246984169389870_a_b_o @ ( pred_f8979122959637397547rm_a_b @ P ) @ ( pred_f8979122959637397547rm_a_b @ Pa ) ) ) ).

% fset.pred_mono
thf(fact_355_fImage__mono,axiom,
    ! [R4: fset_P2527920629275104717rm_a_b,R3: fset_P2527920629275104717rm_a_b,A5: fset_P3577670608877728751rm_a_b,A2: fset_P3577670608877728751rm_a_b] :
      ( ( ord_le2936254066310487405rm_a_b @ R4 @ R3 )
     => ( ( ord_le146453929066646415rm_a_b @ A5 @ A2 )
       => ( ord_le146453929066646415rm_a_b @ ( fSet_f3823212753407577598rm_a_b @ R4 @ A5 ) @ ( fSet_f3823212753407577598rm_a_b @ R3 @ A2 ) ) ) ) ).

% fImage_mono
thf(fact_356_fImage__mono,axiom,
    ! [R4: fset_P3577670608877728751rm_a_b,R3: fset_P3577670608877728751rm_a_b,A5: fset_term_a_b,A2: fset_term_a_b] :
      ( ( ord_le146453929066646415rm_a_b @ R4 @ R3 )
     => ( ( ord_le8417325568038501796rm_a_b @ A5 @ A2 )
       => ( ord_le8417325568038501796rm_a_b @ ( fSet_f8210585146631509224rm_a_b @ R4 @ A5 ) @ ( fSet_f8210585146631509224rm_a_b @ R3 @ A2 ) ) ) ) ).

% fImage_mono
thf(fact_357_less__eq__nat_Osimps_I1_J,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).

% less_eq_nat.simps(1)
thf(fact_358_bot__nat__0_Oextremum__unique,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
      = ( A = zero_zero_nat ) ) ).

% bot_nat_0.extremum_unique
thf(fact_359_bot__nat__0_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( A = zero_zero_nat ) ) ).

% bot_nat_0.extremum_uniqueI
thf(fact_360_le__0__eq,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ N @ zero_zero_nat )
      = ( N = zero_zero_nat ) ) ).

% le_0_eq
thf(fact_361_transitive__stepwise__le,axiom,
    ! [M: nat,N: nat,R5: nat > nat > $o] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ! [X4: nat] : ( R5 @ X4 @ X4 )
       => ( ! [X4: nat,Y4: nat,Z: nat] :
              ( ( R5 @ X4 @ Y4 )
             => ( ( R5 @ Y4 @ Z )
               => ( R5 @ X4 @ Z ) ) )
         => ( ! [N2: nat] : ( R5 @ N2 @ ( suc @ N2 ) )
           => ( R5 @ M @ N ) ) ) ) ) ).

% transitive_stepwise_le
thf(fact_362_nat__induct__at__least,axiom,
    ! [M: nat,N: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( P @ M )
       => ( ! [N2: nat] :
              ( ( ord_less_eq_nat @ M @ N2 )
             => ( ( P @ N2 )
               => ( P @ ( suc @ N2 ) ) ) )
         => ( P @ N ) ) ) ) ).

% nat_induct_at_least
thf(fact_363_full__nat__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ! [N2: nat] :
          ( ! [M2: nat] :
              ( ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 )
             => ( P @ M2 ) )
         => ( P @ N2 ) )
     => ( P @ N ) ) ).

% full_nat_induct
thf(fact_364_not__less__eq__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ~ ( ord_less_eq_nat @ M @ N ) )
      = ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).

% not_less_eq_eq
thf(fact_365_Suc__n__not__le__n,axiom,
    ! [N: nat] :
      ~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).

% Suc_n_not_le_n
thf(fact_366_le__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
      = ( ( ord_less_eq_nat @ M @ N )
        | ( M
          = ( suc @ N ) ) ) ) ).

% le_Suc_eq
thf(fact_367_Suc__le__D,axiom,
    ! [N: nat,M7: nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ M7 )
     => ? [M4: nat] :
          ( M7
          = ( suc @ M4 ) ) ) ).

% Suc_le_D
thf(fact_368_le__SucI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).

% le_SucI
thf(fact_369_le__SucE,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
     => ( ~ ( ord_less_eq_nat @ M @ N )
       => ( M
          = ( suc @ N ) ) ) ) ).

% le_SucE
thf(fact_370_Suc__leD,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% Suc_leD
thf(fact_371_inf__pigeonhole__principle,axiom,
    ! [N: nat,F: nat > nat > $o] :
      ( ! [K2: nat] :
        ? [I2: nat] :
          ( ( ord_less_nat @ I2 @ N )
          & ( F @ K2 @ I2 ) )
     => ? [I: nat] :
          ( ( ord_less_nat @ I @ N )
          & ! [K3: nat] :
            ? [K4: nat] :
              ( ( ord_less_eq_nat @ K3 @ K4 )
              & ( F @ K4 @ I ) ) ) ) ).

% inf_pigeonhole_principle
thf(fact_372_nat__less__le,axiom,
    ( ord_less_nat
    = ( ^ [M3: nat,N3: nat] :
          ( ( ord_less_eq_nat @ M3 @ N3 )
          & ( M3 != N3 ) ) ) ) ).

% nat_less_le
thf(fact_373_less__imp__le__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% less_imp_le_nat
thf(fact_374_le__eq__less__or__eq,axiom,
    ( ord_less_eq_nat
    = ( ^ [M3: nat,N3: nat] :
          ( ( ord_less_nat @ M3 @ N3 )
          | ( M3 = N3 ) ) ) ) ).

% le_eq_less_or_eq
thf(fact_375_less__or__eq__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( ( ord_less_nat @ M @ N )
        | ( M = N ) )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% less_or_eq_imp_le
thf(fact_376_le__neq__implies__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( M != N )
       => ( ord_less_nat @ M @ N ) ) ) ).

% le_neq_implies_less
thf(fact_377_less__mono__imp__le__mono,axiom,
    ! [F: nat > nat,I4: nat,J: nat] :
      ( ! [I: nat,J2: nat] :
          ( ( ord_less_nat @ I @ J2 )
         => ( ord_less_nat @ ( F @ I ) @ ( F @ J2 ) ) )
     => ( ( ord_less_eq_nat @ I4 @ J )
       => ( ord_less_eq_nat @ ( F @ I4 ) @ ( F @ J ) ) ) ) ).

% less_mono_imp_le_mono
thf(fact_378_ex__least__nat__le,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ N )
     => ( ~ ( P @ zero_zero_nat )
       => ? [K2: nat] :
            ( ( ord_less_eq_nat @ K2 @ N )
            & ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ K2 )
               => ~ ( P @ I2 ) )
            & ( P @ K2 ) ) ) ) ).

% ex_least_nat_le
thf(fact_379_Suc__leI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).

% Suc_leI
thf(fact_380_Suc__le__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
      = ( ord_less_nat @ M @ N ) ) ).

% Suc_le_eq
thf(fact_381_dec__induct,axiom,
    ! [I4: nat,J: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ I4 @ J )
     => ( ( P @ I4 )
       => ( ! [N2: nat] :
              ( ( ord_less_eq_nat @ I4 @ N2 )
             => ( ( ord_less_nat @ N2 @ J )
               => ( ( P @ N2 )
                 => ( P @ ( suc @ N2 ) ) ) ) )
         => ( P @ J ) ) ) ) ).

% dec_induct
thf(fact_382_inc__induct,axiom,
    ! [I4: nat,J: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ I4 @ J )
     => ( ( P @ J )
       => ( ! [N2: nat] :
              ( ( ord_less_eq_nat @ I4 @ N2 )
             => ( ( ord_less_nat @ N2 @ J )
               => ( ( P @ ( suc @ N2 ) )
                 => ( P @ N2 ) ) ) )
         => ( P @ I4 ) ) ) ) ).

% inc_induct
thf(fact_383_Suc__le__lessD,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
     => ( ord_less_nat @ M @ N ) ) ).

% Suc_le_lessD
thf(fact_384_le__less__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( ord_less_nat @ N @ ( suc @ M ) )
        = ( N = M ) ) ) ).

% le_less_Suc_eq
thf(fact_385_less__Suc__eq__le,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% less_Suc_eq_le
thf(fact_386_less__eq__Suc__le,axiom,
    ( ord_less_nat
    = ( ^ [N3: nat] : ( ord_less_eq_nat @ ( suc @ N3 ) ) ) ) ).

% less_eq_Suc_le
thf(fact_387_le__imp__less__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).

% le_imp_less_Suc
thf(fact_388_less__eq__fset_Orep__eq,axiom,
    ( ord_le146453929066646415rm_a_b
    = ( ^ [X3: fset_P3577670608877728751rm_a_b,Xa2: fset_P3577670608877728751rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( fset_P6170650533370837496rm_a_b @ X3 ) @ ( fset_P6170650533370837496rm_a_b @ Xa2 ) ) ) ) ).

% less_eq_fset.rep_eq
thf(fact_389_less__eq__fset_Orep__eq,axiom,
    ( ord_le8417325568038501796rm_a_b
    = ( ^ [X3: fset_term_a_b,Xa2: fset_term_a_b] : ( ord_le2705286416250468010rm_a_b @ ( fset_term_a_b2 @ X3 ) @ ( fset_term_a_b2 @ Xa2 ) ) ) ) ).

% less_eq_fset.rep_eq
thf(fact_390_less__eq__fset_Orep__eq,axiom,
    ( ord_le2936254066310487405rm_a_b
    = ( ^ [X3: fset_P2527920629275104717rm_a_b,Xa2: fset_P2527920629275104717rm_a_b] : ( ord_le2593958535671617255rm_a_b @ ( fset_P119123501830535382rm_a_b @ X3 ) @ ( fset_P119123501830535382rm_a_b @ Xa2 ) ) ) ) ).

% less_eq_fset.rep_eq
thf(fact_391_nonneg__int__cases,axiom,
    ! [K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ~ ! [N2: nat] :
            ( K
           != ( semiri1314217659103216013at_int @ N2 ) ) ) ).

% nonneg_int_cases
thf(fact_392_zero__le__imp__eq__int,axiom,
    ! [K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ? [N2: nat] :
          ( K
          = ( semiri1314217659103216013at_int @ N2 ) ) ) ).

% zero_le_imp_eq_int
thf(fact_393_length__removeAll__less__eq,axiom,
    ! [X: fset_P3577670608877728751rm_a_b,Xs: list_f7931864845379430901rm_a_b] : ( ord_less_eq_nat @ ( size_s1967422219310127201rm_a_b @ ( remove3100474213990342139rm_a_b @ X @ Xs ) ) @ ( size_s1967422219310127201rm_a_b @ Xs ) ) ).

% length_removeAll_less_eq
thf(fact_394_remove__nth__id,axiom,
    ! [Xs: list_f7931864845379430901rm_a_b,N: nat] :
      ( ( ord_less_eq_nat @ ( size_s1967422219310127201rm_a_b @ Xs ) @ N )
     => ( ( missin257347171404174568rm_a_b @ N @ Xs )
        = Xs ) ) ).

% remove_nth_id
thf(fact_395_verit__comp__simplify1_I1_J,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_396_verit__comp__simplify1_I1_J,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_397_ex__least__nat__less,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ N )
     => ( ~ ( P @ zero_zero_nat )
       => ? [K2: nat] :
            ( ( ord_less_nat @ K2 @ N )
            & ! [I2: nat] :
                ( ( ord_less_eq_nat @ I2 @ K2 )
               => ~ ( P @ I2 ) )
            & ( P @ ( suc @ K2 ) ) ) ) ) ).

% ex_least_nat_less
thf(fact_398_nat__int__comparison_I1_J,axiom,
    ( ( ^ [Y3: nat,Z2: nat] : ( Y3 = Z2 ) )
    = ( ^ [A4: nat,B3: nat] :
          ( ( semiri1314217659103216013at_int @ A4 )
          = ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).

% nat_int_comparison(1)
thf(fact_399_int__if,axiom,
    ! [P: $o,A: nat,B: nat] :
      ( ( P
       => ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B ) )
          = ( semiri1314217659103216013at_int @ A ) ) )
      & ( ~ P
       => ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B ) )
          = ( semiri1314217659103216013at_int @ B ) ) ) ) ).

% int_if
thf(fact_400_remove__nth__sound__r,axiom,
    ! [N: nat,P3: nat,Xs: list_f7931864845379430901rm_a_b] :
      ( ( ord_less_eq_nat @ N @ P3 )
     => ( ( ord_less_nat @ P3 @ ( size_s1967422219310127201rm_a_b @ Xs ) )
       => ( ( nth_fs3949069584078227574rm_a_b @ ( missin257347171404174568rm_a_b @ N @ Xs ) @ P3 )
          = ( nth_fs3949069584078227574rm_a_b @ Xs @ ( suc @ P3 ) ) ) ) ) ).

% remove_nth_sound_r
thf(fact_401_verit__comp__simplify1_I3_J,axiom,
    ! [B4: nat,A6: nat] :
      ( ( ~ ( ord_less_eq_nat @ B4 @ A6 ) )
      = ( ord_less_nat @ A6 @ B4 ) ) ).

% verit_comp_simplify1(3)
thf(fact_402_verit__comp__simplify1_I3_J,axiom,
    ! [B4: int,A6: int] :
      ( ( ~ ( ord_less_eq_int @ B4 @ A6 ) )
      = ( ord_less_int @ A6 @ B4 ) ) ).

% verit_comp_simplify1(3)
thf(fact_403_le__Nil,axiom,
    ! [X: list_s5629610930313288943rm_a_b] :
      ( ( ord_le616080101116966031rm_a_b @ X @ nil_se7334864872594423151rm_a_b )
      = ( X = nil_se7334864872594423151rm_a_b ) ) ).

% le_Nil
thf(fact_404_le__Nil,axiom,
    ! [X: list_f7931864845379430901rm_a_b] :
      ( ( ord_le3592285473176203669rm_a_b @ X @ nil_fs4020363321024022517rm_a_b )
      = ( X = nil_fs4020363321024022517rm_a_b ) ) ).

% le_Nil
thf(fact_405_nat__descend__induct,axiom,
    ! [N: nat,P: nat > $o,M: nat] :
      ( ! [K2: nat] :
          ( ( ord_less_nat @ N @ K2 )
         => ( P @ K2 ) )
     => ( ! [K2: nat] :
            ( ( ord_less_eq_nat @ K2 @ N )
           => ( ! [I2: nat] :
                  ( ( ord_less_nat @ K2 @ I2 )
                 => ( P @ I2 ) )
             => ( P @ K2 ) ) )
       => ( P @ M ) ) ) ).

% nat_descend_induct
thf(fact_406_order__le__imp__less__or__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_nat @ X @ Y )
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_407_order__le__imp__less__or__eq,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( ord_less_int @ X @ Y )
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_408_linorder__le__less__linear,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
      | ( ord_less_nat @ Y @ X ) ) ).

% linorder_le_less_linear
thf(fact_409_linorder__le__less__linear,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
      | ( ord_less_int @ Y @ X ) ) ).

% linorder_le_less_linear
thf(fact_410_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F: nat > nat,C2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
       => ( ! [X4: nat,Y4: nat] :
              ( ( ord_less_nat @ X4 @ Y4 )
             => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).

% order_less_le_subst2
thf(fact_411_order__less__le__subst2,axiom,
    ! [A: int,B: int,F: int > nat,C2: nat] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
       => ( ! [X4: int,Y4: int] :
              ( ( ord_less_int @ X4 @ Y4 )
             => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).

% order_less_le_subst2
thf(fact_412_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F: nat > int,C2: int] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_int @ ( F @ B ) @ C2 )
       => ( ! [X4: nat,Y4: nat] :
              ( ( ord_less_nat @ X4 @ Y4 )
             => ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).

% order_less_le_subst2
thf(fact_413_order__less__le__subst2,axiom,
    ! [A: int,B: int,F: int > int,C2: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_eq_int @ ( F @ B ) @ C2 )
       => ( ! [X4: int,Y4: int] :
              ( ( ord_less_int @ X4 @ Y4 )
             => ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).

% order_less_le_subst2
thf(fact_414_order__less__imp__not__less,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ~ ( ord_less_nat @ Y @ X ) ) ).

% order_less_imp_not_less
thf(fact_415_order__less__imp__not__less,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ~ ( ord_less_int @ Y @ X ) ) ).

% order_less_imp_not_less
thf(fact_416_order__less__imp__not__eq2,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
thf(fact_417_order__less__imp__not__eq2,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
thf(fact_418_order__less__imp__not__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( X != Y ) ) ).

% order_less_imp_not_eq
thf(fact_419_order__less__imp__not__eq,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ( X != Y ) ) ).

% order_less_imp_not_eq
thf(fact_420_linorder__less__linear,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
      | ( X = Y )
      | ( ord_less_nat @ Y @ X ) ) ).

% linorder_less_linear
thf(fact_421_linorder__less__linear,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
      | ( X = Y )
      | ( ord_less_int @ Y @ X ) ) ).

% linorder_less_linear
thf(fact_422_order__less__imp__triv,axiom,
    ! [X: nat,Y: nat,P: $o] :
      ( ( ord_less_nat @ X @ Y )
     => ( ( ord_less_nat @ Y @ X )
       => P ) ) ).

% order_less_imp_triv
thf(fact_423_order__less__imp__triv,axiom,
    ! [X: int,Y: int,P: $o] :
      ( ( ord_less_int @ X @ Y )
     => ( ( ord_less_int @ Y @ X )
       => P ) ) ).

% order_less_imp_triv
thf(fact_424_order__less__not__sym,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ~ ( ord_less_nat @ Y @ X ) ) ).

% order_less_not_sym
thf(fact_425_order__less__not__sym,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ~ ( ord_less_int @ Y @ X ) ) ).

% order_less_not_sym
thf(fact_426_order__less__subst2,axiom,
    ! [A: nat,B: nat,F: nat > nat,C2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C2 )
       => ( ! [X4: nat,Y4: nat] :
              ( ( ord_less_nat @ X4 @ Y4 )
             => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).

% order_less_subst2
thf(fact_427_order__less__subst2,axiom,
    ! [A: nat,B: nat,F: nat > int,C2: int] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_int @ ( F @ B ) @ C2 )
       => ( ! [X4: nat,Y4: nat] :
              ( ( ord_less_nat @ X4 @ Y4 )
             => ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).

% order_less_subst2
thf(fact_428_order__less__subst2,axiom,
    ! [A: int,B: int,F: int > nat,C2: nat] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C2 )
       => ( ! [X4: int,Y4: int] :
              ( ( ord_less_int @ X4 @ Y4 )
             => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).

% order_less_subst2
thf(fact_429_order__less__subst2,axiom,
    ! [A: int,B: int,F: int > int,C2: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ ( F @ B ) @ C2 )
       => ( ! [X4: int,Y4: int] :
              ( ( ord_less_int @ X4 @ Y4 )
             => ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).

% order_less_subst2
thf(fact_430_order__less__subst1,axiom,
    ! [A: nat,F: nat > nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C2 )
       => ( ! [X4: nat,Y4: nat] :
              ( ( ord_less_nat @ X4 @ Y4 )
             => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).

% order_less_subst1
thf(fact_431_order__less__subst1,axiom,
    ! [A: nat,F: int > nat,B: int,C2: int] :
      ( ( ord_less_nat @ A @ ( F @ B ) )
     => ( ( ord_less_int @ B @ C2 )
       => ( ! [X4: int,Y4: int] :
              ( ( ord_less_int @ X4 @ Y4 )
             => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).

% order_less_subst1
thf(fact_432_order__less__subst1,axiom,
    ! [A: int,F: nat > int,B: nat,C2: nat] :
      ( ( ord_less_int @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C2 )
       => ( ! [X4: nat,Y4: nat] :
              ( ( ord_less_nat @ X4 @ Y4 )
             => ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).

% order_less_subst1
thf(fact_433_order__less__subst1,axiom,
    ! [A: int,F: int > int,B: int,C2: int] :
      ( ( ord_less_int @ A @ ( F @ B ) )
     => ( ( ord_less_int @ B @ C2 )
       => ( ! [X4: int,Y4: int] :
              ( ( ord_less_int @ X4 @ Y4 )
             => ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).

% order_less_subst1
thf(fact_434_order__less__irrefl,axiom,
    ! [X: nat] :
      ~ ( ord_less_nat @ X @ X ) ).

% order_less_irrefl
thf(fact_435_order__less__irrefl,axiom,
    ! [X: int] :
      ~ ( ord_less_int @ X @ X ) ).

% order_less_irrefl
thf(fact_436_ord__less__eq__subst,axiom,
    ! [A: nat,B: nat,F: nat > nat,C2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ( F @ B )
          = C2 )
       => ( ! [X4: nat,Y4: nat] :
              ( ( ord_less_nat @ X4 @ Y4 )
             => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).

% ord_less_eq_subst
thf(fact_437_ord__less__eq__subst,axiom,
    ! [A: nat,B: nat,F: nat > int,C2: int] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ( F @ B )
          = C2 )
       => ( ! [X4: nat,Y4: nat] :
              ( ( ord_less_nat @ X4 @ Y4 )
             => ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).

% ord_less_eq_subst
thf(fact_438_ord__less__eq__subst,axiom,
    ! [A: int,B: int,F: int > nat,C2: nat] :
      ( ( ord_less_int @ A @ B )
     => ( ( ( F @ B )
          = C2 )
       => ( ! [X4: int,Y4: int] :
              ( ( ord_less_int @ X4 @ Y4 )
             => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).

% ord_less_eq_subst
thf(fact_439_ord__less__eq__subst,axiom,
    ! [A: int,B: int,F: int > int,C2: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ( F @ B )
          = C2 )
       => ( ! [X4: int,Y4: int] :
              ( ( ord_less_int @ X4 @ Y4 )
             => ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).

% ord_less_eq_subst
thf(fact_440_ord__eq__less__subst,axiom,
    ! [A: nat,F: nat > nat,B: nat,C2: nat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_nat @ B @ C2 )
       => ( ! [X4: nat,Y4: nat] :
              ( ( ord_less_nat @ X4 @ Y4 )
             => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_441_ord__eq__less__subst,axiom,
    ! [A: int,F: nat > int,B: nat,C2: nat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_nat @ B @ C2 )
       => ( ! [X4: nat,Y4: nat] :
              ( ( ord_less_nat @ X4 @ Y4 )
             => ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_442_ord__eq__less__subst,axiom,
    ! [A: nat,F: int > nat,B: int,C2: int] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_int @ B @ C2 )
       => ( ! [X4: int,Y4: int] :
              ( ( ord_less_int @ X4 @ Y4 )
             => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_443_ord__eq__less__subst,axiom,
    ! [A: int,F: int > int,B: int,C2: int] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_int @ B @ C2 )
       => ( ! [X4: int,Y4: int] :
              ( ( ord_less_int @ X4 @ Y4 )
             => ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_444_order__less__trans,axiom,
    ! [X: nat,Y: nat,Z4: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( ( ord_less_nat @ Y @ Z4 )
       => ( ord_less_nat @ X @ Z4 ) ) ) ).

% order_less_trans
thf(fact_445_order__less__trans,axiom,
    ! [X: int,Y: int,Z4: int] :
      ( ( ord_less_int @ X @ Y )
     => ( ( ord_less_int @ Y @ Z4 )
       => ( ord_less_int @ X @ Z4 ) ) ) ).

% order_less_trans
thf(fact_446_order__less__asym_H,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ( ord_less_nat @ B @ A ) ) ).

% order_less_asym'
thf(fact_447_order__less__asym_H,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ~ ( ord_less_int @ B @ A ) ) ).

% order_less_asym'
thf(fact_448_linorder__neq__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
      = ( ( ord_less_nat @ X @ Y )
        | ( ord_less_nat @ Y @ X ) ) ) ).

% linorder_neq_iff
thf(fact_449_linorder__neq__iff,axiom,
    ! [X: int,Y: int] :
      ( ( X != Y )
      = ( ( ord_less_int @ X @ Y )
        | ( ord_less_int @ Y @ X ) ) ) ).

% linorder_neq_iff
thf(fact_450_order__less__asym,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ~ ( ord_less_nat @ Y @ X ) ) ).

% order_less_asym
thf(fact_451_order__less__asym,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ~ ( ord_less_int @ Y @ X ) ) ).

% order_less_asym
thf(fact_452_linorder__neqE,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
     => ( ~ ( ord_less_nat @ X @ Y )
       => ( ord_less_nat @ Y @ X ) ) ) ).

% linorder_neqE
thf(fact_453_linorder__neqE,axiom,
    ! [X: int,Y: int] :
      ( ( X != Y )
     => ( ~ ( ord_less_int @ X @ Y )
       => ( ord_less_int @ Y @ X ) ) ) ).

% linorder_neqE
thf(fact_454_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_455_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ A )
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_456_order_Ostrict__implies__not__eq,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_457_order_Ostrict__implies__not__eq,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_458_dual__order_Ostrict__trans,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( ord_less_nat @ C2 @ B )
       => ( ord_less_nat @ C2 @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_459_dual__order_Ostrict__trans,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( ord_less_int @ C2 @ B )
       => ( ord_less_int @ C2 @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_460_not__less__iff__gr__or__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ~ ( ord_less_nat @ X @ Y ) )
      = ( ( ord_less_nat @ Y @ X )
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_461_not__less__iff__gr__or__eq,axiom,
    ! [X: int,Y: int] :
      ( ( ~ ( ord_less_int @ X @ Y ) )
      = ( ( ord_less_int @ Y @ X )
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_462_order_Ostrict__trans,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ B @ C2 )
       => ( ord_less_nat @ A @ C2 ) ) ) ).

% order.strict_trans
thf(fact_463_order_Ostrict__trans,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ B @ C2 )
       => ( ord_less_int @ A @ C2 ) ) ) ).

% order.strict_trans
thf(fact_464_linorder__less__wlog,axiom,
    ! [P: nat > nat > $o,A: nat,B: nat] :
      ( ! [A7: nat,B5: nat] :
          ( ( ord_less_nat @ A7 @ B5 )
         => ( P @ A7 @ B5 ) )
     => ( ! [A7: nat] : ( P @ A7 @ A7 )
       => ( ! [A7: nat,B5: nat] :
              ( ( P @ B5 @ A7 )
             => ( P @ A7 @ B5 ) )
         => ( P @ A @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_465_linorder__less__wlog,axiom,
    ! [P: int > int > $o,A: int,B: int] :
      ( ! [A7: int,B5: int] :
          ( ( ord_less_int @ A7 @ B5 )
         => ( P @ A7 @ B5 ) )
     => ( ! [A7: int] : ( P @ A7 @ A7 )
       => ( ! [A7: int,B5: int] :
              ( ( P @ B5 @ A7 )
             => ( P @ A7 @ B5 ) )
         => ( P @ A @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_466_exists__least__iff,axiom,
    ( ( ^ [P4: nat > $o] :
        ? [X7: nat] : ( P4 @ X7 ) )
    = ( ^ [P2: nat > $o] :
        ? [N3: nat] :
          ( ( P2 @ N3 )
          & ! [M3: nat] :
              ( ( ord_less_nat @ M3 @ N3 )
             => ~ ( P2 @ M3 ) ) ) ) ) ).

% exists_least_iff
thf(fact_467_dual__order_Oirrefl,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ A ) ).

% dual_order.irrefl
thf(fact_468_dual__order_Oirrefl,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ A @ A ) ).

% dual_order.irrefl
thf(fact_469_dual__order_Oasym,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ~ ( ord_less_nat @ A @ B ) ) ).

% dual_order.asym
thf(fact_470_dual__order_Oasym,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ A )
     => ~ ( ord_less_int @ A @ B ) ) ).

% dual_order.asym
thf(fact_471_linorder__cases,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X @ Y )
     => ( ( X != Y )
       => ( ord_less_nat @ Y @ X ) ) ) ).

% linorder_cases
thf(fact_472_linorder__cases,axiom,
    ! [X: int,Y: int] :
      ( ~ ( ord_less_int @ X @ Y )
     => ( ( X != Y )
       => ( ord_less_int @ Y @ X ) ) ) ).

% linorder_cases
thf(fact_473_antisym__conv3,axiom,
    ! [Y: nat,X: nat] :
      ( ~ ( ord_less_nat @ Y @ X )
     => ( ( ~ ( ord_less_nat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv3
thf(fact_474_antisym__conv3,axiom,
    ! [Y: int,X: int] :
      ( ~ ( ord_less_int @ Y @ X )
     => ( ( ~ ( ord_less_int @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv3
thf(fact_475_less__induct,axiom,
    ! [P: nat > $o,A: nat] :
      ( ! [X4: nat] :
          ( ! [Y5: nat] :
              ( ( ord_less_nat @ Y5 @ X4 )
             => ( P @ Y5 ) )
         => ( P @ X4 ) )
     => ( P @ A ) ) ).

% less_induct
thf(fact_476_ord__less__eq__trans,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( B = C2 )
       => ( ord_less_nat @ A @ C2 ) ) ) ).

% ord_less_eq_trans
thf(fact_477_ord__less__eq__trans,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( B = C2 )
       => ( ord_less_int @ A @ C2 ) ) ) ).

% ord_less_eq_trans
thf(fact_478_ord__eq__less__trans,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( A = B )
     => ( ( ord_less_nat @ B @ C2 )
       => ( ord_less_nat @ A @ C2 ) ) ) ).

% ord_eq_less_trans
thf(fact_479_ord__eq__less__trans,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( A = B )
     => ( ( ord_less_int @ B @ C2 )
       => ( ord_less_int @ A @ C2 ) ) ) ).

% ord_eq_less_trans
thf(fact_480_order_Oasym,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ( ord_less_nat @ B @ A ) ) ).

% order.asym
thf(fact_481_order_Oasym,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ~ ( ord_less_int @ B @ A ) ) ).

% order.asym
thf(fact_482_less__imp__neq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( X != Y ) ) ).

% less_imp_neq
thf(fact_483_less__imp__neq,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ( X != Y ) ) ).

% less_imp_neq
thf(fact_484_gt__ex,axiom,
    ! [X: nat] :
    ? [X_12: nat] : ( ord_less_nat @ X @ X_12 ) ).

% gt_ex
thf(fact_485_gt__ex,axiom,
    ! [X: int] :
    ? [X_12: int] : ( ord_less_int @ X @ X_12 ) ).

% gt_ex
thf(fact_486_lt__ex,axiom,
    ! [X: int] :
    ? [Y4: int] : ( ord_less_int @ Y4 @ X ) ).

% lt_ex
thf(fact_487_not__less__Nil,axiom,
    ! [X: list_s5629610930313288943rm_a_b] :
      ~ ( ord_le4636864222483984283rm_a_b @ X @ nil_se7334864872594423151rm_a_b ) ).

% not_less_Nil
thf(fact_488_not__less__Nil,axiom,
    ! [X: list_f7931864845379430901rm_a_b] :
      ~ ( ord_le5822847225584068257rm_a_b @ X @ nil_fs4020363321024022517rm_a_b ) ).

% not_less_Nil
thf(fact_489_less__list__code_I1_J,axiom,
    ! [Xs: list_s5629610930313288943rm_a_b] :
      ~ ( ord_le4636864222483984283rm_a_b @ Xs @ nil_se7334864872594423151rm_a_b ) ).

% less_list_code(1)
thf(fact_490_less__list__code_I1_J,axiom,
    ! [Xs: list_f7931864845379430901rm_a_b] :
      ~ ( ord_le5822847225584068257rm_a_b @ Xs @ nil_fs4020363321024022517rm_a_b ) ).

% less_list_code(1)
thf(fact_491_Nil__le__Cons,axiom,
    ! [X: list_s5629610930313288943rm_a_b] : ( ord_le616080101116966031rm_a_b @ nil_se7334864872594423151rm_a_b @ X ) ).

% Nil_le_Cons
thf(fact_492_Nil__le__Cons,axiom,
    ! [X: list_f7931864845379430901rm_a_b] : ( ord_le3592285473176203669rm_a_b @ nil_fs4020363321024022517rm_a_b @ X ) ).

% Nil_le_Cons
thf(fact_493_less__eq__list__code_I2_J,axiom,
    ! [Xs: list_s5629610930313288943rm_a_b] : ( ord_le616080101116966031rm_a_b @ nil_se7334864872594423151rm_a_b @ Xs ) ).

% less_eq_list_code(2)
thf(fact_494_less__eq__list__code_I2_J,axiom,
    ! [Xs: list_f7931864845379430901rm_a_b] : ( ord_le3592285473176203669rm_a_b @ nil_fs4020363321024022517rm_a_b @ Xs ) ).

% less_eq_list_code(2)
thf(fact_495_leD,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ~ ( ord_less_nat @ X @ Y ) ) ).

% leD
thf(fact_496_leD,axiom,
    ! [Y: int,X: int] :
      ( ( ord_less_eq_int @ Y @ X )
     => ~ ( ord_less_int @ X @ Y ) ) ).

% leD
thf(fact_497_leI,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X @ Y )
     => ( ord_less_eq_nat @ Y @ X ) ) ).

% leI
thf(fact_498_leI,axiom,
    ! [X: int,Y: int] :
      ( ~ ( ord_less_int @ X @ Y )
     => ( ord_less_eq_int @ Y @ X ) ) ).

% leI
thf(fact_499_nless__le,axiom,
    ! [A: nat,B: nat] :
      ( ( ~ ( ord_less_nat @ A @ B ) )
      = ( ~ ( ord_less_eq_nat @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_500_nless__le,axiom,
    ! [A: int,B: int] :
      ( ( ~ ( ord_less_int @ A @ B ) )
      = ( ~ ( ord_less_eq_int @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_501_antisym__conv1,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_502_antisym__conv1,axiom,
    ! [X: int,Y: int] :
      ( ~ ( ord_less_int @ X @ Y )
     => ( ( ord_less_eq_int @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_503_antisym__conv2,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ~ ( ord_less_nat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_504_antisym__conv2,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( ~ ( ord_less_int @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_505_less__le__not__le,axiom,
    ( ord_less_nat
    = ( ^ [X3: nat,Y2: nat] :
          ( ( ord_less_eq_nat @ X3 @ Y2 )
          & ~ ( ord_less_eq_nat @ Y2 @ X3 ) ) ) ) ).

% less_le_not_le
thf(fact_506_less__le__not__le,axiom,
    ( ord_less_int
    = ( ^ [X3: int,Y2: int] :
          ( ( ord_less_eq_int @ X3 @ Y2 )
          & ~ ( ord_less_eq_int @ Y2 @ X3 ) ) ) ) ).

% less_le_not_le
thf(fact_507_not__le__imp__less,axiom,
    ! [Y: nat,X: nat] :
      ( ~ ( ord_less_eq_nat @ Y @ X )
     => ( ord_less_nat @ X @ Y ) ) ).

% not_le_imp_less
thf(fact_508_not__le__imp__less,axiom,
    ! [Y: int,X: int] :
      ( ~ ( ord_less_eq_int @ Y @ X )
     => ( ord_less_int @ X @ Y ) ) ).

% not_le_imp_less
thf(fact_509_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [A4: nat,B3: nat] :
          ( ( ord_less_nat @ A4 @ B3 )
          | ( A4 = B3 ) ) ) ) ).

% order.order_iff_strict
thf(fact_510_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_int
    = ( ^ [A4: int,B3: int] :
          ( ( ord_less_int @ A4 @ B3 )
          | ( A4 = B3 ) ) ) ) ).

% order.order_iff_strict
thf(fact_511_order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [A4: nat,B3: nat] :
          ( ( ord_less_eq_nat @ A4 @ B3 )
          & ( A4 != B3 ) ) ) ) ).

% order.strict_iff_order
thf(fact_512_order_Ostrict__iff__order,axiom,
    ( ord_less_int
    = ( ^ [A4: int,B3: int] :
          ( ( ord_less_eq_int @ A4 @ B3 )
          & ( A4 != B3 ) ) ) ) ).

% order.strict_iff_order
thf(fact_513_order_Ostrict__trans1,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ B @ C2 )
       => ( ord_less_nat @ A @ C2 ) ) ) ).

% order.strict_trans1
thf(fact_514_order_Ostrict__trans1,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_int @ B @ C2 )
       => ( ord_less_int @ A @ C2 ) ) ) ).

% order.strict_trans1
thf(fact_515_order_Ostrict__trans2,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ord_less_nat @ A @ C2 ) ) ) ).

% order.strict_trans2
thf(fact_516_order_Ostrict__trans2,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_eq_int @ B @ C2 )
       => ( ord_less_int @ A @ C2 ) ) ) ).

% order.strict_trans2
thf(fact_517_order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [A4: nat,B3: nat] :
          ( ( ord_less_eq_nat @ A4 @ B3 )
          & ~ ( ord_less_eq_nat @ B3 @ A4 ) ) ) ) ).

% order.strict_iff_not
thf(fact_518_order_Ostrict__iff__not,axiom,
    ( ord_less_int
    = ( ^ [A4: int,B3: int] :
          ( ( ord_less_eq_int @ A4 @ B3 )
          & ~ ( ord_less_eq_int @ B3 @ A4 ) ) ) ) ).

% order.strict_iff_not
thf(fact_519_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [B3: nat,A4: nat] :
          ( ( ord_less_nat @ B3 @ A4 )
          | ( A4 = B3 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_520_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_int
    = ( ^ [B3: int,A4: int] :
          ( ( ord_less_int @ B3 @ A4 )
          | ( A4 = B3 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_521_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [B3: nat,A4: nat] :
          ( ( ord_less_eq_nat @ B3 @ A4 )
          & ( A4 != B3 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_522_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_int
    = ( ^ [B3: int,A4: int] :
          ( ( ord_less_eq_int @ B3 @ A4 )
          & ( A4 != B3 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_523_dual__order_Ostrict__trans1,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_nat @ C2 @ B )
       => ( ord_less_nat @ C2 @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_524_dual__order_Ostrict__trans1,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_less_int @ C2 @ B )
       => ( ord_less_int @ C2 @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_525_dual__order_Ostrict__trans2,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( ord_less_eq_nat @ C2 @ B )
       => ( ord_less_nat @ C2 @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_526_dual__order_Ostrict__trans2,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( ord_less_eq_int @ C2 @ B )
       => ( ord_less_int @ C2 @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_527_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [B3: nat,A4: nat] :
          ( ( ord_less_eq_nat @ B3 @ A4 )
          & ~ ( ord_less_eq_nat @ A4 @ B3 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_528_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_int
    = ( ^ [B3: int,A4: int] :
          ( ( ord_less_eq_int @ B3 @ A4 )
          & ~ ( ord_less_eq_int @ A4 @ B3 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_529_order_Ostrict__implies__order,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ord_less_eq_nat @ A @ B ) ) ).

% order.strict_implies_order
thf(fact_530_order_Ostrict__implies__order,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ( ord_less_eq_int @ A @ B ) ) ).

% order.strict_implies_order
thf(fact_531_dual__order_Ostrict__implies__order,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ord_less_eq_nat @ B @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_532_dual__order_Ostrict__implies__order,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ A )
     => ( ord_less_eq_int @ B @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_533_order__le__less,axiom,
    ( ord_less_eq_nat
    = ( ^ [X3: nat,Y2: nat] :
          ( ( ord_less_nat @ X3 @ Y2 )
          | ( X3 = Y2 ) ) ) ) ).

% order_le_less
thf(fact_534_order__le__less,axiom,
    ( ord_less_eq_int
    = ( ^ [X3: int,Y2: int] :
          ( ( ord_less_int @ X3 @ Y2 )
          | ( X3 = Y2 ) ) ) ) ).

% order_le_less
thf(fact_535_order__less__le,axiom,
    ( ord_less_nat
    = ( ^ [X3: nat,Y2: nat] :
          ( ( ord_less_eq_nat @ X3 @ Y2 )
          & ( X3 != Y2 ) ) ) ) ).

% order_less_le
thf(fact_536_order__less__le,axiom,
    ( ord_less_int
    = ( ^ [X3: int,Y2: int] :
          ( ( ord_less_eq_int @ X3 @ Y2 )
          & ( X3 != Y2 ) ) ) ) ).

% order_less_le
thf(fact_537_linorder__not__le,axiom,
    ! [X: nat,Y: nat] :
      ( ( ~ ( ord_less_eq_nat @ X @ Y ) )
      = ( ord_less_nat @ Y @ X ) ) ).

% linorder_not_le
thf(fact_538_linorder__not__le,axiom,
    ! [X: int,Y: int] :
      ( ( ~ ( ord_less_eq_int @ X @ Y ) )
      = ( ord_less_int @ Y @ X ) ) ).

% linorder_not_le
thf(fact_539_linorder__not__less,axiom,
    ! [X: nat,Y: nat] :
      ( ( ~ ( ord_less_nat @ X @ Y ) )
      = ( ord_less_eq_nat @ Y @ X ) ) ).

% linorder_not_less
thf(fact_540_linorder__not__less,axiom,
    ! [X: int,Y: int] :
      ( ( ~ ( ord_less_int @ X @ Y ) )
      = ( ord_less_eq_int @ Y @ X ) ) ).

% linorder_not_less
thf(fact_541_order__less__imp__le,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( ord_less_eq_nat @ X @ Y ) ) ).

% order_less_imp_le
thf(fact_542_order__less__imp__le,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ( ord_less_eq_int @ X @ Y ) ) ).

% order_less_imp_le
thf(fact_543_order__le__neq__trans,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( A != B )
       => ( ord_less_nat @ A @ B ) ) ) ).

% order_le_neq_trans
thf(fact_544_order__le__neq__trans,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( A != B )
       => ( ord_less_int @ A @ B ) ) ) ).

% order_le_neq_trans
thf(fact_545_order__neq__le__trans,axiom,
    ! [A: nat,B: nat] :
      ( ( A != B )
     => ( ( ord_less_eq_nat @ A @ B )
       => ( ord_less_nat @ A @ B ) ) ) ).

% order_neq_le_trans
thf(fact_546_order__neq__le__trans,axiom,
    ! [A: int,B: int] :
      ( ( A != B )
     => ( ( ord_less_eq_int @ A @ B )
       => ( ord_less_int @ A @ B ) ) ) ).

% order_neq_le_trans
thf(fact_547_order__le__less__trans,axiom,
    ! [X: nat,Y: nat,Z4: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_nat @ Y @ Z4 )
       => ( ord_less_nat @ X @ Z4 ) ) ) ).

% order_le_less_trans
thf(fact_548_order__le__less__trans,axiom,
    ! [X: int,Y: int,Z4: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( ord_less_int @ Y @ Z4 )
       => ( ord_less_int @ X @ Z4 ) ) ) ).

% order_le_less_trans
thf(fact_549_order__less__le__trans,axiom,
    ! [X: nat,Y: nat,Z4: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ Y @ Z4 )
       => ( ord_less_nat @ X @ Z4 ) ) ) ).

% order_less_le_trans
thf(fact_550_order__less__le__trans,axiom,
    ! [X: int,Y: int,Z4: int] :
      ( ( ord_less_int @ X @ Y )
     => ( ( ord_less_eq_int @ Y @ Z4 )
       => ( ord_less_int @ X @ Z4 ) ) ) ).

% order_less_le_trans
thf(fact_551_order__le__less__subst1,axiom,
    ! [A: nat,F: nat > nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C2 )
       => ( ! [X4: nat,Y4: nat] :
              ( ( ord_less_nat @ X4 @ Y4 )
             => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_552_order__le__less__subst1,axiom,
    ! [A: nat,F: int > nat,B: int,C2: int] :
      ( ( ord_less_eq_nat @ A @ ( F @ B ) )
     => ( ( ord_less_int @ B @ C2 )
       => ( ! [X4: int,Y4: int] :
              ( ( ord_less_int @ X4 @ Y4 )
             => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_553_order__le__less__subst1,axiom,
    ! [A: int,F: nat > int,B: nat,C2: nat] :
      ( ( ord_less_eq_int @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C2 )
       => ( ! [X4: nat,Y4: nat] :
              ( ( ord_less_nat @ X4 @ Y4 )
             => ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_554_order__le__less__subst1,axiom,
    ! [A: int,F: int > int,B: int,C2: int] :
      ( ( ord_less_eq_int @ A @ ( F @ B ) )
     => ( ( ord_less_int @ B @ C2 )
       => ( ! [X4: int,Y4: int] :
              ( ( ord_less_int @ X4 @ Y4 )
             => ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_555_order__le__less__subst2,axiom,
    ! [A: nat,B: nat,F: nat > nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C2 )
       => ( ! [X4: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).

% order_le_less_subst2
thf(fact_556_order__le__less__subst2,axiom,
    ! [A: nat,B: nat,F: nat > int,C2: int] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_int @ ( F @ B ) @ C2 )
       => ( ! [X4: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y4 )
             => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).

% order_le_less_subst2
thf(fact_557_order__le__less__subst2,axiom,
    ! [A: int,B: int,F: int > nat,C2: nat] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C2 )
       => ( ! [X4: int,Y4: int] :
              ( ( ord_less_eq_int @ X4 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).

% order_le_less_subst2
thf(fact_558_order__le__less__subst2,axiom,
    ! [A: int,B: int,F: int > int,C2: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_int @ ( F @ B ) @ C2 )
       => ( ! [X4: int,Y4: int] :
              ( ( ord_less_eq_int @ X4 @ Y4 )
             => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).

% order_le_less_subst2
thf(fact_559_order__less__le__subst1,axiom,
    ! [A: nat,F: nat > nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ! [X4: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_560_order__less__le__subst1,axiom,
    ! [A: int,F: nat > int,B: nat,C2: nat] :
      ( ( ord_less_int @ A @ ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ! [X4: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y4 )
             => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_561_order__less__le__subst1,axiom,
    ! [A: nat,F: int > nat,B: int,C2: int] :
      ( ( ord_less_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_int @ B @ C2 )
       => ( ! [X4: int,Y4: int] :
              ( ( ord_less_eq_int @ X4 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_562_order__less__le__subst1,axiom,
    ! [A: int,F: int > int,B: int,C2: int] :
      ( ( ord_less_int @ A @ ( F @ B ) )
     => ( ( ord_less_eq_int @ B @ C2 )
       => ( ! [X4: int,Y4: int] :
              ( ( ord_less_eq_int @ X4 @ Y4 )
             => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_563_minf_I8_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X2: nat] :
      ( ( ord_less_nat @ X2 @ Z )
     => ~ ( ord_less_eq_nat @ T @ X2 ) ) ).

% minf(8)
thf(fact_564_minf_I8_J,axiom,
    ! [T: int] :
    ? [Z: int] :
    ! [X2: int] :
      ( ( ord_less_int @ X2 @ Z )
     => ~ ( ord_less_eq_int @ T @ X2 ) ) ).

% minf(8)
thf(fact_565_minf_I6_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X2: nat] :
      ( ( ord_less_nat @ X2 @ Z )
     => ( ord_less_eq_nat @ X2 @ T ) ) ).

% minf(6)
thf(fact_566_minf_I6_J,axiom,
    ! [T: int] :
    ? [Z: int] :
    ! [X2: int] :
      ( ( ord_less_int @ X2 @ Z )
     => ( ord_less_eq_int @ X2 @ T ) ) ).

% minf(6)
thf(fact_567_pinf_I8_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X2: nat] :
      ( ( ord_less_nat @ Z @ X2 )
     => ( ord_less_eq_nat @ T @ X2 ) ) ).

% pinf(8)
thf(fact_568_pinf_I8_J,axiom,
    ! [T: int] :
    ? [Z: int] :
    ! [X2: int] :
      ( ( ord_less_int @ Z @ X2 )
     => ( ord_less_eq_int @ T @ X2 ) ) ).

% pinf(8)
thf(fact_569_pinf_I6_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X2: nat] :
      ( ( ord_less_nat @ Z @ X2 )
     => ~ ( ord_less_eq_nat @ X2 @ T ) ) ).

% pinf(6)
thf(fact_570_pinf_I6_J,axiom,
    ! [T: int] :
    ? [Z: int] :
    ! [X2: int] :
      ( ( ord_less_int @ Z @ X2 )
     => ~ ( ord_less_eq_int @ X2 @ T ) ) ).

% pinf(6)
thf(fact_571_complete__interval,axiom,
    ! [A: nat,B: nat,P: nat > $o] :
      ( ( ord_less_nat @ A @ B )
     => ( ( P @ A )
       => ( ~ ( P @ B )
         => ? [C3: nat] :
              ( ( ord_less_eq_nat @ A @ C3 )
              & ( ord_less_eq_nat @ C3 @ B )
              & ! [X2: nat] :
                  ( ( ( ord_less_eq_nat @ A @ X2 )
                    & ( ord_less_nat @ X2 @ C3 ) )
                 => ( P @ X2 ) )
              & ! [D2: nat] :
                  ( ! [X4: nat] :
                      ( ( ( ord_less_eq_nat @ A @ X4 )
                        & ( ord_less_nat @ X4 @ D2 ) )
                     => ( P @ X4 ) )
                 => ( ord_less_eq_nat @ D2 @ C3 ) ) ) ) ) ) ).

% complete_interval
thf(fact_572_complete__interval,axiom,
    ! [A: int,B: int,P: int > $o] :
      ( ( ord_less_int @ A @ B )
     => ( ( P @ A )
       => ( ~ ( P @ B )
         => ? [C3: int] :
              ( ( ord_less_eq_int @ A @ C3 )
              & ( ord_less_eq_int @ C3 @ B )
              & ! [X2: int] :
                  ( ( ( ord_less_eq_int @ A @ X2 )
                    & ( ord_less_int @ X2 @ C3 ) )
                 => ( P @ X2 ) )
              & ! [D2: int] :
                  ( ! [X4: int] :
                      ( ( ( ord_less_eq_int @ A @ X4 )
                        & ( ord_less_int @ X4 @ D2 ) )
                     => ( P @ X4 ) )
                 => ( ord_less_eq_int @ D2 @ C3 ) ) ) ) ) ) ).

% complete_interval
thf(fact_573_min__list__nth,axiom,
    ! [Xs: list_nat,Ys2: list_nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys2 ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys2 ) )
           => ( ord_less_eq_nat @ ( nth_nat @ Xs @ I ) @ ( nth_nat @ Ys2 @ I ) ) )
       => ( ord_less_eq_nat @ ( missing_min_list_nat @ Xs ) @ ( missing_min_list_nat @ Ys2 ) ) ) ) ).

% min_list_nth
thf(fact_574_min__list__nth,axiom,
    ! [Xs: list_int,Ys2: list_int] :
      ( ( ( size_size_list_int @ Xs )
        = ( size_size_list_int @ Ys2 ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_size_list_int @ Ys2 ) )
           => ( ord_less_eq_int @ ( nth_int @ Xs @ I ) @ ( nth_int @ Ys2 @ I ) ) )
       => ( ord_less_eq_int @ ( missing_min_list_int @ Xs ) @ ( missing_min_list_int @ Ys2 ) ) ) ) ).

% min_list_nth
thf(fact_575_min__list,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
     => ( ord_less_eq_nat @ ( missing_min_list_nat @ Xs ) @ X ) ) ).

% min_list
thf(fact_576_min__list,axiom,
    ! [X: int,Xs: list_int] :
      ( ( member_int @ X @ ( set_int2 @ Xs ) )
     => ( ord_less_eq_int @ ( missing_min_list_int @ Xs ) @ X ) ) ).

% min_list
thf(fact_577_minf_I7_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X2: nat] :
      ( ( ord_less_nat @ X2 @ Z )
     => ~ ( ord_less_nat @ T @ X2 ) ) ).

% minf(7)
thf(fact_578_minf_I7_J,axiom,
    ! [T: int] :
    ? [Z: int] :
    ! [X2: int] :
      ( ( ord_less_int @ X2 @ Z )
     => ~ ( ord_less_int @ T @ X2 ) ) ).

% minf(7)
thf(fact_579_minf_I5_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X2: nat] :
      ( ( ord_less_nat @ X2 @ Z )
     => ( ord_less_nat @ X2 @ T ) ) ).

% minf(5)
thf(fact_580_minf_I5_J,axiom,
    ! [T: int] :
    ? [Z: int] :
    ! [X2: int] :
      ( ( ord_less_int @ X2 @ Z )
     => ( ord_less_int @ X2 @ T ) ) ).

% minf(5)
thf(fact_581_minf_I4_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X2: nat] :
      ( ( ord_less_nat @ X2 @ Z )
     => ( X2 != T ) ) ).

% minf(4)
thf(fact_582_minf_I4_J,axiom,
    ! [T: int] :
    ? [Z: int] :
    ! [X2: int] :
      ( ( ord_less_int @ X2 @ Z )
     => ( X2 != T ) ) ).

% minf(4)
thf(fact_583_minf_I3_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X2: nat] :
      ( ( ord_less_nat @ X2 @ Z )
     => ( X2 != T ) ) ).

% minf(3)
thf(fact_584_minf_I3_J,axiom,
    ! [T: int] :
    ? [Z: int] :
    ! [X2: int] :
      ( ( ord_less_int @ X2 @ Z )
     => ( X2 != T ) ) ).

% minf(3)
thf(fact_585_minf_I2_J,axiom,
    ! [P: nat > $o,P5: nat > $o,Q: nat > $o,Q2: nat > $o] :
      ( ? [Z5: nat] :
        ! [X4: nat] :
          ( ( ord_less_nat @ X4 @ Z5 )
         => ( ( P @ X4 )
            = ( P5 @ X4 ) ) )
     => ( ? [Z5: nat] :
          ! [X4: nat] :
            ( ( ord_less_nat @ X4 @ Z5 )
           => ( ( Q @ X4 )
              = ( Q2 @ X4 ) ) )
       => ? [Z: nat] :
          ! [X2: nat] :
            ( ( ord_less_nat @ X2 @ Z )
           => ( ( ( P @ X2 )
                | ( Q @ X2 ) )
              = ( ( P5 @ X2 )
                | ( Q2 @ X2 ) ) ) ) ) ) ).

% minf(2)
thf(fact_586_minf_I2_J,axiom,
    ! [P: int > $o,P5: int > $o,Q: int > $o,Q2: int > $o] :
      ( ? [Z5: int] :
        ! [X4: int] :
          ( ( ord_less_int @ X4 @ Z5 )
         => ( ( P @ X4 )
            = ( P5 @ X4 ) ) )
     => ( ? [Z5: int] :
          ! [X4: int] :
            ( ( ord_less_int @ X4 @ Z5 )
           => ( ( Q @ X4 )
              = ( Q2 @ X4 ) ) )
       => ? [Z: int] :
          ! [X2: int] :
            ( ( ord_less_int @ X2 @ Z )
           => ( ( ( P @ X2 )
                | ( Q @ X2 ) )
              = ( ( P5 @ X2 )
                | ( Q2 @ X2 ) ) ) ) ) ) ).

% minf(2)
thf(fact_587_minf_I1_J,axiom,
    ! [P: nat > $o,P5: nat > $o,Q: nat > $o,Q2: nat > $o] :
      ( ? [Z5: nat] :
        ! [X4: nat] :
          ( ( ord_less_nat @ X4 @ Z5 )
         => ( ( P @ X4 )
            = ( P5 @ X4 ) ) )
     => ( ? [Z5: nat] :
          ! [X4: nat] :
            ( ( ord_less_nat @ X4 @ Z5 )
           => ( ( Q @ X4 )
              = ( Q2 @ X4 ) ) )
       => ? [Z: nat] :
          ! [X2: nat] :
            ( ( ord_less_nat @ X2 @ Z )
           => ( ( ( P @ X2 )
                & ( Q @ X2 ) )
              = ( ( P5 @ X2 )
                & ( Q2 @ X2 ) ) ) ) ) ) ).

% minf(1)
thf(fact_588_minf_I1_J,axiom,
    ! [P: int > $o,P5: int > $o,Q: int > $o,Q2: int > $o] :
      ( ? [Z5: int] :
        ! [X4: int] :
          ( ( ord_less_int @ X4 @ Z5 )
         => ( ( P @ X4 )
            = ( P5 @ X4 ) ) )
     => ( ? [Z5: int] :
          ! [X4: int] :
            ( ( ord_less_int @ X4 @ Z5 )
           => ( ( Q @ X4 )
              = ( Q2 @ X4 ) ) )
       => ? [Z: int] :
          ! [X2: int] :
            ( ( ord_less_int @ X2 @ Z )
           => ( ( ( P @ X2 )
                & ( Q @ X2 ) )
              = ( ( P5 @ X2 )
                & ( Q2 @ X2 ) ) ) ) ) ) ).

% minf(1)
thf(fact_589_pinf_I7_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X2: nat] :
      ( ( ord_less_nat @ Z @ X2 )
     => ( ord_less_nat @ T @ X2 ) ) ).

% pinf(7)
thf(fact_590_pinf_I7_J,axiom,
    ! [T: int] :
    ? [Z: int] :
    ! [X2: int] :
      ( ( ord_less_int @ Z @ X2 )
     => ( ord_less_int @ T @ X2 ) ) ).

% pinf(7)
thf(fact_591_pinf_I5_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X2: nat] :
      ( ( ord_less_nat @ Z @ X2 )
     => ~ ( ord_less_nat @ X2 @ T ) ) ).

% pinf(5)
thf(fact_592_pinf_I5_J,axiom,
    ! [T: int] :
    ? [Z: int] :
    ! [X2: int] :
      ( ( ord_less_int @ Z @ X2 )
     => ~ ( ord_less_int @ X2 @ T ) ) ).

% pinf(5)
thf(fact_593_pinf_I4_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X2: nat] :
      ( ( ord_less_nat @ Z @ X2 )
     => ( X2 != T ) ) ).

% pinf(4)
thf(fact_594_pinf_I4_J,axiom,
    ! [T: int] :
    ? [Z: int] :
    ! [X2: int] :
      ( ( ord_less_int @ Z @ X2 )
     => ( X2 != T ) ) ).

% pinf(4)
thf(fact_595_pinf_I3_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X2: nat] :
      ( ( ord_less_nat @ Z @ X2 )
     => ( X2 != T ) ) ).

% pinf(3)
thf(fact_596_pinf_I3_J,axiom,
    ! [T: int] :
    ? [Z: int] :
    ! [X2: int] :
      ( ( ord_less_int @ Z @ X2 )
     => ( X2 != T ) ) ).

% pinf(3)
thf(fact_597_pinf_I2_J,axiom,
    ! [P: nat > $o,P5: nat > $o,Q: nat > $o,Q2: nat > $o] :
      ( ? [Z5: nat] :
        ! [X4: nat] :
          ( ( ord_less_nat @ Z5 @ X4 )
         => ( ( P @ X4 )
            = ( P5 @ X4 ) ) )
     => ( ? [Z5: nat] :
          ! [X4: nat] :
            ( ( ord_less_nat @ Z5 @ X4 )
           => ( ( Q @ X4 )
              = ( Q2 @ X4 ) ) )
       => ? [Z: nat] :
          ! [X2: nat] :
            ( ( ord_less_nat @ Z @ X2 )
           => ( ( ( P @ X2 )
                | ( Q @ X2 ) )
              = ( ( P5 @ X2 )
                | ( Q2 @ X2 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_598_pinf_I2_J,axiom,
    ! [P: int > $o,P5: int > $o,Q: int > $o,Q2: int > $o] :
      ( ? [Z5: int] :
        ! [X4: int] :
          ( ( ord_less_int @ Z5 @ X4 )
         => ( ( P @ X4 )
            = ( P5 @ X4 ) ) )
     => ( ? [Z5: int] :
          ! [X4: int] :
            ( ( ord_less_int @ Z5 @ X4 )
           => ( ( Q @ X4 )
              = ( Q2 @ X4 ) ) )
       => ? [Z: int] :
          ! [X2: int] :
            ( ( ord_less_int @ Z @ X2 )
           => ( ( ( P @ X2 )
                | ( Q @ X2 ) )
              = ( ( P5 @ X2 )
                | ( Q2 @ X2 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_599_pinf_I1_J,axiom,
    ! [P: nat > $o,P5: nat > $o,Q: nat > $o,Q2: nat > $o] :
      ( ? [Z5: nat] :
        ! [X4: nat] :
          ( ( ord_less_nat @ Z5 @ X4 )
         => ( ( P @ X4 )
            = ( P5 @ X4 ) ) )
     => ( ? [Z5: nat] :
          ! [X4: nat] :
            ( ( ord_less_nat @ Z5 @ X4 )
           => ( ( Q @ X4 )
              = ( Q2 @ X4 ) ) )
       => ? [Z: nat] :
          ! [X2: nat] :
            ( ( ord_less_nat @ Z @ X2 )
           => ( ( ( P @ X2 )
                & ( Q @ X2 ) )
              = ( ( P5 @ X2 )
                & ( Q2 @ X2 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_600_pinf_I1_J,axiom,
    ! [P: int > $o,P5: int > $o,Q: int > $o,Q2: int > $o] :
      ( ? [Z5: int] :
        ! [X4: int] :
          ( ( ord_less_int @ Z5 @ X4 )
         => ( ( P @ X4 )
            = ( P5 @ X4 ) ) )
     => ( ? [Z5: int] :
          ! [X4: int] :
            ( ( ord_less_int @ Z5 @ X4 )
           => ( ( Q @ X4 )
              = ( Q2 @ X4 ) ) )
       => ? [Z: int] :
          ! [X2: int] :
            ( ( ord_less_int @ Z @ X2 )
           => ( ( ( P @ X2 )
                & ( Q @ X2 ) )
              = ( ( P5 @ X2 )
                & ( Q2 @ X2 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_601_neg__int__cases,axiom,
    ! [K: int] :
      ( ( ord_less_int @ K @ zero_zero_int )
     => ~ ! [N2: nat] :
            ( ( K
              = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) )
           => ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% neg_int_cases
thf(fact_602_nth__equal__first__eq,axiom,
    ! [X: produc357393685978478089rm_a_b,Xs: list_P8875379029341186191rm_a_b,N: nat] :
      ( ~ ( member5869715511025134514rm_a_b @ X @ ( set_Pr2370880052973118494rm_a_b @ Xs ) )
     => ( ( ord_less_eq_nat @ N @ ( size_s8600120110137556219rm_a_b @ Xs ) )
       => ( ( ( nth_Pr1098598711982373392rm_a_b @ ( cons_P3536305108106557631rm_a_b @ X @ Xs ) @ N )
            = X )
          = ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
thf(fact_603_nth__equal__first__eq,axiom,
    ! [X: fOR_ftrs,Xs: list_FOR_ftrs,N: nat] :
      ( ~ ( member_FOR_ftrs @ X @ ( set_FOR_ftrs2 @ Xs ) )
     => ( ( ord_less_eq_nat @ N @ ( size_s7038034906319482769R_ftrs @ Xs ) )
       => ( ( ( nth_FOR_ftrs @ ( cons_FOR_ftrs @ X @ Xs ) @ N )
            = X )
          = ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
thf(fact_604_nth__equal__first__eq,axiom,
    ! [X: fset_P3577670608877728751rm_a_b,Xs: list_f7931864845379430901rm_a_b,N: nat] :
      ( ~ ( member2349228652077105432rm_a_b @ X @ ( set_fs2829528993146950532rm_a_b @ Xs ) )
     => ( ( ord_less_eq_nat @ N @ ( size_s1967422219310127201rm_a_b @ Xs ) )
       => ( ( ( nth_fs3949069584078227574rm_a_b @ ( cons_f831858257744602021rm_a_b @ X @ Xs ) @ N )
            = X )
          = ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
thf(fact_605_nth__rotate1,axiom,
    ! [N: nat,Xs: list_s5629610930313288943rm_a_b] :
      ( ( ord_less_nat @ N @ ( size_s3663623692424790363rm_a_b @ Xs ) )
     => ( ( nth_se1992752789547675248rm_a_b @ ( rotate753530590164781500rm_a_b @ Xs ) @ N )
        = ( nth_se1992752789547675248rm_a_b @ Xs @ ( modulo_modulo_nat @ ( suc @ N ) @ ( size_s3663623692424790363rm_a_b @ Xs ) ) ) ) ) ).

% nth_rotate1
thf(fact_606_nth__rotate1,axiom,
    ! [N: nat,Xs: list_f7931864845379430901rm_a_b] :
      ( ( ord_less_nat @ N @ ( size_s1967422219310127201rm_a_b @ Xs ) )
     => ( ( nth_fs3949069584078227574rm_a_b @ ( rotate8515649208237718978rm_a_b @ Xs ) @ N )
        = ( nth_fs3949069584078227574rm_a_b @ Xs @ ( modulo_modulo_nat @ ( suc @ N ) @ ( size_s1967422219310127201rm_a_b @ Xs ) ) ) ) ) ).

% nth_rotate1
thf(fact_607_remdups__adj__length__ge1,axiom,
    ! [Xs: list_s5629610930313288943rm_a_b] :
      ( ( Xs != nil_se7334864872594423151rm_a_b )
     => ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( size_s3663623692424790363rm_a_b @ ( remdup7997809740792892069rm_a_b @ Xs ) ) ) ) ).

% remdups_adj_length_ge1
thf(fact_608_remdups__adj__length__ge1,axiom,
    ! [Xs: list_f7931864845379430901rm_a_b] :
      ( ( Xs != nil_fs4020363321024022517rm_a_b )
     => ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( size_s1967422219310127201rm_a_b @ ( remdup2633883733652511275rm_a_b @ Xs ) ) ) ) ).

% remdups_adj_length_ge1
thf(fact_609_subset__code_I3_J,axiom,
    ~ ( ord_le9025091529776396905rm_a_b @ ( coset_6808587370936601180rm_a_b @ nil_se7334864872594423151rm_a_b ) @ ( set_se5504503741092831358rm_a_b @ nil_se7334864872594423151rm_a_b ) ) ).

% subset_code(3)
thf(fact_610_subset__code_I3_J,axiom,
    ~ ( ord_le8400580047904139887rm_a_b @ ( coset_2940879206044380770rm_a_b @ nil_fs4020363321024022517rm_a_b ) @ ( set_fs2829528993146950532rm_a_b @ nil_fs4020363321024022517rm_a_b ) ) ).

% subset_code(3)
thf(fact_611_subset__code_I3_J,axiom,
    ~ ( ord_le1454117246873690627R_ftrs @ ( coset_FOR_ftrs @ nil_FOR_ftrs ) @ ( set_FOR_ftrs2 @ nil_FOR_ftrs ) ) ).

% subset_code(3)
thf(fact_612_add_Oinverse__inverse,axiom,
    ! [A: int] :
      ( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
      = A ) ).

% add.inverse_inverse
thf(fact_613_neg__equal__iff__equal,axiom,
    ! [A: int,B: int] :
      ( ( ( uminus_uminus_int @ A )
        = ( uminus_uminus_int @ B ) )
      = ( A = B ) ) ).

% neg_equal_iff_equal
thf(fact_614_neg__le__iff__le,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
      = ( ord_less_eq_int @ A @ B ) ) ).

% neg_le_iff_le
thf(fact_615_neg__equal__zero,axiom,
    ! [A: int] :
      ( ( ( uminus_uminus_int @ A )
        = A )
      = ( A = zero_zero_int ) ) ).

% neg_equal_zero
thf(fact_616_equal__neg__zero,axiom,
    ! [A: int] :
      ( ( A
        = ( uminus_uminus_int @ A ) )
      = ( A = zero_zero_int ) ) ).

% equal_neg_zero
thf(fact_617_neg__equal__0__iff__equal,axiom,
    ! [A: int] :
      ( ( ( uminus_uminus_int @ A )
        = zero_zero_int )
      = ( A = zero_zero_int ) ) ).

% neg_equal_0_iff_equal
thf(fact_618_neg__0__equal__iff__equal,axiom,
    ! [A: int] :
      ( ( zero_zero_int
        = ( uminus_uminus_int @ A ) )
      = ( zero_zero_int = A ) ) ).

% neg_0_equal_iff_equal
thf(fact_619_add_Oinverse__neutral,axiom,
    ( ( uminus_uminus_int @ zero_zero_int )
    = zero_zero_int ) ).

% add.inverse_neutral
thf(fact_620_neg__less__iff__less,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
      = ( ord_less_int @ A @ B ) ) ).

% neg_less_iff_less
thf(fact_621_remdups__adj__Nil__iff,axiom,
    ! [Xs: list_s5629610930313288943rm_a_b] :
      ( ( ( remdup7997809740792892069rm_a_b @ Xs )
        = nil_se7334864872594423151rm_a_b )
      = ( Xs = nil_se7334864872594423151rm_a_b ) ) ).

% remdups_adj_Nil_iff
thf(fact_622_remdups__adj__Nil__iff,axiom,
    ! [Xs: list_f7931864845379430901rm_a_b] :
      ( ( ( remdup2633883733652511275rm_a_b @ Xs )
        = nil_fs4020363321024022517rm_a_b )
      = ( Xs = nil_fs4020363321024022517rm_a_b ) ) ).

% remdups_adj_Nil_iff
thf(fact_623_remdups__adj__set,axiom,
    ! [Xs: list_FOR_ftrs] :
      ( ( set_FOR_ftrs2 @ ( remdups_adj_FOR_ftrs @ Xs ) )
      = ( set_FOR_ftrs2 @ Xs ) ) ).

% remdups_adj_set
thf(fact_624_neg__less__eq__nonneg,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ A )
      = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).

% neg_less_eq_nonneg
thf(fact_625_less__eq__neg__nonpos,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ A ) )
      = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).

% less_eq_neg_nonpos
thf(fact_626_neg__le__0__iff__le,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
      = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).

% neg_le_0_iff_le
thf(fact_627_neg__0__le__iff__le,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
      = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).

% neg_0_le_iff_le
thf(fact_628_less__neg__neg,axiom,
    ! [A: int] :
      ( ( ord_less_int @ A @ ( uminus_uminus_int @ A ) )
      = ( ord_less_int @ A @ zero_zero_int ) ) ).

% less_neg_neg
thf(fact_629_neg__less__pos,axiom,
    ! [A: int] :
      ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ A )
      = ( ord_less_int @ zero_zero_int @ A ) ) ).

% neg_less_pos
thf(fact_630_neg__0__less__iff__less,axiom,
    ! [A: int] :
      ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
      = ( ord_less_int @ A @ zero_zero_int ) ) ).

% neg_0_less_iff_less
thf(fact_631_neg__less__0__iff__less,axiom,
    ! [A: int] :
      ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
      = ( ord_less_int @ zero_zero_int @ A ) ) ).

% neg_less_0_iff_less
thf(fact_632_negative__eq__positive,axiom,
    ! [N: nat,M: nat] :
      ( ( ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) )
        = ( semiri1314217659103216013at_int @ M ) )
      = ( ( N = zero_zero_nat )
        & ( M = zero_zero_nat ) ) ) ).

% negative_eq_positive
thf(fact_633_Cons__less__Cons,axiom,
    ! [A: nat,X: list_nat,B: nat,Y: list_nat] :
      ( ( ord_less_list_nat @ ( cons_nat @ A @ X ) @ ( cons_nat @ B @ Y ) )
      = ( ( ord_less_nat @ A @ B )
        | ( ( A = B )
          & ( ord_less_list_nat @ X @ Y ) ) ) ) ).

% Cons_less_Cons
thf(fact_634_Cons__less__Cons,axiom,
    ! [A: int,X: list_int,B: int,Y: list_int] :
      ( ( ord_less_list_int @ ( cons_int @ A @ X ) @ ( cons_int @ B @ Y ) )
      = ( ( ord_less_int @ A @ B )
        | ( ( A = B )
          & ( ord_less_list_int @ X @ Y ) ) ) ) ).

% Cons_less_Cons
thf(fact_635_negative__zle,axiom,
    ! [N: nat,M: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).

% negative_zle
thf(fact_636_Cons__le__Cons,axiom,
    ! [A: nat,X: list_nat,B: nat,Y: list_nat] :
      ( ( ord_less_eq_list_nat @ ( cons_nat @ A @ X ) @ ( cons_nat @ B @ Y ) )
      = ( ( ord_less_nat @ A @ B )
        | ( ( A = B )
          & ( ord_less_eq_list_nat @ X @ Y ) ) ) ) ).

% Cons_le_Cons
thf(fact_637_Cons__le__Cons,axiom,
    ! [A: int,X: list_int,B: int,Y: list_int] :
      ( ( ord_less_eq_list_int @ ( cons_int @ A @ X ) @ ( cons_int @ B @ Y ) )
      = ( ( ord_less_int @ A @ B )
        | ( ( A = B )
          & ( ord_less_eq_list_int @ X @ Y ) ) ) ) ).

% Cons_le_Cons
thf(fact_638_negative__zless,axiom,
    ! [N: nat,M: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).

% negative_zless
thf(fact_639_le__imp__neg__le,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).

% le_imp_neg_le
thf(fact_640_minus__le__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
      = ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ A ) ) ).

% minus_le_iff
thf(fact_641_le__minus__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ B ) )
      = ( ord_less_eq_int @ B @ ( uminus_uminus_int @ A ) ) ) ).

% le_minus_iff
thf(fact_642_verit__negate__coefficient_I2_J,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).

% verit_negate_coefficient(2)
thf(fact_643_less__minus__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ ( uminus_uminus_int @ B ) )
      = ( ord_less_int @ B @ ( uminus_uminus_int @ A ) ) ) ).

% less_minus_iff
thf(fact_644_minus__less__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ B )
      = ( ord_less_int @ ( uminus_uminus_int @ B ) @ A ) ) ).

% minus_less_iff
thf(fact_645_remdups__adj_Osimps_I1_J,axiom,
    ( ( remdup7997809740792892069rm_a_b @ nil_se7334864872594423151rm_a_b )
    = nil_se7334864872594423151rm_a_b ) ).

% remdups_adj.simps(1)
thf(fact_646_remdups__adj_Osimps_I1_J,axiom,
    ( ( remdup2633883733652511275rm_a_b @ nil_fs4020363321024022517rm_a_b )
    = nil_fs4020363321024022517rm_a_b ) ).

% remdups_adj.simps(1)
thf(fact_647_list_Odistinct_I1_J,axiom,
    ! [X21: set_Pr4386577575007340137rm_a_b,X222: list_s5629610930313288943rm_a_b] :
      ( nil_se7334864872594423151rm_a_b
     != ( cons_s7847606222307733791rm_a_b @ X21 @ X222 ) ) ).

% list.distinct(1)
thf(fact_648_list_Odistinct_I1_J,axiom,
    ! [X21: fset_P3577670608877728751rm_a_b,X222: list_f7931864845379430901rm_a_b] :
      ( nil_fs4020363321024022517rm_a_b
     != ( cons_f831858257744602021rm_a_b @ X21 @ X222 ) ) ).

% list.distinct(1)
thf(fact_649_list_OdiscI,axiom,
    ! [List: list_s5629610930313288943rm_a_b,X21: set_Pr4386577575007340137rm_a_b,X222: list_s5629610930313288943rm_a_b] :
      ( ( List
        = ( cons_s7847606222307733791rm_a_b @ X21 @ X222 ) )
     => ( List != nil_se7334864872594423151rm_a_b ) ) ).

% list.discI
thf(fact_650_list_OdiscI,axiom,
    ! [List: list_f7931864845379430901rm_a_b,X21: fset_P3577670608877728751rm_a_b,X222: list_f7931864845379430901rm_a_b] :
      ( ( List
        = ( cons_f831858257744602021rm_a_b @ X21 @ X222 ) )
     => ( List != nil_fs4020363321024022517rm_a_b ) ) ).

% list.discI
thf(fact_651_list_Oexhaust,axiom,
    ! [Y: list_s5629610930313288943rm_a_b] :
      ( ( Y != nil_se7334864872594423151rm_a_b )
     => ~ ! [X212: set_Pr4386577575007340137rm_a_b,X223: list_s5629610930313288943rm_a_b] :
            ( Y
           != ( cons_s7847606222307733791rm_a_b @ X212 @ X223 ) ) ) ).

% list.exhaust
thf(fact_652_list_Oexhaust,axiom,
    ! [Y: list_f7931864845379430901rm_a_b] :
      ( ( Y != nil_fs4020363321024022517rm_a_b )
     => ~ ! [X212: fset_P3577670608877728751rm_a_b,X223: list_f7931864845379430901rm_a_b] :
            ( Y
           != ( cons_f831858257744602021rm_a_b @ X212 @ X223 ) ) ) ).

% list.exhaust
thf(fact_653_List_Omin__list_Ocases,axiom,
    ! [X: list_s5629610930313288943rm_a_b] :
      ( ! [X4: set_Pr4386577575007340137rm_a_b,Xs2: list_s5629610930313288943rm_a_b] :
          ( X
         != ( cons_s7847606222307733791rm_a_b @ X4 @ Xs2 ) )
     => ( X = nil_se7334864872594423151rm_a_b ) ) ).

% List.min_list.cases
thf(fact_654_List_Omin__list_Ocases,axiom,
    ! [X: list_f7931864845379430901rm_a_b] :
      ( ! [X4: fset_P3577670608877728751rm_a_b,Xs2: list_f7931864845379430901rm_a_b] :
          ( X
         != ( cons_f831858257744602021rm_a_b @ X4 @ Xs2 ) )
     => ( X = nil_fs4020363321024022517rm_a_b ) ) ).

% List.min_list.cases
thf(fact_655_neq__Nil__conv,axiom,
    ! [Xs: list_s5629610930313288943rm_a_b] :
      ( ( Xs != nil_se7334864872594423151rm_a_b )
      = ( ? [Y2: set_Pr4386577575007340137rm_a_b,Ys3: list_s5629610930313288943rm_a_b] :
            ( Xs
            = ( cons_s7847606222307733791rm_a_b @ Y2 @ Ys3 ) ) ) ) ).

% neq_Nil_conv
thf(fact_656_neq__Nil__conv,axiom,
    ! [Xs: list_f7931864845379430901rm_a_b] :
      ( ( Xs != nil_fs4020363321024022517rm_a_b )
      = ( ? [Y2: fset_P3577670608877728751rm_a_b,Ys3: list_f7931864845379430901rm_a_b] :
            ( Xs
            = ( cons_f831858257744602021rm_a_b @ Y2 @ Ys3 ) ) ) ) ).

% neq_Nil_conv
thf(fact_657_list__induct2_H,axiom,
    ! [P: list_s5629610930313288943rm_a_b > list_s5629610930313288943rm_a_b > $o,Xs: list_s5629610930313288943rm_a_b,Ys2: list_s5629610930313288943rm_a_b] :
      ( ( P @ nil_se7334864872594423151rm_a_b @ nil_se7334864872594423151rm_a_b )
     => ( ! [X4: set_Pr4386577575007340137rm_a_b,Xs2: list_s5629610930313288943rm_a_b] : ( P @ ( cons_s7847606222307733791rm_a_b @ X4 @ Xs2 ) @ nil_se7334864872594423151rm_a_b )
       => ( ! [Y4: set_Pr4386577575007340137rm_a_b,Ys4: list_s5629610930313288943rm_a_b] : ( P @ nil_se7334864872594423151rm_a_b @ ( cons_s7847606222307733791rm_a_b @ Y4 @ Ys4 ) )
         => ( ! [X4: set_Pr4386577575007340137rm_a_b,Xs2: list_s5629610930313288943rm_a_b,Y4: set_Pr4386577575007340137rm_a_b,Ys4: list_s5629610930313288943rm_a_b] :
                ( ( P @ Xs2 @ Ys4 )
               => ( P @ ( cons_s7847606222307733791rm_a_b @ X4 @ Xs2 ) @ ( cons_s7847606222307733791rm_a_b @ Y4 @ Ys4 ) ) )
           => ( P @ Xs @ Ys2 ) ) ) ) ) ).

% list_induct2'
thf(fact_658_list__induct2_H,axiom,
    ! [P: list_s5629610930313288943rm_a_b > list_f7931864845379430901rm_a_b > $o,Xs: list_s5629610930313288943rm_a_b,Ys2: list_f7931864845379430901rm_a_b] :
      ( ( P @ nil_se7334864872594423151rm_a_b @ nil_fs4020363321024022517rm_a_b )
     => ( ! [X4: set_Pr4386577575007340137rm_a_b,Xs2: list_s5629610930313288943rm_a_b] : ( P @ ( cons_s7847606222307733791rm_a_b @ X4 @ Xs2 ) @ nil_fs4020363321024022517rm_a_b )
       => ( ! [Y4: fset_P3577670608877728751rm_a_b,Ys4: list_f7931864845379430901rm_a_b] : ( P @ nil_se7334864872594423151rm_a_b @ ( cons_f831858257744602021rm_a_b @ Y4 @ Ys4 ) )
         => ( ! [X4: set_Pr4386577575007340137rm_a_b,Xs2: list_s5629610930313288943rm_a_b,Y4: fset_P3577670608877728751rm_a_b,Ys4: list_f7931864845379430901rm_a_b] :
                ( ( P @ Xs2 @ Ys4 )
               => ( P @ ( cons_s7847606222307733791rm_a_b @ X4 @ Xs2 ) @ ( cons_f831858257744602021rm_a_b @ Y4 @ Ys4 ) ) )
           => ( P @ Xs @ Ys2 ) ) ) ) ) ).

% list_induct2'
thf(fact_659_list__induct2_H,axiom,
    ! [P: list_f7931864845379430901rm_a_b > list_s5629610930313288943rm_a_b > $o,Xs: list_f7931864845379430901rm_a_b,Ys2: list_s5629610930313288943rm_a_b] :
      ( ( P @ nil_fs4020363321024022517rm_a_b @ nil_se7334864872594423151rm_a_b )
     => ( ! [X4: fset_P3577670608877728751rm_a_b,Xs2: list_f7931864845379430901rm_a_b] : ( P @ ( cons_f831858257744602021rm_a_b @ X4 @ Xs2 ) @ nil_se7334864872594423151rm_a_b )
       => ( ! [Y4: set_Pr4386577575007340137rm_a_b,Ys4: list_s5629610930313288943rm_a_b] : ( P @ nil_fs4020363321024022517rm_a_b @ ( cons_s7847606222307733791rm_a_b @ Y4 @ Ys4 ) )
         => ( ! [X4: fset_P3577670608877728751rm_a_b,Xs2: list_f7931864845379430901rm_a_b,Y4: set_Pr4386577575007340137rm_a_b,Ys4: list_s5629610930313288943rm_a_b] :
                ( ( P @ Xs2 @ Ys4 )
               => ( P @ ( cons_f831858257744602021rm_a_b @ X4 @ Xs2 ) @ ( cons_s7847606222307733791rm_a_b @ Y4 @ Ys4 ) ) )
           => ( P @ Xs @ Ys2 ) ) ) ) ) ).

% list_induct2'
thf(fact_660_list__induct2_H,axiom,
    ! [P: list_f7931864845379430901rm_a_b > list_f7931864845379430901rm_a_b > $o,Xs: list_f7931864845379430901rm_a_b,Ys2: list_f7931864845379430901rm_a_b] :
      ( ( P @ nil_fs4020363321024022517rm_a_b @ nil_fs4020363321024022517rm_a_b )
     => ( ! [X4: fset_P3577670608877728751rm_a_b,Xs2: list_f7931864845379430901rm_a_b] : ( P @ ( cons_f831858257744602021rm_a_b @ X4 @ Xs2 ) @ nil_fs4020363321024022517rm_a_b )
       => ( ! [Y4: fset_P3577670608877728751rm_a_b,Ys4: list_f7931864845379430901rm_a_b] : ( P @ nil_fs4020363321024022517rm_a_b @ ( cons_f831858257744602021rm_a_b @ Y4 @ Ys4 ) )
         => ( ! [X4: fset_P3577670608877728751rm_a_b,Xs2: list_f7931864845379430901rm_a_b,Y4: fset_P3577670608877728751rm_a_b,Ys4: list_f7931864845379430901rm_a_b] :
                ( ( P @ Xs2 @ Ys4 )
               => ( P @ ( cons_f831858257744602021rm_a_b @ X4 @ Xs2 ) @ ( cons_f831858257744602021rm_a_b @ Y4 @ Ys4 ) ) )
           => ( P @ Xs @ Ys2 ) ) ) ) ) ).

% list_induct2'
thf(fact_661_list__nonempty__induct,axiom,
    ! [Xs: list_s5629610930313288943rm_a_b,P: list_s5629610930313288943rm_a_b > $o] :
      ( ( Xs != nil_se7334864872594423151rm_a_b )
     => ( ! [X4: set_Pr4386577575007340137rm_a_b] : ( P @ ( cons_s7847606222307733791rm_a_b @ X4 @ nil_se7334864872594423151rm_a_b ) )
       => ( ! [X4: set_Pr4386577575007340137rm_a_b,Xs2: list_s5629610930313288943rm_a_b] :
              ( ( Xs2 != nil_se7334864872594423151rm_a_b )
             => ( ( P @ Xs2 )
               => ( P @ ( cons_s7847606222307733791rm_a_b @ X4 @ Xs2 ) ) ) )
         => ( P @ Xs ) ) ) ) ).

% list_nonempty_induct
thf(fact_662_list__nonempty__induct,axiom,
    ! [Xs: list_f7931864845379430901rm_a_b,P: list_f7931864845379430901rm_a_b > $o] :
      ( ( Xs != nil_fs4020363321024022517rm_a_b )
     => ( ! [X4: fset_P3577670608877728751rm_a_b] : ( P @ ( cons_f831858257744602021rm_a_b @ X4 @ nil_fs4020363321024022517rm_a_b ) )
       => ( ! [X4: fset_P3577670608877728751rm_a_b,Xs2: list_f7931864845379430901rm_a_b] :
              ( ( Xs2 != nil_fs4020363321024022517rm_a_b )
             => ( ( P @ Xs2 )
               => ( P @ ( cons_f831858257744602021rm_a_b @ X4 @ Xs2 ) ) ) )
         => ( P @ Xs ) ) ) ) ).

% list_nonempty_induct
thf(fact_663_list__3__cases,axiom,
    ! [Xs: list_s5629610930313288943rm_a_b] :
      ( ( Xs != nil_se7334864872594423151rm_a_b )
     => ( ! [X4: set_Pr4386577575007340137rm_a_b] :
            ( Xs
           != ( cons_s7847606222307733791rm_a_b @ X4 @ nil_se7334864872594423151rm_a_b ) )
       => ~ ! [X4: set_Pr4386577575007340137rm_a_b,Y4: set_Pr4386577575007340137rm_a_b,Ys4: list_s5629610930313288943rm_a_b] :
              ( Xs
             != ( cons_s7847606222307733791rm_a_b @ X4 @ ( cons_s7847606222307733791rm_a_b @ Y4 @ Ys4 ) ) ) ) ) ).

% list_3_cases
thf(fact_664_list__3__cases,axiom,
    ! [Xs: list_f7931864845379430901rm_a_b] :
      ( ( Xs != nil_fs4020363321024022517rm_a_b )
     => ( ! [X4: fset_P3577670608877728751rm_a_b] :
            ( Xs
           != ( cons_f831858257744602021rm_a_b @ X4 @ nil_fs4020363321024022517rm_a_b ) )
       => ~ ! [X4: fset_P3577670608877728751rm_a_b,Y4: fset_P3577670608877728751rm_a_b,Ys4: list_f7931864845379430901rm_a_b] :
              ( Xs
             != ( cons_f831858257744602021rm_a_b @ X4 @ ( cons_f831858257744602021rm_a_b @ Y4 @ Ys4 ) ) ) ) ) ).

% list_3_cases
thf(fact_665_list__4__cases,axiom,
    ! [Xs: list_s5629610930313288943rm_a_b] :
      ( ( Xs != nil_se7334864872594423151rm_a_b )
     => ( ! [X4: set_Pr4386577575007340137rm_a_b] :
            ( Xs
           != ( cons_s7847606222307733791rm_a_b @ X4 @ nil_se7334864872594423151rm_a_b ) )
       => ( ! [X4: set_Pr4386577575007340137rm_a_b,Y4: set_Pr4386577575007340137rm_a_b] :
              ( Xs
             != ( cons_s7847606222307733791rm_a_b @ X4 @ ( cons_s7847606222307733791rm_a_b @ Y4 @ nil_se7334864872594423151rm_a_b ) ) )
         => ~ ! [X4: set_Pr4386577575007340137rm_a_b,Y4: set_Pr4386577575007340137rm_a_b,Z: set_Pr4386577575007340137rm_a_b,Zs: list_s5629610930313288943rm_a_b] :
                ( Xs
               != ( cons_s7847606222307733791rm_a_b @ X4 @ ( cons_s7847606222307733791rm_a_b @ Y4 @ ( cons_s7847606222307733791rm_a_b @ Z @ Zs ) ) ) ) ) ) ) ).

% list_4_cases
thf(fact_666_list__4__cases,axiom,
    ! [Xs: list_f7931864845379430901rm_a_b] :
      ( ( Xs != nil_fs4020363321024022517rm_a_b )
     => ( ! [X4: fset_P3577670608877728751rm_a_b] :
            ( Xs
           != ( cons_f831858257744602021rm_a_b @ X4 @ nil_fs4020363321024022517rm_a_b ) )
       => ( ! [X4: fset_P3577670608877728751rm_a_b,Y4: fset_P3577670608877728751rm_a_b] :
              ( Xs
             != ( cons_f831858257744602021rm_a_b @ X4 @ ( cons_f831858257744602021rm_a_b @ Y4 @ nil_fs4020363321024022517rm_a_b ) ) )
         => ~ ! [X4: fset_P3577670608877728751rm_a_b,Y4: fset_P3577670608877728751rm_a_b,Z: fset_P3577670608877728751rm_a_b,Zs: list_f7931864845379430901rm_a_b] :
                ( Xs
               != ( cons_f831858257744602021rm_a_b @ X4 @ ( cons_f831858257744602021rm_a_b @ Y4 @ ( cons_f831858257744602021rm_a_b @ Z @ Zs ) ) ) ) ) ) ) ).

% list_4_cases
thf(fact_667_equation__minus__iff,axiom,
    ! [A: int,B: int] :
      ( ( A
        = ( uminus_uminus_int @ B ) )
      = ( B
        = ( uminus_uminus_int @ A ) ) ) ).

% equation_minus_iff
thf(fact_668_minus__equation__iff,axiom,
    ! [A: int,B: int] :
      ( ( ( uminus_uminus_int @ A )
        = B )
      = ( ( uminus_uminus_int @ B )
        = A ) ) ).

% minus_equation_iff
thf(fact_669_remdups__adj_Osimps_I2_J,axiom,
    ! [X: set_Pr4386577575007340137rm_a_b] :
      ( ( remdup7997809740792892069rm_a_b @ ( cons_s7847606222307733791rm_a_b @ X @ nil_se7334864872594423151rm_a_b ) )
      = ( cons_s7847606222307733791rm_a_b @ X @ nil_se7334864872594423151rm_a_b ) ) ).

% remdups_adj.simps(2)
thf(fact_670_remdups__adj_Osimps_I2_J,axiom,
    ! [X: fset_P3577670608877728751rm_a_b] :
      ( ( remdup2633883733652511275rm_a_b @ ( cons_f831858257744602021rm_a_b @ X @ nil_fs4020363321024022517rm_a_b ) )
      = ( cons_f831858257744602021rm_a_b @ X @ nil_fs4020363321024022517rm_a_b ) ) ).

% remdups_adj.simps(2)
thf(fact_671_remdups__adj_Oelims,axiom,
    ! [X: list_s5629610930313288943rm_a_b,Y: list_s5629610930313288943rm_a_b] :
      ( ( ( remdup7997809740792892069rm_a_b @ X )
        = Y )
     => ( ( ( X = nil_se7334864872594423151rm_a_b )
         => ( Y != nil_se7334864872594423151rm_a_b ) )
       => ( ! [X4: set_Pr4386577575007340137rm_a_b] :
              ( ( X
                = ( cons_s7847606222307733791rm_a_b @ X4 @ nil_se7334864872594423151rm_a_b ) )
             => ( Y
               != ( cons_s7847606222307733791rm_a_b @ X4 @ nil_se7334864872594423151rm_a_b ) ) )
         => ~ ! [X4: set_Pr4386577575007340137rm_a_b,Y4: set_Pr4386577575007340137rm_a_b,Xs2: list_s5629610930313288943rm_a_b] :
                ( ( X
                  = ( cons_s7847606222307733791rm_a_b @ X4 @ ( cons_s7847606222307733791rm_a_b @ Y4 @ Xs2 ) ) )
               => ~ ( ( ( X4 = Y4 )
                     => ( Y
                        = ( remdup7997809740792892069rm_a_b @ ( cons_s7847606222307733791rm_a_b @ X4 @ Xs2 ) ) ) )
                    & ( ( X4 != Y4 )
                     => ( Y
                        = ( cons_s7847606222307733791rm_a_b @ X4 @ ( remdup7997809740792892069rm_a_b @ ( cons_s7847606222307733791rm_a_b @ Y4 @ Xs2 ) ) ) ) ) ) ) ) ) ) ).

% remdups_adj.elims
thf(fact_672_remdups__adj_Oelims,axiom,
    ! [X: list_f7931864845379430901rm_a_b,Y: list_f7931864845379430901rm_a_b] :
      ( ( ( remdup2633883733652511275rm_a_b @ X )
        = Y )
     => ( ( ( X = nil_fs4020363321024022517rm_a_b )
         => ( Y != nil_fs4020363321024022517rm_a_b ) )
       => ( ! [X4: fset_P3577670608877728751rm_a_b] :
              ( ( X
                = ( cons_f831858257744602021rm_a_b @ X4 @ nil_fs4020363321024022517rm_a_b ) )
             => ( Y
               != ( cons_f831858257744602021rm_a_b @ X4 @ nil_fs4020363321024022517rm_a_b ) ) )
         => ~ ! [X4: fset_P3577670608877728751rm_a_b,Y4: fset_P3577670608877728751rm_a_b,Xs2: list_f7931864845379430901rm_a_b] :
                ( ( X
                  = ( cons_f831858257744602021rm_a_b @ X4 @ ( cons_f831858257744602021rm_a_b @ Y4 @ Xs2 ) ) )
               => ~ ( ( ( X4 = Y4 )
                     => ( Y
                        = ( remdup2633883733652511275rm_a_b @ ( cons_f831858257744602021rm_a_b @ X4 @ Xs2 ) ) ) )
                    & ( ( X4 != Y4 )
                     => ( Y
                        = ( cons_f831858257744602021rm_a_b @ X4 @ ( remdup2633883733652511275rm_a_b @ ( cons_f831858257744602021rm_a_b @ Y4 @ Xs2 ) ) ) ) ) ) ) ) ) ) ).

% remdups_adj.elims
thf(fact_673_transpose_Ocases,axiom,
    ! [X: list_l6192929068897674997rm_a_b] :
      ( ( X != nil_li1283984768393418997rm_a_b )
     => ( ! [Xss: list_l6192929068897674997rm_a_b] :
            ( X
           != ( cons_l92702159317333157rm_a_b @ nil_se7334864872594423151rm_a_b @ Xss ) )
       => ~ ! [X4: set_Pr4386577575007340137rm_a_b,Xs2: list_s5629610930313288943rm_a_b,Xss: list_l6192929068897674997rm_a_b] :
              ( X
             != ( cons_l92702159317333157rm_a_b @ ( cons_s7847606222307733791rm_a_b @ X4 @ Xs2 ) @ Xss ) ) ) ) ).

% transpose.cases
thf(fact_674_transpose_Ocases,axiom,
    ! [X: list_l3960025731926578043rm_a_b] :
      ( ( X != nil_li2074289066168648443rm_a_b )
     => ( ! [Xss: list_l3960025731926578043rm_a_b] :
            ( X
           != ( cons_l2079606646850324139rm_a_b @ nil_fs4020363321024022517rm_a_b @ Xss ) )
       => ~ ! [X4: fset_P3577670608877728751rm_a_b,Xs2: list_f7931864845379430901rm_a_b,Xss: list_l3960025731926578043rm_a_b] :
              ( X
             != ( cons_l2079606646850324139rm_a_b @ ( cons_f831858257744602021rm_a_b @ X4 @ Xs2 ) @ Xss ) ) ) ) ).

% transpose.cases
thf(fact_675_in__set__simps_I1_J,axiom,
    ! [X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,Z4: produc357393685978478089rm_a_b,Ys2: list_P8875379029341186191rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ X @ ( set_Pr2370880052973118494rm_a_b @ ( cons_P3536305108106557631rm_a_b @ Y @ ( cons_P3536305108106557631rm_a_b @ Z4 @ Ys2 ) ) ) )
      = ( ( X = Y )
        | ( member5869715511025134514rm_a_b @ X @ ( set_Pr2370880052973118494rm_a_b @ ( cons_P3536305108106557631rm_a_b @ Z4 @ Ys2 ) ) ) ) ) ).

% in_set_simps(1)
thf(fact_676_in__set__simps_I1_J,axiom,
    ! [X: fOR_ftrs,Y: fOR_ftrs,Z4: fOR_ftrs,Ys2: list_FOR_ftrs] :
      ( ( member_FOR_ftrs @ X @ ( set_FOR_ftrs2 @ ( cons_FOR_ftrs @ Y @ ( cons_FOR_ftrs @ Z4 @ Ys2 ) ) ) )
      = ( ( X = Y )
        | ( member_FOR_ftrs @ X @ ( set_FOR_ftrs2 @ ( cons_FOR_ftrs @ Z4 @ Ys2 ) ) ) ) ) ).

% in_set_simps(1)
thf(fact_677_set__ConsD,axiom,
    ! [Y: produc357393685978478089rm_a_b,X: produc357393685978478089rm_a_b,Xs: list_P8875379029341186191rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ Y @ ( set_Pr2370880052973118494rm_a_b @ ( cons_P3536305108106557631rm_a_b @ X @ Xs ) ) )
     => ( ( Y = X )
        | ( member5869715511025134514rm_a_b @ Y @ ( set_Pr2370880052973118494rm_a_b @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_678_set__ConsD,axiom,
    ! [Y: fOR_ftrs,X: fOR_ftrs,Xs: list_FOR_ftrs] :
      ( ( member_FOR_ftrs @ Y @ ( set_FOR_ftrs2 @ ( cons_FOR_ftrs @ X @ Xs ) ) )
     => ( ( Y = X )
        | ( member_FOR_ftrs @ Y @ ( set_FOR_ftrs2 @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_679_list_Oset__cases,axiom,
    ! [E: produc357393685978478089rm_a_b,A: list_P8875379029341186191rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ E @ ( set_Pr2370880052973118494rm_a_b @ A ) )
     => ( ! [Z22: list_P8875379029341186191rm_a_b] :
            ( A
           != ( cons_P3536305108106557631rm_a_b @ E @ Z22 ) )
       => ~ ! [Z1: produc357393685978478089rm_a_b,Z22: list_P8875379029341186191rm_a_b] :
              ( ( A
                = ( cons_P3536305108106557631rm_a_b @ Z1 @ Z22 ) )
             => ~ ( member5869715511025134514rm_a_b @ E @ ( set_Pr2370880052973118494rm_a_b @ Z22 ) ) ) ) ) ).

% list.set_cases
thf(fact_680_list_Oset__cases,axiom,
    ! [E: fOR_ftrs,A: list_FOR_ftrs] :
      ( ( member_FOR_ftrs @ E @ ( set_FOR_ftrs2 @ A ) )
     => ( ! [Z22: list_FOR_ftrs] :
            ( A
           != ( cons_FOR_ftrs @ E @ Z22 ) )
       => ~ ! [Z1: fOR_ftrs,Z22: list_FOR_ftrs] :
              ( ( A
                = ( cons_FOR_ftrs @ Z1 @ Z22 ) )
             => ~ ( member_FOR_ftrs @ E @ ( set_FOR_ftrs2 @ Z22 ) ) ) ) ) ).

% list.set_cases
thf(fact_681_list_Oset__intros_I1_J,axiom,
    ! [X21: produc357393685978478089rm_a_b,X222: list_P8875379029341186191rm_a_b] : ( member5869715511025134514rm_a_b @ X21 @ ( set_Pr2370880052973118494rm_a_b @ ( cons_P3536305108106557631rm_a_b @ X21 @ X222 ) ) ) ).

% list.set_intros(1)
thf(fact_682_list_Oset__intros_I1_J,axiom,
    ! [X21: fOR_ftrs,X222: list_FOR_ftrs] : ( member_FOR_ftrs @ X21 @ ( set_FOR_ftrs2 @ ( cons_FOR_ftrs @ X21 @ X222 ) ) ) ).

% list.set_intros(1)
thf(fact_683_list_Oset__intros_I2_J,axiom,
    ! [Y: produc357393685978478089rm_a_b,X222: list_P8875379029341186191rm_a_b,X21: produc357393685978478089rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ Y @ ( set_Pr2370880052973118494rm_a_b @ X222 ) )
     => ( member5869715511025134514rm_a_b @ Y @ ( set_Pr2370880052973118494rm_a_b @ ( cons_P3536305108106557631rm_a_b @ X21 @ X222 ) ) ) ) ).

% list.set_intros(2)
thf(fact_684_list_Oset__intros_I2_J,axiom,
    ! [Y: fOR_ftrs,X222: list_FOR_ftrs,X21: fOR_ftrs] :
      ( ( member_FOR_ftrs @ Y @ ( set_FOR_ftrs2 @ X222 ) )
     => ( member_FOR_ftrs @ Y @ ( set_FOR_ftrs2 @ ( cons_FOR_ftrs @ X21 @ X222 ) ) ) ) ).

% list.set_intros(2)
thf(fact_685_list_Osimps_I9_J,axiom,
    ! [F: fset_P3577670608877728751rm_a_b > set_Pr4386577575007340137rm_a_b,X21: fset_P3577670608877728751rm_a_b,X222: list_f7931864845379430901rm_a_b] :
      ( ( map_fs2147491833908496126rm_a_b @ F @ ( cons_f831858257744602021rm_a_b @ X21 @ X222 ) )
      = ( cons_s7847606222307733791rm_a_b @ ( F @ X21 ) @ ( map_fs2147491833908496126rm_a_b @ F @ X222 ) ) ) ).

% list.simps(9)
thf(fact_686_Cons__eq__map__D,axiom,
    ! [X: set_Pr4386577575007340137rm_a_b,Xs: list_s5629610930313288943rm_a_b,F: fset_P3577670608877728751rm_a_b > set_Pr4386577575007340137rm_a_b,Ys2: list_f7931864845379430901rm_a_b] :
      ( ( ( cons_s7847606222307733791rm_a_b @ X @ Xs )
        = ( map_fs2147491833908496126rm_a_b @ F @ Ys2 ) )
     => ? [Z: fset_P3577670608877728751rm_a_b,Zs: list_f7931864845379430901rm_a_b] :
          ( ( Ys2
            = ( cons_f831858257744602021rm_a_b @ Z @ Zs ) )
          & ( X
            = ( F @ Z ) )
          & ( Xs
            = ( map_fs2147491833908496126rm_a_b @ F @ Zs ) ) ) ) ).

% Cons_eq_map_D
thf(fact_687_map__eq__Cons__D,axiom,
    ! [F: fset_P3577670608877728751rm_a_b > set_Pr4386577575007340137rm_a_b,Xs: list_f7931864845379430901rm_a_b,Y: set_Pr4386577575007340137rm_a_b,Ys2: list_s5629610930313288943rm_a_b] :
      ( ( ( map_fs2147491833908496126rm_a_b @ F @ Xs )
        = ( cons_s7847606222307733791rm_a_b @ Y @ Ys2 ) )
     => ? [Z: fset_P3577670608877728751rm_a_b,Zs: list_f7931864845379430901rm_a_b] :
          ( ( Xs
            = ( cons_f831858257744602021rm_a_b @ Z @ Zs ) )
          & ( ( F @ Z )
            = Y )
          & ( ( map_fs2147491833908496126rm_a_b @ F @ Zs )
            = Ys2 ) ) ) ).

% map_eq_Cons_D
thf(fact_688_Cons__eq__map__conv,axiom,
    ! [X: set_Pr4386577575007340137rm_a_b,Xs: list_s5629610930313288943rm_a_b,F: fset_P3577670608877728751rm_a_b > set_Pr4386577575007340137rm_a_b,Ys2: list_f7931864845379430901rm_a_b] :
      ( ( ( cons_s7847606222307733791rm_a_b @ X @ Xs )
        = ( map_fs2147491833908496126rm_a_b @ F @ Ys2 ) )
      = ( ? [Z3: fset_P3577670608877728751rm_a_b,Zs2: list_f7931864845379430901rm_a_b] :
            ( ( Ys2
              = ( cons_f831858257744602021rm_a_b @ Z3 @ Zs2 ) )
            & ( X
              = ( F @ Z3 ) )
            & ( Xs
              = ( map_fs2147491833908496126rm_a_b @ F @ Zs2 ) ) ) ) ) ).

% Cons_eq_map_conv
thf(fact_689_map__eq__Cons__conv,axiom,
    ! [F: fset_P3577670608877728751rm_a_b > set_Pr4386577575007340137rm_a_b,Xs: list_f7931864845379430901rm_a_b,Y: set_Pr4386577575007340137rm_a_b,Ys2: list_s5629610930313288943rm_a_b] :
      ( ( ( map_fs2147491833908496126rm_a_b @ F @ Xs )
        = ( cons_s7847606222307733791rm_a_b @ Y @ Ys2 ) )
      = ( ? [Z3: fset_P3577670608877728751rm_a_b,Zs2: list_f7931864845379430901rm_a_b] :
            ( ( Xs
              = ( cons_f831858257744602021rm_a_b @ Z3 @ Zs2 ) )
            & ( ( F @ Z3 )
              = Y )
            & ( ( map_fs2147491833908496126rm_a_b @ F @ Zs2 )
              = Ys2 ) ) ) ) ).

% map_eq_Cons_conv
thf(fact_690_int__cases2,axiom,
    ! [Z4: int] :
      ( ! [N2: nat] :
          ( Z4
         != ( semiri1314217659103216013at_int @ N2 ) )
     => ~ ! [N2: nat] :
            ( Z4
           != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).

% int_cases2
thf(fact_691_remdups__adj__length,axiom,
    ! [Xs: list_f7931864845379430901rm_a_b] : ( ord_less_eq_nat @ ( size_s1967422219310127201rm_a_b @ ( remdup2633883733652511275rm_a_b @ Xs ) ) @ ( size_s1967422219310127201rm_a_b @ Xs ) ) ).

% remdups_adj_length
thf(fact_692_list__induct4,axiom,
    ! [Xs: list_s5629610930313288943rm_a_b,Ys2: list_s5629610930313288943rm_a_b,Zs3: list_s5629610930313288943rm_a_b,Ws: list_s5629610930313288943rm_a_b,P: list_s5629610930313288943rm_a_b > list_s5629610930313288943rm_a_b > list_s5629610930313288943rm_a_b > list_s5629610930313288943rm_a_b > $o] :
      ( ( ( size_s3663623692424790363rm_a_b @ Xs )
        = ( size_s3663623692424790363rm_a_b @ Ys2 ) )
     => ( ( ( size_s3663623692424790363rm_a_b @ Ys2 )
          = ( size_s3663623692424790363rm_a_b @ Zs3 ) )
       => ( ( ( size_s3663623692424790363rm_a_b @ Zs3 )
            = ( size_s3663623692424790363rm_a_b @ Ws ) )
         => ( ( P @ nil_se7334864872594423151rm_a_b @ nil_se7334864872594423151rm_a_b @ nil_se7334864872594423151rm_a_b @ nil_se7334864872594423151rm_a_b )
           => ( ! [X4: set_Pr4386577575007340137rm_a_b,Xs2: list_s5629610930313288943rm_a_b,Y4: set_Pr4386577575007340137rm_a_b,Ys4: list_s5629610930313288943rm_a_b,Z: set_Pr4386577575007340137rm_a_b,Zs: list_s5629610930313288943rm_a_b,W: set_Pr4386577575007340137rm_a_b,Ws2: list_s5629610930313288943rm_a_b] :
                  ( ( ( size_s3663623692424790363rm_a_b @ Xs2 )
                    = ( size_s3663623692424790363rm_a_b @ Ys4 ) )
                 => ( ( ( size_s3663623692424790363rm_a_b @ Ys4 )
                      = ( size_s3663623692424790363rm_a_b @ Zs ) )
                   => ( ( ( size_s3663623692424790363rm_a_b @ Zs )
                        = ( size_s3663623692424790363rm_a_b @ Ws2 ) )
                     => ( ( P @ Xs2 @ Ys4 @ Zs @ Ws2 )
                       => ( P @ ( cons_s7847606222307733791rm_a_b @ X4 @ Xs2 ) @ ( cons_s7847606222307733791rm_a_b @ Y4 @ Ys4 ) @ ( cons_s7847606222307733791rm_a_b @ Z @ Zs ) @ ( cons_s7847606222307733791rm_a_b @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys2 @ Zs3 @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_693_list__induct4,axiom,
    ! [Xs: list_s5629610930313288943rm_a_b,Ys2: list_s5629610930313288943rm_a_b,Zs3: list_s5629610930313288943rm_a_b,Ws: list_f7931864845379430901rm_a_b,P: list_s5629610930313288943rm_a_b > list_s5629610930313288943rm_a_b > list_s5629610930313288943rm_a_b > list_f7931864845379430901rm_a_b > $o] :
      ( ( ( size_s3663623692424790363rm_a_b @ Xs )
        = ( size_s3663623692424790363rm_a_b @ Ys2 ) )
     => ( ( ( size_s3663623692424790363rm_a_b @ Ys2 )
          = ( size_s3663623692424790363rm_a_b @ Zs3 ) )
       => ( ( ( size_s3663623692424790363rm_a_b @ Zs3 )
            = ( size_s1967422219310127201rm_a_b @ Ws ) )
         => ( ( P @ nil_se7334864872594423151rm_a_b @ nil_se7334864872594423151rm_a_b @ nil_se7334864872594423151rm_a_b @ nil_fs4020363321024022517rm_a_b )
           => ( ! [X4: set_Pr4386577575007340137rm_a_b,Xs2: list_s5629610930313288943rm_a_b,Y4: set_Pr4386577575007340137rm_a_b,Ys4: list_s5629610930313288943rm_a_b,Z: set_Pr4386577575007340137rm_a_b,Zs: list_s5629610930313288943rm_a_b,W: fset_P3577670608877728751rm_a_b,Ws2: list_f7931864845379430901rm_a_b] :
                  ( ( ( size_s3663623692424790363rm_a_b @ Xs2 )
                    = ( size_s3663623692424790363rm_a_b @ Ys4 ) )
                 => ( ( ( size_s3663623692424790363rm_a_b @ Ys4 )
                      = ( size_s3663623692424790363rm_a_b @ Zs ) )
                   => ( ( ( size_s3663623692424790363rm_a_b @ Zs )
                        = ( size_s1967422219310127201rm_a_b @ Ws2 ) )
                     => ( ( P @ Xs2 @ Ys4 @ Zs @ Ws2 )
                       => ( P @ ( cons_s7847606222307733791rm_a_b @ X4 @ Xs2 ) @ ( cons_s7847606222307733791rm_a_b @ Y4 @ Ys4 ) @ ( cons_s7847606222307733791rm_a_b @ Z @ Zs ) @ ( cons_f831858257744602021rm_a_b @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys2 @ Zs3 @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_694_list__induct4,axiom,
    ! [Xs: list_s5629610930313288943rm_a_b,Ys2: list_s5629610930313288943rm_a_b,Zs3: list_f7931864845379430901rm_a_b,Ws: list_s5629610930313288943rm_a_b,P: list_s5629610930313288943rm_a_b > list_s5629610930313288943rm_a_b > list_f7931864845379430901rm_a_b > list_s5629610930313288943rm_a_b > $o] :
      ( ( ( size_s3663623692424790363rm_a_b @ Xs )
        = ( size_s3663623692424790363rm_a_b @ Ys2 ) )
     => ( ( ( size_s3663623692424790363rm_a_b @ Ys2 )
          = ( size_s1967422219310127201rm_a_b @ Zs3 ) )
       => ( ( ( size_s1967422219310127201rm_a_b @ Zs3 )
            = ( size_s3663623692424790363rm_a_b @ Ws ) )
         => ( ( P @ nil_se7334864872594423151rm_a_b @ nil_se7334864872594423151rm_a_b @ nil_fs4020363321024022517rm_a_b @ nil_se7334864872594423151rm_a_b )
           => ( ! [X4: set_Pr4386577575007340137rm_a_b,Xs2: list_s5629610930313288943rm_a_b,Y4: set_Pr4386577575007340137rm_a_b,Ys4: list_s5629610930313288943rm_a_b,Z: fset_P3577670608877728751rm_a_b,Zs: list_f7931864845379430901rm_a_b,W: set_Pr4386577575007340137rm_a_b,Ws2: list_s5629610930313288943rm_a_b] :
                  ( ( ( size_s3663623692424790363rm_a_b @ Xs2 )
                    = ( size_s3663623692424790363rm_a_b @ Ys4 ) )
                 => ( ( ( size_s3663623692424790363rm_a_b @ Ys4 )
                      = ( size_s1967422219310127201rm_a_b @ Zs ) )
                   => ( ( ( size_s1967422219310127201rm_a_b @ Zs )
                        = ( size_s3663623692424790363rm_a_b @ Ws2 ) )
                     => ( ( P @ Xs2 @ Ys4 @ Zs @ Ws2 )
                       => ( P @ ( cons_s7847606222307733791rm_a_b @ X4 @ Xs2 ) @ ( cons_s7847606222307733791rm_a_b @ Y4 @ Ys4 ) @ ( cons_f831858257744602021rm_a_b @ Z @ Zs ) @ ( cons_s7847606222307733791rm_a_b @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys2 @ Zs3 @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_695_list__induct4,axiom,
    ! [Xs: list_s5629610930313288943rm_a_b,Ys2: list_s5629610930313288943rm_a_b,Zs3: list_f7931864845379430901rm_a_b,Ws: list_f7931864845379430901rm_a_b,P: list_s5629610930313288943rm_a_b > list_s5629610930313288943rm_a_b > list_f7931864845379430901rm_a_b > list_f7931864845379430901rm_a_b > $o] :
      ( ( ( size_s3663623692424790363rm_a_b @ Xs )
        = ( size_s3663623692424790363rm_a_b @ Ys2 ) )
     => ( ( ( size_s3663623692424790363rm_a_b @ Ys2 )
          = ( size_s1967422219310127201rm_a_b @ Zs3 ) )
       => ( ( ( size_s1967422219310127201rm_a_b @ Zs3 )
            = ( size_s1967422219310127201rm_a_b @ Ws ) )
         => ( ( P @ nil_se7334864872594423151rm_a_b @ nil_se7334864872594423151rm_a_b @ nil_fs4020363321024022517rm_a_b @ nil_fs4020363321024022517rm_a_b )
           => ( ! [X4: set_Pr4386577575007340137rm_a_b,Xs2: list_s5629610930313288943rm_a_b,Y4: set_Pr4386577575007340137rm_a_b,Ys4: list_s5629610930313288943rm_a_b,Z: fset_P3577670608877728751rm_a_b,Zs: list_f7931864845379430901rm_a_b,W: fset_P3577670608877728751rm_a_b,Ws2: list_f7931864845379430901rm_a_b] :
                  ( ( ( size_s3663623692424790363rm_a_b @ Xs2 )
                    = ( size_s3663623692424790363rm_a_b @ Ys4 ) )
                 => ( ( ( size_s3663623692424790363rm_a_b @ Ys4 )
                      = ( size_s1967422219310127201rm_a_b @ Zs ) )
                   => ( ( ( size_s1967422219310127201rm_a_b @ Zs )
                        = ( size_s1967422219310127201rm_a_b @ Ws2 ) )
                     => ( ( P @ Xs2 @ Ys4 @ Zs @ Ws2 )
                       => ( P @ ( cons_s7847606222307733791rm_a_b @ X4 @ Xs2 ) @ ( cons_s7847606222307733791rm_a_b @ Y4 @ Ys4 ) @ ( cons_f831858257744602021rm_a_b @ Z @ Zs ) @ ( cons_f831858257744602021rm_a_b @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys2 @ Zs3 @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_696_list__induct4,axiom,
    ! [Xs: list_s5629610930313288943rm_a_b,Ys2: list_f7931864845379430901rm_a_b,Zs3: list_s5629610930313288943rm_a_b,Ws: list_s5629610930313288943rm_a_b,P: list_s5629610930313288943rm_a_b > list_f7931864845379430901rm_a_b > list_s5629610930313288943rm_a_b > list_s5629610930313288943rm_a_b > $o] :
      ( ( ( size_s3663623692424790363rm_a_b @ Xs )
        = ( size_s1967422219310127201rm_a_b @ Ys2 ) )
     => ( ( ( size_s1967422219310127201rm_a_b @ Ys2 )
          = ( size_s3663623692424790363rm_a_b @ Zs3 ) )
       => ( ( ( size_s3663623692424790363rm_a_b @ Zs3 )
            = ( size_s3663623692424790363rm_a_b @ Ws ) )
         => ( ( P @ nil_se7334864872594423151rm_a_b @ nil_fs4020363321024022517rm_a_b @ nil_se7334864872594423151rm_a_b @ nil_se7334864872594423151rm_a_b )
           => ( ! [X4: set_Pr4386577575007340137rm_a_b,Xs2: list_s5629610930313288943rm_a_b,Y4: fset_P3577670608877728751rm_a_b,Ys4: list_f7931864845379430901rm_a_b,Z: set_Pr4386577575007340137rm_a_b,Zs: list_s5629610930313288943rm_a_b,W: set_Pr4386577575007340137rm_a_b,Ws2: list_s5629610930313288943rm_a_b] :
                  ( ( ( size_s3663623692424790363rm_a_b @ Xs2 )
                    = ( size_s1967422219310127201rm_a_b @ Ys4 ) )
                 => ( ( ( size_s1967422219310127201rm_a_b @ Ys4 )
                      = ( size_s3663623692424790363rm_a_b @ Zs ) )
                   => ( ( ( size_s3663623692424790363rm_a_b @ Zs )
                        = ( size_s3663623692424790363rm_a_b @ Ws2 ) )
                     => ( ( P @ Xs2 @ Ys4 @ Zs @ Ws2 )
                       => ( P @ ( cons_s7847606222307733791rm_a_b @ X4 @ Xs2 ) @ ( cons_f831858257744602021rm_a_b @ Y4 @ Ys4 ) @ ( cons_s7847606222307733791rm_a_b @ Z @ Zs ) @ ( cons_s7847606222307733791rm_a_b @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys2 @ Zs3 @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_697_list__induct4,axiom,
    ! [Xs: list_s5629610930313288943rm_a_b,Ys2: list_f7931864845379430901rm_a_b,Zs3: list_s5629610930313288943rm_a_b,Ws: list_f7931864845379430901rm_a_b,P: list_s5629610930313288943rm_a_b > list_f7931864845379430901rm_a_b > list_s5629610930313288943rm_a_b > list_f7931864845379430901rm_a_b > $o] :
      ( ( ( size_s3663623692424790363rm_a_b @ Xs )
        = ( size_s1967422219310127201rm_a_b @ Ys2 ) )
     => ( ( ( size_s1967422219310127201rm_a_b @ Ys2 )
          = ( size_s3663623692424790363rm_a_b @ Zs3 ) )
       => ( ( ( size_s3663623692424790363rm_a_b @ Zs3 )
            = ( size_s1967422219310127201rm_a_b @ Ws ) )
         => ( ( P @ nil_se7334864872594423151rm_a_b @ nil_fs4020363321024022517rm_a_b @ nil_se7334864872594423151rm_a_b @ nil_fs4020363321024022517rm_a_b )
           => ( ! [X4: set_Pr4386577575007340137rm_a_b,Xs2: list_s5629610930313288943rm_a_b,Y4: fset_P3577670608877728751rm_a_b,Ys4: list_f7931864845379430901rm_a_b,Z: set_Pr4386577575007340137rm_a_b,Zs: list_s5629610930313288943rm_a_b,W: fset_P3577670608877728751rm_a_b,Ws2: list_f7931864845379430901rm_a_b] :
                  ( ( ( size_s3663623692424790363rm_a_b @ Xs2 )
                    = ( size_s1967422219310127201rm_a_b @ Ys4 ) )
                 => ( ( ( size_s1967422219310127201rm_a_b @ Ys4 )
                      = ( size_s3663623692424790363rm_a_b @ Zs ) )
                   => ( ( ( size_s3663623692424790363rm_a_b @ Zs )
                        = ( size_s1967422219310127201rm_a_b @ Ws2 ) )
                     => ( ( P @ Xs2 @ Ys4 @ Zs @ Ws2 )
                       => ( P @ ( cons_s7847606222307733791rm_a_b @ X4 @ Xs2 ) @ ( cons_f831858257744602021rm_a_b @ Y4 @ Ys4 ) @ ( cons_s7847606222307733791rm_a_b @ Z @ Zs ) @ ( cons_f831858257744602021rm_a_b @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys2 @ Zs3 @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_698_list__induct4,axiom,
    ! [Xs: list_s5629610930313288943rm_a_b,Ys2: list_f7931864845379430901rm_a_b,Zs3: list_f7931864845379430901rm_a_b,Ws: list_s5629610930313288943rm_a_b,P: list_s5629610930313288943rm_a_b > list_f7931864845379430901rm_a_b > list_f7931864845379430901rm_a_b > list_s5629610930313288943rm_a_b > $o] :
      ( ( ( size_s3663623692424790363rm_a_b @ Xs )
        = ( size_s1967422219310127201rm_a_b @ Ys2 ) )
     => ( ( ( size_s1967422219310127201rm_a_b @ Ys2 )
          = ( size_s1967422219310127201rm_a_b @ Zs3 ) )
       => ( ( ( size_s1967422219310127201rm_a_b @ Zs3 )
            = ( size_s3663623692424790363rm_a_b @ Ws ) )
         => ( ( P @ nil_se7334864872594423151rm_a_b @ nil_fs4020363321024022517rm_a_b @ nil_fs4020363321024022517rm_a_b @ nil_se7334864872594423151rm_a_b )
           => ( ! [X4: set_Pr4386577575007340137rm_a_b,Xs2: list_s5629610930313288943rm_a_b,Y4: fset_P3577670608877728751rm_a_b,Ys4: list_f7931864845379430901rm_a_b,Z: fset_P3577670608877728751rm_a_b,Zs: list_f7931864845379430901rm_a_b,W: set_Pr4386577575007340137rm_a_b,Ws2: list_s5629610930313288943rm_a_b] :
                  ( ( ( size_s3663623692424790363rm_a_b @ Xs2 )
                    = ( size_s1967422219310127201rm_a_b @ Ys4 ) )
                 => ( ( ( size_s1967422219310127201rm_a_b @ Ys4 )
                      = ( size_s1967422219310127201rm_a_b @ Zs ) )
                   => ( ( ( size_s1967422219310127201rm_a_b @ Zs )
                        = ( size_s3663623692424790363rm_a_b @ Ws2 ) )
                     => ( ( P @ Xs2 @ Ys4 @ Zs @ Ws2 )
                       => ( P @ ( cons_s7847606222307733791rm_a_b @ X4 @ Xs2 ) @ ( cons_f831858257744602021rm_a_b @ Y4 @ Ys4 ) @ ( cons_f831858257744602021rm_a_b @ Z @ Zs ) @ ( cons_s7847606222307733791rm_a_b @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys2 @ Zs3 @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_699_list__induct4,axiom,
    ! [Xs: list_s5629610930313288943rm_a_b,Ys2: list_f7931864845379430901rm_a_b,Zs3: list_f7931864845379430901rm_a_b,Ws: list_f7931864845379430901rm_a_b,P: list_s5629610930313288943rm_a_b > list_f7931864845379430901rm_a_b > list_f7931864845379430901rm_a_b > list_f7931864845379430901rm_a_b > $o] :
      ( ( ( size_s3663623692424790363rm_a_b @ Xs )
        = ( size_s1967422219310127201rm_a_b @ Ys2 ) )
     => ( ( ( size_s1967422219310127201rm_a_b @ Ys2 )
          = ( size_s1967422219310127201rm_a_b @ Zs3 ) )
       => ( ( ( size_s1967422219310127201rm_a_b @ Zs3 )
            = ( size_s1967422219310127201rm_a_b @ Ws ) )
         => ( ( P @ nil_se7334864872594423151rm_a_b @ nil_fs4020363321024022517rm_a_b @ nil_fs4020363321024022517rm_a_b @ nil_fs4020363321024022517rm_a_b )
           => ( ! [X4: set_Pr4386577575007340137rm_a_b,Xs2: list_s5629610930313288943rm_a_b,Y4: fset_P3577670608877728751rm_a_b,Ys4: list_f7931864845379430901rm_a_b,Z: fset_P3577670608877728751rm_a_b,Zs: list_f7931864845379430901rm_a_b,W: fset_P3577670608877728751rm_a_b,Ws2: list_f7931864845379430901rm_a_b] :
                  ( ( ( size_s3663623692424790363rm_a_b @ Xs2 )
                    = ( size_s1967422219310127201rm_a_b @ Ys4 ) )
                 => ( ( ( size_s1967422219310127201rm_a_b @ Ys4 )
                      = ( size_s1967422219310127201rm_a_b @ Zs ) )
                   => ( ( ( size_s1967422219310127201rm_a_b @ Zs )
                        = ( size_s1967422219310127201rm_a_b @ Ws2 ) )
                     => ( ( P @ Xs2 @ Ys4 @ Zs @ Ws2 )
                       => ( P @ ( cons_s7847606222307733791rm_a_b @ X4 @ Xs2 ) @ ( cons_f831858257744602021rm_a_b @ Y4 @ Ys4 ) @ ( cons_f831858257744602021rm_a_b @ Z @ Zs ) @ ( cons_f831858257744602021rm_a_b @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys2 @ Zs3 @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_700_list__induct4,axiom,
    ! [Xs: list_f7931864845379430901rm_a_b,Ys2: list_s5629610930313288943rm_a_b,Zs3: list_s5629610930313288943rm_a_b,Ws: list_s5629610930313288943rm_a_b,P: list_f7931864845379430901rm_a_b > list_s5629610930313288943rm_a_b > list_s5629610930313288943rm_a_b > list_s5629610930313288943rm_a_b > $o] :
      ( ( ( size_s1967422219310127201rm_a_b @ Xs )
        = ( size_s3663623692424790363rm_a_b @ Ys2 ) )
     => ( ( ( size_s3663623692424790363rm_a_b @ Ys2 )
          = ( size_s3663623692424790363rm_a_b @ Zs3 ) )
       => ( ( ( size_s3663623692424790363rm_a_b @ Zs3 )
            = ( size_s3663623692424790363rm_a_b @ Ws ) )
         => ( ( P @ nil_fs4020363321024022517rm_a_b @ nil_se7334864872594423151rm_a_b @ nil_se7334864872594423151rm_a_b @ nil_se7334864872594423151rm_a_b )
           => ( ! [X4: fset_P3577670608877728751rm_a_b,Xs2: list_f7931864845379430901rm_a_b,Y4: set_Pr4386577575007340137rm_a_b,Ys4: list_s5629610930313288943rm_a_b,Z: set_Pr4386577575007340137rm_a_b,Zs: list_s5629610930313288943rm_a_b,W: set_Pr4386577575007340137rm_a_b,Ws2: list_s5629610930313288943rm_a_b] :
                  ( ( ( size_s1967422219310127201rm_a_b @ Xs2 )
                    = ( size_s3663623692424790363rm_a_b @ Ys4 ) )
                 => ( ( ( size_s3663623692424790363rm_a_b @ Ys4 )
                      = ( size_s3663623692424790363rm_a_b @ Zs ) )
                   => ( ( ( size_s3663623692424790363rm_a_b @ Zs )
                        = ( size_s3663623692424790363rm_a_b @ Ws2 ) )
                     => ( ( P @ Xs2 @ Ys4 @ Zs @ Ws2 )
                       => ( P @ ( cons_f831858257744602021rm_a_b @ X4 @ Xs2 ) @ ( cons_s7847606222307733791rm_a_b @ Y4 @ Ys4 ) @ ( cons_s7847606222307733791rm_a_b @ Z @ Zs ) @ ( cons_s7847606222307733791rm_a_b @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys2 @ Zs3 @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_701_list__induct4,axiom,
    ! [Xs: list_f7931864845379430901rm_a_b,Ys2: list_s5629610930313288943rm_a_b,Zs3: list_s5629610930313288943rm_a_b,Ws: list_f7931864845379430901rm_a_b,P: list_f7931864845379430901rm_a_b > list_s5629610930313288943rm_a_b > list_s5629610930313288943rm_a_b > list_f7931864845379430901rm_a_b > $o] :
      ( ( ( size_s1967422219310127201rm_a_b @ Xs )
        = ( size_s3663623692424790363rm_a_b @ Ys2 ) )
     => ( ( ( size_s3663623692424790363rm_a_b @ Ys2 )
          = ( size_s3663623692424790363rm_a_b @ Zs3 ) )
       => ( ( ( size_s3663623692424790363rm_a_b @ Zs3 )
            = ( size_s1967422219310127201rm_a_b @ Ws ) )
         => ( ( P @ nil_fs4020363321024022517rm_a_b @ nil_se7334864872594423151rm_a_b @ nil_se7334864872594423151rm_a_b @ nil_fs4020363321024022517rm_a_b )
           => ( ! [X4: fset_P3577670608877728751rm_a_b,Xs2: list_f7931864845379430901rm_a_b,Y4: set_Pr4386577575007340137rm_a_b,Ys4: list_s5629610930313288943rm_a_b,Z: set_Pr4386577575007340137rm_a_b,Zs: list_s5629610930313288943rm_a_b,W: fset_P3577670608877728751rm_a_b,Ws2: list_f7931864845379430901rm_a_b] :
                  ( ( ( size_s1967422219310127201rm_a_b @ Xs2 )
                    = ( size_s3663623692424790363rm_a_b @ Ys4 ) )
                 => ( ( ( size_s3663623692424790363rm_a_b @ Ys4 )
                      = ( size_s3663623692424790363rm_a_b @ Zs ) )
                   => ( ( ( size_s3663623692424790363rm_a_b @ Zs )
                        = ( size_s1967422219310127201rm_a_b @ Ws2 ) )
                     => ( ( P @ Xs2 @ Ys4 @ Zs @ Ws2 )
                       => ( P @ ( cons_f831858257744602021rm_a_b @ X4 @ Xs2 ) @ ( cons_s7847606222307733791rm_a_b @ Y4 @ Ys4 ) @ ( cons_s7847606222307733791rm_a_b @ Z @ Zs ) @ ( cons_f831858257744602021rm_a_b @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys2 @ Zs3 @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_702_list__induct3,axiom,
    ! [Xs: list_s5629610930313288943rm_a_b,Ys2: list_s5629610930313288943rm_a_b,Zs3: list_s5629610930313288943rm_a_b,P: list_s5629610930313288943rm_a_b > list_s5629610930313288943rm_a_b > list_s5629610930313288943rm_a_b > $o] :
      ( ( ( size_s3663623692424790363rm_a_b @ Xs )
        = ( size_s3663623692424790363rm_a_b @ Ys2 ) )
     => ( ( ( size_s3663623692424790363rm_a_b @ Ys2 )
          = ( size_s3663623692424790363rm_a_b @ Zs3 ) )
       => ( ( P @ nil_se7334864872594423151rm_a_b @ nil_se7334864872594423151rm_a_b @ nil_se7334864872594423151rm_a_b )
         => ( ! [X4: set_Pr4386577575007340137rm_a_b,Xs2: list_s5629610930313288943rm_a_b,Y4: set_Pr4386577575007340137rm_a_b,Ys4: list_s5629610930313288943rm_a_b,Z: set_Pr4386577575007340137rm_a_b,Zs: list_s5629610930313288943rm_a_b] :
                ( ( ( size_s3663623692424790363rm_a_b @ Xs2 )
                  = ( size_s3663623692424790363rm_a_b @ Ys4 ) )
               => ( ( ( size_s3663623692424790363rm_a_b @ Ys4 )
                    = ( size_s3663623692424790363rm_a_b @ Zs ) )
                 => ( ( P @ Xs2 @ Ys4 @ Zs )
                   => ( P @ ( cons_s7847606222307733791rm_a_b @ X4 @ Xs2 ) @ ( cons_s7847606222307733791rm_a_b @ Y4 @ Ys4 ) @ ( cons_s7847606222307733791rm_a_b @ Z @ Zs ) ) ) ) )
           => ( P @ Xs @ Ys2 @ Zs3 ) ) ) ) ) ).

% list_induct3
thf(fact_703_list__induct3,axiom,
    ! [Xs: list_s5629610930313288943rm_a_b,Ys2: list_s5629610930313288943rm_a_b,Zs3: list_f7931864845379430901rm_a_b,P: list_s5629610930313288943rm_a_b > list_s5629610930313288943rm_a_b > list_f7931864845379430901rm_a_b > $o] :
      ( ( ( size_s3663623692424790363rm_a_b @ Xs )
        = ( size_s3663623692424790363rm_a_b @ Ys2 ) )
     => ( ( ( size_s3663623692424790363rm_a_b @ Ys2 )
          = ( size_s1967422219310127201rm_a_b @ Zs3 ) )
       => ( ( P @ nil_se7334864872594423151rm_a_b @ nil_se7334864872594423151rm_a_b @ nil_fs4020363321024022517rm_a_b )
         => ( ! [X4: set_Pr4386577575007340137rm_a_b,Xs2: list_s5629610930313288943rm_a_b,Y4: set_Pr4386577575007340137rm_a_b,Ys4: list_s5629610930313288943rm_a_b,Z: fset_P3577670608877728751rm_a_b,Zs: list_f7931864845379430901rm_a_b] :
                ( ( ( size_s3663623692424790363rm_a_b @ Xs2 )
                  = ( size_s3663623692424790363rm_a_b @ Ys4 ) )
               => ( ( ( size_s3663623692424790363rm_a_b @ Ys4 )
                    = ( size_s1967422219310127201rm_a_b @ Zs ) )
                 => ( ( P @ Xs2 @ Ys4 @ Zs )
                   => ( P @ ( cons_s7847606222307733791rm_a_b @ X4 @ Xs2 ) @ ( cons_s7847606222307733791rm_a_b @ Y4 @ Ys4 ) @ ( cons_f831858257744602021rm_a_b @ Z @ Zs ) ) ) ) )
           => ( P @ Xs @ Ys2 @ Zs3 ) ) ) ) ) ).

% list_induct3
thf(fact_704_list__induct3,axiom,
    ! [Xs: list_s5629610930313288943rm_a_b,Ys2: list_f7931864845379430901rm_a_b,Zs3: list_s5629610930313288943rm_a_b,P: list_s5629610930313288943rm_a_b > list_f7931864845379430901rm_a_b > list_s5629610930313288943rm_a_b > $o] :
      ( ( ( size_s3663623692424790363rm_a_b @ Xs )
        = ( size_s1967422219310127201rm_a_b @ Ys2 ) )
     => ( ( ( size_s1967422219310127201rm_a_b @ Ys2 )
          = ( size_s3663623692424790363rm_a_b @ Zs3 ) )
       => ( ( P @ nil_se7334864872594423151rm_a_b @ nil_fs4020363321024022517rm_a_b @ nil_se7334864872594423151rm_a_b )
         => ( ! [X4: set_Pr4386577575007340137rm_a_b,Xs2: list_s5629610930313288943rm_a_b,Y4: fset_P3577670608877728751rm_a_b,Ys4: list_f7931864845379430901rm_a_b,Z: set_Pr4386577575007340137rm_a_b,Zs: list_s5629610930313288943rm_a_b] :
                ( ( ( size_s3663623692424790363rm_a_b @ Xs2 )
                  = ( size_s1967422219310127201rm_a_b @ Ys4 ) )
               => ( ( ( size_s1967422219310127201rm_a_b @ Ys4 )
                    = ( size_s3663623692424790363rm_a_b @ Zs ) )
                 => ( ( P @ Xs2 @ Ys4 @ Zs )
                   => ( P @ ( cons_s7847606222307733791rm_a_b @ X4 @ Xs2 ) @ ( cons_f831858257744602021rm_a_b @ Y4 @ Ys4 ) @ ( cons_s7847606222307733791rm_a_b @ Z @ Zs ) ) ) ) )
           => ( P @ Xs @ Ys2 @ Zs3 ) ) ) ) ) ).

% list_induct3
thf(fact_705_list__induct3,axiom,
    ! [Xs: list_s5629610930313288943rm_a_b,Ys2: list_f7931864845379430901rm_a_b,Zs3: list_f7931864845379430901rm_a_b,P: list_s5629610930313288943rm_a_b > list_f7931864845379430901rm_a_b > list_f7931864845379430901rm_a_b > $o] :
      ( ( ( size_s3663623692424790363rm_a_b @ Xs )
        = ( size_s1967422219310127201rm_a_b @ Ys2 ) )
     => ( ( ( size_s1967422219310127201rm_a_b @ Ys2 )
          = ( size_s1967422219310127201rm_a_b @ Zs3 ) )
       => ( ( P @ nil_se7334864872594423151rm_a_b @ nil_fs4020363321024022517rm_a_b @ nil_fs4020363321024022517rm_a_b )
         => ( ! [X4: set_Pr4386577575007340137rm_a_b,Xs2: list_s5629610930313288943rm_a_b,Y4: fset_P3577670608877728751rm_a_b,Ys4: list_f7931864845379430901rm_a_b,Z: fset_P3577670608877728751rm_a_b,Zs: list_f7931864845379430901rm_a_b] :
                ( ( ( size_s3663623692424790363rm_a_b @ Xs2 )
                  = ( size_s1967422219310127201rm_a_b @ Ys4 ) )
               => ( ( ( size_s1967422219310127201rm_a_b @ Ys4 )
                    = ( size_s1967422219310127201rm_a_b @ Zs ) )
                 => ( ( P @ Xs2 @ Ys4 @ Zs )
                   => ( P @ ( cons_s7847606222307733791rm_a_b @ X4 @ Xs2 ) @ ( cons_f831858257744602021rm_a_b @ Y4 @ Ys4 ) @ ( cons_f831858257744602021rm_a_b @ Z @ Zs ) ) ) ) )
           => ( P @ Xs @ Ys2 @ Zs3 ) ) ) ) ) ).

% list_induct3
thf(fact_706_list__induct3,axiom,
    ! [Xs: list_f7931864845379430901rm_a_b,Ys2: list_s5629610930313288943rm_a_b,Zs3: list_s5629610930313288943rm_a_b,P: list_f7931864845379430901rm_a_b > list_s5629610930313288943rm_a_b > list_s5629610930313288943rm_a_b > $o] :
      ( ( ( size_s1967422219310127201rm_a_b @ Xs )
        = ( size_s3663623692424790363rm_a_b @ Ys2 ) )
     => ( ( ( size_s3663623692424790363rm_a_b @ Ys2 )
          = ( size_s3663623692424790363rm_a_b @ Zs3 ) )
       => ( ( P @ nil_fs4020363321024022517rm_a_b @ nil_se7334864872594423151rm_a_b @ nil_se7334864872594423151rm_a_b )
         => ( ! [X4: fset_P3577670608877728751rm_a_b,Xs2: list_f7931864845379430901rm_a_b,Y4: set_Pr4386577575007340137rm_a_b,Ys4: list_s5629610930313288943rm_a_b,Z: set_Pr4386577575007340137rm_a_b,Zs: list_s5629610930313288943rm_a_b] :
                ( ( ( size_s1967422219310127201rm_a_b @ Xs2 )
                  = ( size_s3663623692424790363rm_a_b @ Ys4 ) )
               => ( ( ( size_s3663623692424790363rm_a_b @ Ys4 )
                    = ( size_s3663623692424790363rm_a_b @ Zs ) )
                 => ( ( P @ Xs2 @ Ys4 @ Zs )
                   => ( P @ ( cons_f831858257744602021rm_a_b @ X4 @ Xs2 ) @ ( cons_s7847606222307733791rm_a_b @ Y4 @ Ys4 ) @ ( cons_s7847606222307733791rm_a_b @ Z @ Zs ) ) ) ) )
           => ( P @ Xs @ Ys2 @ Zs3 ) ) ) ) ) ).

% list_induct3
thf(fact_707_list__induct3,axiom,
    ! [Xs: list_f7931864845379430901rm_a_b,Ys2: list_s5629610930313288943rm_a_b,Zs3: list_f7931864845379430901rm_a_b,P: list_f7931864845379430901rm_a_b > list_s5629610930313288943rm_a_b > list_f7931864845379430901rm_a_b > $o] :
      ( ( ( size_s1967422219310127201rm_a_b @ Xs )
        = ( size_s3663623692424790363rm_a_b @ Ys2 ) )
     => ( ( ( size_s3663623692424790363rm_a_b @ Ys2 )
          = ( size_s1967422219310127201rm_a_b @ Zs3 ) )
       => ( ( P @ nil_fs4020363321024022517rm_a_b @ nil_se7334864872594423151rm_a_b @ nil_fs4020363321024022517rm_a_b )
         => ( ! [X4: fset_P3577670608877728751rm_a_b,Xs2: list_f7931864845379430901rm_a_b,Y4: set_Pr4386577575007340137rm_a_b,Ys4: list_s5629610930313288943rm_a_b,Z: fset_P3577670608877728751rm_a_b,Zs: list_f7931864845379430901rm_a_b] :
                ( ( ( size_s1967422219310127201rm_a_b @ Xs2 )
                  = ( size_s3663623692424790363rm_a_b @ Ys4 ) )
               => ( ( ( size_s3663623692424790363rm_a_b @ Ys4 )
                    = ( size_s1967422219310127201rm_a_b @ Zs ) )
                 => ( ( P @ Xs2 @ Ys4 @ Zs )
                   => ( P @ ( cons_f831858257744602021rm_a_b @ X4 @ Xs2 ) @ ( cons_s7847606222307733791rm_a_b @ Y4 @ Ys4 ) @ ( cons_f831858257744602021rm_a_b @ Z @ Zs ) ) ) ) )
           => ( P @ Xs @ Ys2 @ Zs3 ) ) ) ) ) ).

% list_induct3
thf(fact_708_list__induct3,axiom,
    ! [Xs: list_f7931864845379430901rm_a_b,Ys2: list_f7931864845379430901rm_a_b,Zs3: list_s5629610930313288943rm_a_b,P: list_f7931864845379430901rm_a_b > list_f7931864845379430901rm_a_b > list_s5629610930313288943rm_a_b > $o] :
      ( ( ( size_s1967422219310127201rm_a_b @ Xs )
        = ( size_s1967422219310127201rm_a_b @ Ys2 ) )
     => ( ( ( size_s1967422219310127201rm_a_b @ Ys2 )
          = ( size_s3663623692424790363rm_a_b @ Zs3 ) )
       => ( ( P @ nil_fs4020363321024022517rm_a_b @ nil_fs4020363321024022517rm_a_b @ nil_se7334864872594423151rm_a_b )
         => ( ! [X4: fset_P3577670608877728751rm_a_b,Xs2: list_f7931864845379430901rm_a_b,Y4: fset_P3577670608877728751rm_a_b,Ys4: list_f7931864845379430901rm_a_b,Z: set_Pr4386577575007340137rm_a_b,Zs: list_s5629610930313288943rm_a_b] :
                ( ( ( size_s1967422219310127201rm_a_b @ Xs2 )
                  = ( size_s1967422219310127201rm_a_b @ Ys4 ) )
               => ( ( ( size_s1967422219310127201rm_a_b @ Ys4 )
                    = ( size_s3663623692424790363rm_a_b @ Zs ) )
                 => ( ( P @ Xs2 @ Ys4 @ Zs )
                   => ( P @ ( cons_f831858257744602021rm_a_b @ X4 @ Xs2 ) @ ( cons_f831858257744602021rm_a_b @ Y4 @ Ys4 ) @ ( cons_s7847606222307733791rm_a_b @ Z @ Zs ) ) ) ) )
           => ( P @ Xs @ Ys2 @ Zs3 ) ) ) ) ) ).

% list_induct3
thf(fact_709_list__induct3,axiom,
    ! [Xs: list_f7931864845379430901rm_a_b,Ys2: list_f7931864845379430901rm_a_b,Zs3: list_f7931864845379430901rm_a_b,P: list_f7931864845379430901rm_a_b > list_f7931864845379430901rm_a_b > list_f7931864845379430901rm_a_b > $o] :
      ( ( ( size_s1967422219310127201rm_a_b @ Xs )
        = ( size_s1967422219310127201rm_a_b @ Ys2 ) )
     => ( ( ( size_s1967422219310127201rm_a_b @ Ys2 )
          = ( size_s1967422219310127201rm_a_b @ Zs3 ) )
       => ( ( P @ nil_fs4020363321024022517rm_a_b @ nil_fs4020363321024022517rm_a_b @ nil_fs4020363321024022517rm_a_b )
         => ( ! [X4: fset_P3577670608877728751rm_a_b,Xs2: list_f7931864845379430901rm_a_b,Y4: fset_P3577670608877728751rm_a_b,Ys4: list_f7931864845379430901rm_a_b,Z: fset_P3577670608877728751rm_a_b,Zs: list_f7931864845379430901rm_a_b] :
                ( ( ( size_s1967422219310127201rm_a_b @ Xs2 )
                  = ( size_s1967422219310127201rm_a_b @ Ys4 ) )
               => ( ( ( size_s1967422219310127201rm_a_b @ Ys4 )
                    = ( size_s1967422219310127201rm_a_b @ Zs ) )
                 => ( ( P @ Xs2 @ Ys4 @ Zs )
                   => ( P @ ( cons_f831858257744602021rm_a_b @ X4 @ Xs2 ) @ ( cons_f831858257744602021rm_a_b @ Y4 @ Ys4 ) @ ( cons_f831858257744602021rm_a_b @ Z @ Zs ) ) ) ) )
           => ( P @ Xs @ Ys2 @ Zs3 ) ) ) ) ) ).

% list_induct3
thf(fact_710_list__induct2,axiom,
    ! [Xs: list_s5629610930313288943rm_a_b,Ys2: list_s5629610930313288943rm_a_b,P: list_s5629610930313288943rm_a_b > list_s5629610930313288943rm_a_b > $o] :
      ( ( ( size_s3663623692424790363rm_a_b @ Xs )
        = ( size_s3663623692424790363rm_a_b @ Ys2 ) )
     => ( ( P @ nil_se7334864872594423151rm_a_b @ nil_se7334864872594423151rm_a_b )
       => ( ! [X4: set_Pr4386577575007340137rm_a_b,Xs2: list_s5629610930313288943rm_a_b,Y4: set_Pr4386577575007340137rm_a_b,Ys4: list_s5629610930313288943rm_a_b] :
              ( ( ( size_s3663623692424790363rm_a_b @ Xs2 )
                = ( size_s3663623692424790363rm_a_b @ Ys4 ) )
             => ( ( P @ Xs2 @ Ys4 )
               => ( P @ ( cons_s7847606222307733791rm_a_b @ X4 @ Xs2 ) @ ( cons_s7847606222307733791rm_a_b @ Y4 @ Ys4 ) ) ) )
         => ( P @ Xs @ Ys2 ) ) ) ) ).

% list_induct2
thf(fact_711_list__induct2,axiom,
    ! [Xs: list_s5629610930313288943rm_a_b,Ys2: list_f7931864845379430901rm_a_b,P: list_s5629610930313288943rm_a_b > list_f7931864845379430901rm_a_b > $o] :
      ( ( ( size_s3663623692424790363rm_a_b @ Xs )
        = ( size_s1967422219310127201rm_a_b @ Ys2 ) )
     => ( ( P @ nil_se7334864872594423151rm_a_b @ nil_fs4020363321024022517rm_a_b )
       => ( ! [X4: set_Pr4386577575007340137rm_a_b,Xs2: list_s5629610930313288943rm_a_b,Y4: fset_P3577670608877728751rm_a_b,Ys4: list_f7931864845379430901rm_a_b] :
              ( ( ( size_s3663623692424790363rm_a_b @ Xs2 )
                = ( size_s1967422219310127201rm_a_b @ Ys4 ) )
             => ( ( P @ Xs2 @ Ys4 )
               => ( P @ ( cons_s7847606222307733791rm_a_b @ X4 @ Xs2 ) @ ( cons_f831858257744602021rm_a_b @ Y4 @ Ys4 ) ) ) )
         => ( P @ Xs @ Ys2 ) ) ) ) ).

% list_induct2
thf(fact_712_list__induct2,axiom,
    ! [Xs: list_f7931864845379430901rm_a_b,Ys2: list_s5629610930313288943rm_a_b,P: list_f7931864845379430901rm_a_b > list_s5629610930313288943rm_a_b > $o] :
      ( ( ( size_s1967422219310127201rm_a_b @ Xs )
        = ( size_s3663623692424790363rm_a_b @ Ys2 ) )
     => ( ( P @ nil_fs4020363321024022517rm_a_b @ nil_se7334864872594423151rm_a_b )
       => ( ! [X4: fset_P3577670608877728751rm_a_b,Xs2: list_f7931864845379430901rm_a_b,Y4: set_Pr4386577575007340137rm_a_b,Ys4: list_s5629610930313288943rm_a_b] :
              ( ( ( size_s1967422219310127201rm_a_b @ Xs2 )
                = ( size_s3663623692424790363rm_a_b @ Ys4 ) )
             => ( ( P @ Xs2 @ Ys4 )
               => ( P @ ( cons_f831858257744602021rm_a_b @ X4 @ Xs2 ) @ ( cons_s7847606222307733791rm_a_b @ Y4 @ Ys4 ) ) ) )
         => ( P @ Xs @ Ys2 ) ) ) ) ).

% list_induct2
thf(fact_713_list__induct2,axiom,
    ! [Xs: list_f7931864845379430901rm_a_b,Ys2: list_f7931864845379430901rm_a_b,P: list_f7931864845379430901rm_a_b > list_f7931864845379430901rm_a_b > $o] :
      ( ( ( size_s1967422219310127201rm_a_b @ Xs )
        = ( size_s1967422219310127201rm_a_b @ Ys2 ) )
     => ( ( P @ nil_fs4020363321024022517rm_a_b @ nil_fs4020363321024022517rm_a_b )
       => ( ! [X4: fset_P3577670608877728751rm_a_b,Xs2: list_f7931864845379430901rm_a_b,Y4: fset_P3577670608877728751rm_a_b,Ys4: list_f7931864845379430901rm_a_b] :
              ( ( ( size_s1967422219310127201rm_a_b @ Xs2 )
                = ( size_s1967422219310127201rm_a_b @ Ys4 ) )
             => ( ( P @ Xs2 @ Ys4 )
               => ( P @ ( cons_f831858257744602021rm_a_b @ X4 @ Xs2 ) @ ( cons_f831858257744602021rm_a_b @ Y4 @ Ys4 ) ) ) )
         => ( P @ Xs @ Ys2 ) ) ) ) ).

% list_induct2
thf(fact_714_in__set__simps_I2_J,axiom,
    ! [X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ X @ ( set_Pr2370880052973118494rm_a_b @ ( cons_P3536305108106557631rm_a_b @ Y @ nil_Pr6942174756412032271rm_a_b ) ) )
      = ( X = Y ) ) ).

% in_set_simps(2)
thf(fact_715_in__set__simps_I2_J,axiom,
    ! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b] :
      ( ( member8823642877475723282rm_a_b @ X @ ( set_se5504503741092831358rm_a_b @ ( cons_s7847606222307733791rm_a_b @ Y @ nil_se7334864872594423151rm_a_b ) ) )
      = ( X = Y ) ) ).

% in_set_simps(2)
thf(fact_716_in__set__simps_I2_J,axiom,
    ! [X: fset_P3577670608877728751rm_a_b,Y: fset_P3577670608877728751rm_a_b] :
      ( ( member2349228652077105432rm_a_b @ X @ ( set_fs2829528993146950532rm_a_b @ ( cons_f831858257744602021rm_a_b @ Y @ nil_fs4020363321024022517rm_a_b ) ) )
      = ( X = Y ) ) ).

% in_set_simps(2)
thf(fact_717_in__set__simps_I2_J,axiom,
    ! [X: fOR_ftrs,Y: fOR_ftrs] :
      ( ( member_FOR_ftrs @ X @ ( set_FOR_ftrs2 @ ( cons_FOR_ftrs @ Y @ nil_FOR_ftrs ) ) )
      = ( X = Y ) ) ).

% in_set_simps(2)
thf(fact_718_length__Suc__conv,axiom,
    ! [Xs: list_f7931864845379430901rm_a_b,N: nat] :
      ( ( ( size_s1967422219310127201rm_a_b @ Xs )
        = ( suc @ N ) )
      = ( ? [Y2: fset_P3577670608877728751rm_a_b,Ys3: list_f7931864845379430901rm_a_b] :
            ( ( Xs
              = ( cons_f831858257744602021rm_a_b @ Y2 @ Ys3 ) )
            & ( ( size_s1967422219310127201rm_a_b @ Ys3 )
              = N ) ) ) ) ).

% length_Suc_conv
thf(fact_719_Suc__length__conv,axiom,
    ! [N: nat,Xs: list_f7931864845379430901rm_a_b] :
      ( ( ( suc @ N )
        = ( size_s1967422219310127201rm_a_b @ Xs ) )
      = ( ? [Y2: fset_P3577670608877728751rm_a_b,Ys3: list_f7931864845379430901rm_a_b] :
            ( ( Xs
              = ( cons_f831858257744602021rm_a_b @ Y2 @ Ys3 ) )
            & ( ( size_s1967422219310127201rm_a_b @ Ys3 )
              = N ) ) ) ) ).

% Suc_length_conv
thf(fact_720_length__nth__simps_I2_J,axiom,
    ! [X: fset_P3577670608877728751rm_a_b,Xs: list_f7931864845379430901rm_a_b] :
      ( ( size_s1967422219310127201rm_a_b @ ( cons_f831858257744602021rm_a_b @ X @ Xs ) )
      = ( suc @ ( size_s1967422219310127201rm_a_b @ Xs ) ) ) ).

% length_nth_simps(2)
thf(fact_721_set__subset__Cons,axiom,
    ! [Xs: list_FOR_ftrs,X: fOR_ftrs] : ( ord_le1454117246873690627R_ftrs @ ( set_FOR_ftrs2 @ Xs ) @ ( set_FOR_ftrs2 @ ( cons_FOR_ftrs @ X @ Xs ) ) ) ).

% set_subset_Cons
thf(fact_722_impossible__Cons,axiom,
    ! [Xs: list_f7931864845379430901rm_a_b,Ys2: list_f7931864845379430901rm_a_b,X: fset_P3577670608877728751rm_a_b] :
      ( ( ord_less_eq_nat @ ( size_s1967422219310127201rm_a_b @ Xs ) @ ( size_s1967422219310127201rm_a_b @ Ys2 ) )
     => ( Xs
       != ( cons_f831858257744602021rm_a_b @ X @ Ys2 ) ) ) ).

% impossible_Cons
thf(fact_723_int__of__nat__induct,axiom,
    ! [P: int > $o,Z4: int] :
      ( ! [N2: nat] : ( P @ ( semiri1314217659103216013at_int @ N2 ) )
     => ( ! [N2: nat] : ( P @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) )
       => ( P @ Z4 ) ) ) ).

% int_of_nat_induct
thf(fact_724_int__cases,axiom,
    ! [Z4: int] :
      ( ! [N2: nat] :
          ( Z4
         != ( semiri1314217659103216013at_int @ N2 ) )
     => ~ ! [N2: nat] :
            ( Z4
           != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ).

% int_cases
thf(fact_725_not__int__zless__negative,axiom,
    ! [N: nat,M: nat] :
      ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).

% not_int_zless_negative
thf(fact_726_less__list__code_I3_J,axiom,
    ! [X: nat,Xs: list_nat,Y: nat,Ys2: list_nat] :
      ( ( ord_less_list_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Ys2 ) )
      = ( ( ord_less_nat @ X @ Y )
        | ( ( X = Y )
          & ( ord_less_list_nat @ Xs @ Ys2 ) ) ) ) ).

% less_list_code(3)
thf(fact_727_less__list__code_I3_J,axiom,
    ! [X: int,Xs: list_int,Y: int,Ys2: list_int] :
      ( ( ord_less_list_int @ ( cons_int @ X @ Xs ) @ ( cons_int @ Y @ Ys2 ) )
      = ( ( ord_less_int @ X @ Y )
        | ( ( X = Y )
          & ( ord_less_list_int @ Xs @ Ys2 ) ) ) ) ).

% less_list_code(3)
thf(fact_728_less__list__code_I2_J,axiom,
    ! [X: set_Pr4386577575007340137rm_a_b,Xs: list_s5629610930313288943rm_a_b] : ( ord_le4636864222483984283rm_a_b @ nil_se7334864872594423151rm_a_b @ ( cons_s7847606222307733791rm_a_b @ X @ Xs ) ) ).

% less_list_code(2)
thf(fact_729_less__list__code_I2_J,axiom,
    ! [X: fset_P3577670608877728751rm_a_b,Xs: list_f7931864845379430901rm_a_b] : ( ord_le5822847225584068257rm_a_b @ nil_fs4020363321024022517rm_a_b @ ( cons_f831858257744602021rm_a_b @ X @ Xs ) ) ).

% less_list_code(2)
thf(fact_730_Nil__less__Cons,axiom,
    ! [A: set_Pr4386577575007340137rm_a_b,X: list_s5629610930313288943rm_a_b] : ( ord_le4636864222483984283rm_a_b @ nil_se7334864872594423151rm_a_b @ ( cons_s7847606222307733791rm_a_b @ A @ X ) ) ).

% Nil_less_Cons
thf(fact_731_Nil__less__Cons,axiom,
    ! [A: fset_P3577670608877728751rm_a_b,X: list_f7931864845379430901rm_a_b] : ( ord_le5822847225584068257rm_a_b @ nil_fs4020363321024022517rm_a_b @ ( cons_f831858257744602021rm_a_b @ A @ X ) ) ).

% Nil_less_Cons
thf(fact_732_distinct__adj__singleton,axiom,
    ! [X: set_Pr4386577575007340137rm_a_b] : ( distin160423016700913479rm_a_b @ ( cons_s7847606222307733791rm_a_b @ X @ nil_se7334864872594423151rm_a_b ) ) ).

% distinct_adj_singleton
thf(fact_733_distinct__adj__singleton,axiom,
    ! [X: fset_P3577670608877728751rm_a_b] : ( distin8264758955140711885rm_a_b @ ( cons_f831858257744602021rm_a_b @ X @ nil_fs4020363321024022517rm_a_b ) ) ).

% distinct_adj_singleton
thf(fact_734_less__eq__list__code_I3_J,axiom,
    ! [X: nat,Xs: list_nat,Y: nat,Ys2: list_nat] :
      ( ( ord_less_eq_list_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Ys2 ) )
      = ( ( ord_less_nat @ X @ Y )
        | ( ( X = Y )
          & ( ord_less_eq_list_nat @ Xs @ Ys2 ) ) ) ) ).

% less_eq_list_code(3)
thf(fact_735_less__eq__list__code_I3_J,axiom,
    ! [X: int,Xs: list_int,Y: int,Ys2: list_int] :
      ( ( ord_less_eq_list_int @ ( cons_int @ X @ Xs ) @ ( cons_int @ Y @ Ys2 ) )
      = ( ( ord_less_int @ X @ Y )
        | ( ( X = Y )
          & ( ord_less_eq_list_int @ Xs @ Ys2 ) ) ) ) ).

% less_eq_list_code(3)
thf(fact_736_less__eq__list__code_I1_J,axiom,
    ! [X: set_Pr4386577575007340137rm_a_b,Xs: list_s5629610930313288943rm_a_b] :
      ~ ( ord_le616080101116966031rm_a_b @ ( cons_s7847606222307733791rm_a_b @ X @ Xs ) @ nil_se7334864872594423151rm_a_b ) ).

% less_eq_list_code(1)
thf(fact_737_less__eq__list__code_I1_J,axiom,
    ! [X: fset_P3577670608877728751rm_a_b,Xs: list_f7931864845379430901rm_a_b] :
      ~ ( ord_le3592285473176203669rm_a_b @ ( cons_f831858257744602021rm_a_b @ X @ Xs ) @ nil_fs4020363321024022517rm_a_b ) ).

% less_eq_list_code(1)
thf(fact_738_int__cases4,axiom,
    ! [M: int] :
      ( ! [N2: nat] :
          ( M
         != ( semiri1314217659103216013at_int @ N2 ) )
     => ~ ! [N2: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N2 )
           => ( M
             != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).

% int_cases4
thf(fact_739_Suc__le__length__iff,axiom,
    ! [N: nat,Xs: list_f7931864845379430901rm_a_b] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_s1967422219310127201rm_a_b @ Xs ) )
      = ( ? [X3: fset_P3577670608877728751rm_a_b,Ys3: list_f7931864845379430901rm_a_b] :
            ( ( Xs
              = ( cons_f831858257744602021rm_a_b @ X3 @ Ys3 ) )
            & ( ord_less_eq_nat @ N @ ( size_s1967422219310127201rm_a_b @ Ys3 ) ) ) ) ) ).

% Suc_le_length_iff
thf(fact_740_int__zle__neg,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) )
      = ( ( N = zero_zero_nat )
        & ( M = zero_zero_nat ) ) ) ).

% int_zle_neg
thf(fact_741_nonpos__int__cases,axiom,
    ! [K: int] :
      ( ( ord_less_eq_int @ K @ zero_zero_int )
     => ~ ! [N2: nat] :
            ( K
           != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).

% nonpos_int_cases
thf(fact_742_negative__zle__0,axiom,
    ! [N: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ zero_zero_int ) ).

% negative_zle_0
thf(fact_743_subset__code_I2_J,axiom,
    ! [A2: set_Pr4386577575007340137rm_a_b,Ys2: list_P8875379029341186191rm_a_b] :
      ( ( ord_le118470702582115849rm_a_b @ A2 @ ( coset_9213161874213406204rm_a_b @ Ys2 ) )
      = ( ! [X3: produc357393685978478089rm_a_b] :
            ( ( member5869715511025134514rm_a_b @ X3 @ ( set_Pr2370880052973118494rm_a_b @ Ys2 ) )
           => ~ ( member5869715511025134514rm_a_b @ X3 @ A2 ) ) ) ) ).

% subset_code(2)
thf(fact_744_subset__code_I2_J,axiom,
    ! [A2: set_FOR_ftrs,Ys2: list_FOR_ftrs] :
      ( ( ord_le1454117246873690627R_ftrs @ A2 @ ( coset_FOR_ftrs @ Ys2 ) )
      = ( ! [X3: fOR_ftrs] :
            ( ( member_FOR_ftrs @ X3 @ ( set_FOR_ftrs2 @ Ys2 ) )
           => ~ ( member_FOR_ftrs @ X3 @ A2 ) ) ) ) ).

% subset_code(2)
thf(fact_745_min__list__Cons,axiom,
    ! [X: nat,Y: nat,Xs: list_nat,Ys2: list_nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ( size_size_list_nat @ Xs )
          = ( size_size_list_nat @ Ys2 ) )
       => ( ( ord_less_eq_nat @ ( missing_min_list_nat @ Xs ) @ ( missing_min_list_nat @ Ys2 ) )
         => ( ord_less_eq_nat @ ( missing_min_list_nat @ ( cons_nat @ X @ Xs ) ) @ ( missing_min_list_nat @ ( cons_nat @ Y @ Ys2 ) ) ) ) ) ) ).

% min_list_Cons
thf(fact_746_min__list__Cons,axiom,
    ! [X: int,Y: int,Xs: list_int,Ys2: list_int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( ( size_size_list_int @ Xs )
          = ( size_size_list_int @ Ys2 ) )
       => ( ( ord_less_eq_int @ ( missing_min_list_int @ Xs ) @ ( missing_min_list_int @ Ys2 ) )
         => ( ord_less_eq_int @ ( missing_min_list_int @ ( cons_int @ X @ Xs ) ) @ ( missing_min_list_int @ ( cons_int @ Y @ Ys2 ) ) ) ) ) ) ).

% min_list_Cons
thf(fact_747_remdups__adj__adjacent,axiom,
    ! [I4: nat,Xs: list_f7931864845379430901rm_a_b] :
      ( ( ord_less_nat @ ( suc @ I4 ) @ ( size_s1967422219310127201rm_a_b @ ( remdup2633883733652511275rm_a_b @ Xs ) ) )
     => ( ( nth_fs3949069584078227574rm_a_b @ ( remdup2633883733652511275rm_a_b @ Xs ) @ I4 )
       != ( nth_fs3949069584078227574rm_a_b @ ( remdup2633883733652511275rm_a_b @ Xs ) @ ( suc @ I4 ) ) ) ) ).

% remdups_adj_adjacent
thf(fact_748_int__cases3,axiom,
    ! [K: int] :
      ( ( K != zero_zero_int )
     => ( ! [N2: nat] :
            ( ( K
              = ( semiri1314217659103216013at_int @ N2 ) )
           => ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
       => ~ ! [N2: nat] :
              ( ( K
                = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) )
             => ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ) ).

% int_cases3
thf(fact_749_not__zle__0__negative,axiom,
    ! [N: nat] :
      ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) ) ).

% not_zle_0_negative
thf(fact_750_negD,axiom,
    ! [X: int] :
      ( ( ord_less_int @ X @ zero_zero_int )
     => ? [N2: nat] :
          ( X
          = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ).

% negD
thf(fact_751_negative__zless__0,axiom,
    ! [N: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ zero_zero_int ) ).

% negative_zless_0
thf(fact_752_mod__by__Suc__0,axiom,
    ! [M: nat] :
      ( ( modulo_modulo_nat @ M @ ( suc @ zero_zero_nat ) )
      = zero_zero_nat ) ).

% mod_by_Suc_0
thf(fact_753_mod__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ( modulo_modulo_nat @ M @ N )
        = M ) ) ).

% mod_less
thf(fact_754_mod__0,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% mod_0
thf(fact_755_mod__0,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% mod_0
thf(fact_756_mod__by__0,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ A @ zero_zero_nat )
      = A ) ).

% mod_by_0
thf(fact_757_mod__by__0,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ zero_zero_int )
      = A ) ).

% mod_by_0
thf(fact_758_mod__self,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ A @ A )
      = zero_zero_nat ) ).

% mod_self
thf(fact_759_mod__self,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ A )
      = zero_zero_int ) ).

% mod_self
thf(fact_760_zmod__int,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N ) )
      = ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).

% zmod_int
thf(fact_761_int__ops_I9_J,axiom,
    ! [A: nat,B: nat] :
      ( ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ A @ B ) )
      = ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).

% int_ops(9)
thf(fact_762_compl__coset,axiom,
    ! [Xs: list_FOR_ftrs] :
      ( ( uminus4397402216021623788R_ftrs @ ( coset_FOR_ftrs @ Xs ) )
      = ( set_FOR_ftrs2 @ Xs ) ) ).

% compl_coset
thf(fact_763_coset__def,axiom,
    ( coset_FOR_ftrs
    = ( ^ [Xs3: list_FOR_ftrs] : ( uminus4397402216021623788R_ftrs @ ( set_FOR_ftrs2 @ Xs3 ) ) ) ) ).

% coset_def
thf(fact_764_linorder__neqE__linordered__idom,axiom,
    ! [X: int,Y: int] :
      ( ( X != Y )
     => ( ~ ( ord_less_int @ X @ Y )
       => ( ord_less_int @ Y @ X ) ) ) ).

% linorder_neqE_linordered_idom
thf(fact_765_mod__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( ( ( suc @ ( modulo_modulo_nat @ M @ N ) )
          = N )
       => ( ( modulo_modulo_nat @ ( suc @ M ) @ N )
          = zero_zero_nat ) )
      & ( ( ( suc @ ( modulo_modulo_nat @ M @ N ) )
         != N )
       => ( ( modulo_modulo_nat @ ( suc @ M ) @ N )
          = ( suc @ ( modulo_modulo_nat @ M @ N ) ) ) ) ) ).

% mod_Suc
thf(fact_766_mod__less__divisor,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ord_less_nat @ ( modulo_modulo_nat @ M @ N ) @ N ) ) ).

% mod_less_divisor
thf(fact_767_mod__induct,axiom,
    ! [P: nat > $o,N: nat,P3: nat,M: nat] :
      ( ( P @ N )
     => ( ( ord_less_nat @ N @ P3 )
       => ( ( ord_less_nat @ M @ P3 )
         => ( ! [N2: nat] :
                ( ( ord_less_nat @ N2 @ P3 )
               => ( ( P @ N2 )
                 => ( P @ ( modulo_modulo_nat @ ( suc @ N2 ) @ P3 ) ) ) )
           => ( P @ M ) ) ) ) ) ).

% mod_induct
thf(fact_768_mod__le__divisor,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N ) @ N ) ) ).

% mod_le_divisor
thf(fact_769_bits__mod__0,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% bits_mod_0
thf(fact_770_bits__mod__0,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% bits_mod_0
thf(fact_771_list__length1E,axiom,
    ! [Xs: list_s5629610930313288943rm_a_b] :
      ( ( ( size_s3663623692424790363rm_a_b @ Xs )
        = ( suc @ zero_zero_nat ) )
     => ~ ! [X4: set_Pr4386577575007340137rm_a_b] :
            ( Xs
           != ( cons_s7847606222307733791rm_a_b @ X4 @ nil_se7334864872594423151rm_a_b ) ) ) ).

% list_length1E
thf(fact_772_list__length1E,axiom,
    ! [Xs: list_f7931864845379430901rm_a_b] :
      ( ( ( size_s1967422219310127201rm_a_b @ Xs )
        = ( suc @ zero_zero_nat ) )
     => ~ ! [X4: fset_P3577670608877728751rm_a_b] :
            ( Xs
           != ( cons_f831858257744602021rm_a_b @ X4 @ nil_fs4020363321024022517rm_a_b ) ) ) ).

% list_length1E
thf(fact_773_of__nat__mod,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri1316708129612266289at_nat @ ( modulo_modulo_nat @ M @ N ) )
      = ( modulo_modulo_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).

% of_nat_mod
thf(fact_774_of__nat__mod,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N ) )
      = ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).

% of_nat_mod
thf(fact_775_gcd__nat__induct,axiom,
    ! [P: nat > nat > $o,M: nat,N: nat] :
      ( ! [M4: nat] : ( P @ M4 @ zero_zero_nat )
     => ( ! [M4: nat,N2: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N2 )
           => ( ( P @ N2 @ ( modulo_modulo_nat @ M4 @ N2 ) )
             => ( P @ M4 @ N2 ) ) )
       => ( P @ M @ N ) ) ) ).

% gcd_nat_induct
thf(fact_776_subset__cons,axiom,
    ! [Z4: list_FOR_ftrs,Z6: list_FOR_ftrs,P3: fOR_ftrs] :
      ( ( ord_le1454117246873690627R_ftrs @ ( set_FOR_ftrs2 @ Z4 ) @ ( set_FOR_ftrs2 @ Z6 ) )
     => ( ord_le1454117246873690627R_ftrs @ ( set_FOR_ftrs2 @ ( cons_FOR_ftrs @ P3 @ Z4 ) ) @ ( set_FOR_ftrs2 @ ( cons_FOR_ftrs @ P3 @ Z6 ) ) ) ) ).

% subset_cons
thf(fact_777_length__Cons,axiom,
    ! [X: fset_P3577670608877728751rm_a_b,Xs: list_f7931864845379430901rm_a_b] :
      ( ( size_s1967422219310127201rm_a_b @ ( cons_f831858257744602021rm_a_b @ X @ Xs ) )
      = ( suc @ ( size_s1967422219310127201rm_a_b @ Xs ) ) ) ).

% length_Cons
thf(fact_778_n__lists__Nil,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( n_list6731552439858285943rm_a_b @ N @ nil_se7334864872594423151rm_a_b )
          = ( cons_l92702159317333157rm_a_b @ nil_se7334864872594423151rm_a_b @ nil_li1283984768393418997rm_a_b ) ) )
      & ( ( N != zero_zero_nat )
       => ( ( n_list6731552439858285943rm_a_b @ N @ nil_se7334864872594423151rm_a_b )
          = nil_li1283984768393418997rm_a_b ) ) ) ).

% n_lists_Nil
thf(fact_779_n__lists__Nil,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( n_list7988791480787244029rm_a_b @ N @ nil_fs4020363321024022517rm_a_b )
          = ( cons_l2079606646850324139rm_a_b @ nil_fs4020363321024022517rm_a_b @ nil_li2074289066168648443rm_a_b ) ) )
      & ( ( N != zero_zero_nat )
       => ( ( n_list7988791480787244029rm_a_b @ N @ nil_fs4020363321024022517rm_a_b )
          = nil_li2074289066168648443rm_a_b ) ) ) ).

% n_lists_Nil
thf(fact_780_length__n__lists__elem,axiom,
    ! [Ys2: list_f7931864845379430901rm_a_b,N: nat,Xs: list_f7931864845379430901rm_a_b] :
      ( ( member3062333693255306910rm_a_b @ Ys2 @ ( set_li8657584070121143050rm_a_b @ ( n_list7988791480787244029rm_a_b @ N @ Xs ) ) )
     => ( ( size_s1967422219310127201rm_a_b @ Ys2 )
        = N ) ) ).

% length_n_lists_elem
thf(fact_781_ord_Oremdups__sorted_Ocases,axiom,
    ! [X: list_s5629610930313288943rm_a_b] :
      ( ( X != nil_se7334864872594423151rm_a_b )
     => ( ! [X4: set_Pr4386577575007340137rm_a_b] :
            ( X
           != ( cons_s7847606222307733791rm_a_b @ X4 @ nil_se7334864872594423151rm_a_b ) )
       => ~ ! [X4: set_Pr4386577575007340137rm_a_b,Y4: set_Pr4386577575007340137rm_a_b,Xs2: list_s5629610930313288943rm_a_b] :
              ( X
             != ( cons_s7847606222307733791rm_a_b @ X4 @ ( cons_s7847606222307733791rm_a_b @ Y4 @ Xs2 ) ) ) ) ) ).

% ord.remdups_sorted.cases
thf(fact_782_ord_Oremdups__sorted_Ocases,axiom,
    ! [X: list_f7931864845379430901rm_a_b] :
      ( ( X != nil_fs4020363321024022517rm_a_b )
     => ( ! [X4: fset_P3577670608877728751rm_a_b] :
            ( X
           != ( cons_f831858257744602021rm_a_b @ X4 @ nil_fs4020363321024022517rm_a_b ) )
       => ~ ! [X4: fset_P3577670608877728751rm_a_b,Y4: fset_P3577670608877728751rm_a_b,Xs2: list_f7931864845379430901rm_a_b] :
              ( X
             != ( cons_f831858257744602021rm_a_b @ X4 @ ( cons_f831858257744602021rm_a_b @ Y4 @ Xs2 ) ) ) ) ) ).

% ord.remdups_sorted.cases
thf(fact_783_remdups__sorted_Ocases,axiom,
    ! [X: list_s5629610930313288943rm_a_b] :
      ( ( X != nil_se7334864872594423151rm_a_b )
     => ( ! [X4: set_Pr4386577575007340137rm_a_b] :
            ( X
           != ( cons_s7847606222307733791rm_a_b @ X4 @ nil_se7334864872594423151rm_a_b ) )
       => ~ ! [X4: set_Pr4386577575007340137rm_a_b,Y4: set_Pr4386577575007340137rm_a_b,Xs2: list_s5629610930313288943rm_a_b] :
              ( X
             != ( cons_s7847606222307733791rm_a_b @ X4 @ ( cons_s7847606222307733791rm_a_b @ Y4 @ Xs2 ) ) ) ) ) ).

% remdups_sorted.cases
thf(fact_784_remdups__sorted_Ocases,axiom,
    ! [X: list_f7931864845379430901rm_a_b] :
      ( ( X != nil_fs4020363321024022517rm_a_b )
     => ( ! [X4: fset_P3577670608877728751rm_a_b] :
            ( X
           != ( cons_f831858257744602021rm_a_b @ X4 @ nil_fs4020363321024022517rm_a_b ) )
       => ~ ! [X4: fset_P3577670608877728751rm_a_b,Y4: fset_P3577670608877728751rm_a_b,Xs2: list_f7931864845379430901rm_a_b] :
              ( X
             != ( cons_f831858257744602021rm_a_b @ X4 @ ( cons_f831858257744602021rm_a_b @ Y4 @ Xs2 ) ) ) ) ) ).

% remdups_sorted.cases
thf(fact_785_proper__intrvl_Oexhaustive_Ocases,axiom,
    ! [X: list_s5629610930313288943rm_a_b] :
      ( ( X != nil_se7334864872594423151rm_a_b )
     => ~ ! [X4: set_Pr4386577575007340137rm_a_b,Xs2: list_s5629610930313288943rm_a_b] :
            ( X
           != ( cons_s7847606222307733791rm_a_b @ X4 @ Xs2 ) ) ) ).

% proper_intrvl.exhaustive.cases
thf(fact_786_proper__intrvl_Oexhaustive_Ocases,axiom,
    ! [X: list_f7931864845379430901rm_a_b] :
      ( ( X != nil_fs4020363321024022517rm_a_b )
     => ~ ! [X4: fset_P3577670608877728751rm_a_b,Xs2: list_f7931864845379430901rm_a_b] :
            ( X
           != ( cons_f831858257744602021rm_a_b @ X4 @ Xs2 ) ) ) ).

% proper_intrvl.exhaustive.cases
thf(fact_787_n__lists_Osimps_I1_J,axiom,
    ! [Xs: list_s5629610930313288943rm_a_b] :
      ( ( n_list6731552439858285943rm_a_b @ zero_zero_nat @ Xs )
      = ( cons_l92702159317333157rm_a_b @ nil_se7334864872594423151rm_a_b @ nil_li1283984768393418997rm_a_b ) ) ).

% n_lists.simps(1)
thf(fact_788_n__lists_Osimps_I1_J,axiom,
    ! [Xs: list_f7931864845379430901rm_a_b] :
      ( ( n_list7988791480787244029rm_a_b @ zero_zero_nat @ Xs )
      = ( cons_l2079606646850324139rm_a_b @ nil_fs4020363321024022517rm_a_b @ nil_li2074289066168648443rm_a_b ) ) ).

% n_lists.simps(1)
thf(fact_789_list__of__permutation__element__n_Osimps_I1_J,axiom,
    ! [X: set_Pr4386577575007340137rm_a_b,L3: list_s5629610930313288943rm_a_b] :
      ( ( basic_1071435344846903220rm_a_b @ X @ zero_zero_nat @ L3 )
      = ( cons_l92702159317333157rm_a_b @ nil_se7334864872594423151rm_a_b @ nil_li1283984768393418997rm_a_b ) ) ).

% list_of_permutation_element_n.simps(1)
thf(fact_790_list__of__permutation__element__n_Osimps_I1_J,axiom,
    ! [X: fset_P3577670608877728751rm_a_b,L3: list_f7931864845379430901rm_a_b] :
      ( ( basic_205589215891509178rm_a_b @ X @ zero_zero_nat @ L3 )
      = ( cons_l2079606646850324139rm_a_b @ nil_fs4020363321024022517rm_a_b @ nil_li2074289066168648443rm_a_b ) ) ).

% list_of_permutation_element_n.simps(1)
thf(fact_791_add__elem__list__lists_Osimps_I1_J,axiom,
    ! [X: set_Pr4386577575007340137rm_a_b] :
      ( ( basic_6852907471535255442rm_a_b @ X @ nil_se7334864872594423151rm_a_b )
      = ( cons_l92702159317333157rm_a_b @ ( cons_s7847606222307733791rm_a_b @ X @ nil_se7334864872594423151rm_a_b ) @ nil_li1283984768393418997rm_a_b ) ) ).

% add_elem_list_lists.simps(1)
thf(fact_792_add__elem__list__lists_Osimps_I1_J,axiom,
    ! [X: fset_P3577670608877728751rm_a_b] :
      ( ( basic_5157831931618400408rm_a_b @ X @ nil_fs4020363321024022517rm_a_b )
      = ( cons_l2079606646850324139rm_a_b @ ( cons_f831858257744602021rm_a_b @ X @ nil_fs4020363321024022517rm_a_b ) @ nil_li2074289066168648443rm_a_b ) ) ).

% add_elem_list_lists.simps(1)
thf(fact_793_length__add__elem__list__lists,axiom,
    ! [Ys2: list_f7931864845379430901rm_a_b,X: fset_P3577670608877728751rm_a_b,Xs: list_f7931864845379430901rm_a_b] :
      ( ( member3062333693255306910rm_a_b @ Ys2 @ ( set_li8657584070121143050rm_a_b @ ( basic_5157831931618400408rm_a_b @ X @ Xs ) ) )
     => ( ( size_s1967422219310127201rm_a_b @ Ys2 )
        = ( suc @ ( size_s1967422219310127201rm_a_b @ Xs ) ) ) ) ).

% length_add_elem_list_lists
thf(fact_794_list__of__permutation__element__n_Oelims,axiom,
    ! [X: set_Pr4386577575007340137rm_a_b,Xa: nat,Xb: list_s5629610930313288943rm_a_b,Y: list_l6192929068897674997rm_a_b] :
      ( ( ( basic_1071435344846903220rm_a_b @ X @ Xa @ Xb )
        = Y )
     => ( ( ( Xa = zero_zero_nat )
         => ( Y
           != ( cons_l92702159317333157rm_a_b @ nil_se7334864872594423151rm_a_b @ nil_li1283984768393418997rm_a_b ) ) )
       => ~ ! [N2: nat] :
              ( ( Xa
                = ( suc @ N2 ) )
             => ( Y
               != ( concat3770752595006032944rm_a_b @ ( map_li5449371397125115274rm_a_b @ ( basic_6852907471535255442rm_a_b @ X ) @ ( n_list6731552439858285943rm_a_b @ N2 @ Xb ) ) ) ) ) ) ) ).

% list_of_permutation_element_n.elims
thf(fact_795_list__of__permutation__element__n_Oelims,axiom,
    ! [X: fset_P3577670608877728751rm_a_b,Xa: nat,Xb: list_f7931864845379430901rm_a_b,Y: list_l3960025731926578043rm_a_b] :
      ( ( ( basic_205589215891509178rm_a_b @ X @ Xa @ Xb )
        = Y )
     => ( ( ( Xa = zero_zero_nat )
         => ( Y
           != ( cons_l2079606646850324139rm_a_b @ nil_fs4020363321024022517rm_a_b @ nil_li2074289066168648443rm_a_b ) ) )
       => ~ ! [N2: nat] :
              ( ( Xa
                = ( suc @ N2 ) )
             => ( Y
               != ( concat7128574750897932470rm_a_b @ ( map_li8124276902986747786rm_a_b @ ( basic_5157831931618400408rm_a_b @ X ) @ ( n_list7988791480787244029rm_a_b @ N2 @ Xb ) ) ) ) ) ) ) ).

% list_of_permutation_element_n.elims
thf(fact_796_zip__induct,axiom,
    ! [P: list_s5629610930313288943rm_a_b > list_s5629610930313288943rm_a_b > $o,Xs: list_s5629610930313288943rm_a_b,Ys2: list_s5629610930313288943rm_a_b] :
      ( ! [X4: set_Pr4386577575007340137rm_a_b,Xs2: list_s5629610930313288943rm_a_b,Y4: set_Pr4386577575007340137rm_a_b,Ys4: list_s5629610930313288943rm_a_b] :
          ( ( P @ Xs2 @ Ys4 )
         => ( P @ ( cons_s7847606222307733791rm_a_b @ X4 @ Xs2 ) @ ( cons_s7847606222307733791rm_a_b @ Y4 @ Ys4 ) ) )
     => ( ! [X_12: list_s5629610930313288943rm_a_b] : ( P @ nil_se7334864872594423151rm_a_b @ X_12 )
       => ( ! [Xs2: list_s5629610930313288943rm_a_b] : ( P @ Xs2 @ nil_se7334864872594423151rm_a_b )
         => ( P @ Xs @ Ys2 ) ) ) ) ).

% zip_induct
thf(fact_797_zip__induct,axiom,
    ! [P: list_s5629610930313288943rm_a_b > list_f7931864845379430901rm_a_b > $o,Xs: list_s5629610930313288943rm_a_b,Ys2: list_f7931864845379430901rm_a_b] :
      ( ! [X4: set_Pr4386577575007340137rm_a_b,Xs2: list_s5629610930313288943rm_a_b,Y4: fset_P3577670608877728751rm_a_b,Ys4: list_f7931864845379430901rm_a_b] :
          ( ( P @ Xs2 @ Ys4 )
         => ( P @ ( cons_s7847606222307733791rm_a_b @ X4 @ Xs2 ) @ ( cons_f831858257744602021rm_a_b @ Y4 @ Ys4 ) ) )
     => ( ! [X_12: list_f7931864845379430901rm_a_b] : ( P @ nil_se7334864872594423151rm_a_b @ X_12 )
       => ( ! [Xs2: list_s5629610930313288943rm_a_b] : ( P @ Xs2 @ nil_fs4020363321024022517rm_a_b )
         => ( P @ Xs @ Ys2 ) ) ) ) ).

% zip_induct
thf(fact_798_zip__induct,axiom,
    ! [P: list_f7931864845379430901rm_a_b > list_s5629610930313288943rm_a_b > $o,Xs: list_f7931864845379430901rm_a_b,Ys2: list_s5629610930313288943rm_a_b] :
      ( ! [X4: fset_P3577670608877728751rm_a_b,Xs2: list_f7931864845379430901rm_a_b,Y4: set_Pr4386577575007340137rm_a_b,Ys4: list_s5629610930313288943rm_a_b] :
          ( ( P @ Xs2 @ Ys4 )
         => ( P @ ( cons_f831858257744602021rm_a_b @ X4 @ Xs2 ) @ ( cons_s7847606222307733791rm_a_b @ Y4 @ Ys4 ) ) )
     => ( ! [X_12: list_s5629610930313288943rm_a_b] : ( P @ nil_fs4020363321024022517rm_a_b @ X_12 )
       => ( ! [Xs2: list_f7931864845379430901rm_a_b] : ( P @ Xs2 @ nil_se7334864872594423151rm_a_b )
         => ( P @ Xs @ Ys2 ) ) ) ) ).

% zip_induct
thf(fact_799_zip__induct,axiom,
    ! [P: list_f7931864845379430901rm_a_b > list_f7931864845379430901rm_a_b > $o,Xs: list_f7931864845379430901rm_a_b,Ys2: list_f7931864845379430901rm_a_b] :
      ( ! [X4: fset_P3577670608877728751rm_a_b,Xs2: list_f7931864845379430901rm_a_b,Y4: fset_P3577670608877728751rm_a_b,Ys4: list_f7931864845379430901rm_a_b] :
          ( ( P @ Xs2 @ Ys4 )
         => ( P @ ( cons_f831858257744602021rm_a_b @ X4 @ Xs2 ) @ ( cons_f831858257744602021rm_a_b @ Y4 @ Ys4 ) ) )
     => ( ! [X_12: list_f7931864845379430901rm_a_b] : ( P @ nil_fs4020363321024022517rm_a_b @ X_12 )
       => ( ! [Xs2: list_f7931864845379430901rm_a_b] : ( P @ Xs2 @ nil_fs4020363321024022517rm_a_b )
         => ( P @ Xs @ Ys2 ) ) ) ) ).

% zip_induct
thf(fact_800_concat__eq__Nil__conv,axiom,
    ! [Xss2: list_l6192929068897674997rm_a_b] :
      ( ( ( concat3969203760323611178rm_a_b @ Xss2 )
        = nil_se7334864872594423151rm_a_b )
      = ( ! [X3: list_s5629610930313288943rm_a_b] :
            ( ( member857836065738737176rm_a_b @ X3 @ ( set_li7688249504041543812rm_a_b @ Xss2 ) )
           => ( X3 = nil_se7334864872594423151rm_a_b ) ) ) ) ).

% concat_eq_Nil_conv
thf(fact_801_concat__eq__Nil__conv,axiom,
    ! [Xss2: list_l3960025731926578043rm_a_b] :
      ( ( ( concat8439565143799850800rm_a_b @ Xss2 )
        = nil_fs4020363321024022517rm_a_b )
      = ( ! [X3: list_f7931864845379430901rm_a_b] :
            ( ( member3062333693255306910rm_a_b @ X3 @ ( set_li8657584070121143050rm_a_b @ Xss2 ) )
           => ( X3 = nil_fs4020363321024022517rm_a_b ) ) ) ) ).

% concat_eq_Nil_conv
thf(fact_802_Nil__eq__concat__conv,axiom,
    ! [Xss2: list_l6192929068897674997rm_a_b] :
      ( ( nil_se7334864872594423151rm_a_b
        = ( concat3969203760323611178rm_a_b @ Xss2 ) )
      = ( ! [X3: list_s5629610930313288943rm_a_b] :
            ( ( member857836065738737176rm_a_b @ X3 @ ( set_li7688249504041543812rm_a_b @ Xss2 ) )
           => ( X3 = nil_se7334864872594423151rm_a_b ) ) ) ) ).

% Nil_eq_concat_conv
thf(fact_803_Nil__eq__concat__conv,axiom,
    ! [Xss2: list_l3960025731926578043rm_a_b] :
      ( ( nil_fs4020363321024022517rm_a_b
        = ( concat8439565143799850800rm_a_b @ Xss2 ) )
      = ( ! [X3: list_f7931864845379430901rm_a_b] :
            ( ( member3062333693255306910rm_a_b @ X3 @ ( set_li8657584070121143050rm_a_b @ Xss2 ) )
           => ( X3 = nil_fs4020363321024022517rm_a_b ) ) ) ) ).

% Nil_eq_concat_conv
thf(fact_804_map__concat,axiom,
    ! [F: fset_P3577670608877728751rm_a_b > set_Pr4386577575007340137rm_a_b,Xs: list_l3960025731926578043rm_a_b] :
      ( ( map_fs2147491833908496126rm_a_b @ F @ ( concat8439565143799850800rm_a_b @ Xs ) )
      = ( concat3969203760323611178rm_a_b @ ( map_li1380862427395245054rm_a_b @ ( map_fs2147491833908496126rm_a_b @ F ) @ Xs ) ) ) ).

% map_concat
thf(fact_805_eq__length__concat__nth,axiom,
    ! [Xs: list_l3960025731926578043rm_a_b,Ys2: list_l3960025731926578043rm_a_b] :
      ( ( ( size_s7600236048634125287rm_a_b @ Xs )
        = ( size_s7600236048634125287rm_a_b @ Ys2 ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s7600236048634125287rm_a_b @ Xs ) )
           => ( ( size_s1967422219310127201rm_a_b @ ( nth_li947802356562890492rm_a_b @ Xs @ I ) )
              = ( size_s1967422219310127201rm_a_b @ ( nth_li947802356562890492rm_a_b @ Ys2 @ I ) ) ) )
       => ( ( size_s1967422219310127201rm_a_b @ ( concat8439565143799850800rm_a_b @ Xs ) )
          = ( size_s1967422219310127201rm_a_b @ ( concat8439565143799850800rm_a_b @ Ys2 ) ) ) ) ) ).

% eq_length_concat_nth
thf(fact_806_concat_Osimps_I1_J,axiom,
    ( ( concat3969203760323611178rm_a_b @ nil_li1283984768393418997rm_a_b )
    = nil_se7334864872594423151rm_a_b ) ).

% concat.simps(1)
thf(fact_807_concat_Osimps_I1_J,axiom,
    ( ( concat8439565143799850800rm_a_b @ nil_li2074289066168648443rm_a_b )
    = nil_fs4020363321024022517rm_a_b ) ).

% concat.simps(1)
thf(fact_808_concat__all__nth,axiom,
    ! [Xs: list_l3960025731926578043rm_a_b,Ys2: list_l3960025731926578043rm_a_b,P: fset_P3577670608877728751rm_a_b > fset_P3577670608877728751rm_a_b > $o] :
      ( ( ( size_s7600236048634125287rm_a_b @ Xs )
        = ( size_s7600236048634125287rm_a_b @ Ys2 ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s7600236048634125287rm_a_b @ Xs ) )
           => ( ( size_s1967422219310127201rm_a_b @ ( nth_li947802356562890492rm_a_b @ Xs @ I ) )
              = ( size_s1967422219310127201rm_a_b @ ( nth_li947802356562890492rm_a_b @ Ys2 @ I ) ) ) )
       => ( ! [I: nat,J2: nat] :
              ( ( ord_less_nat @ I @ ( size_s7600236048634125287rm_a_b @ Xs ) )
             => ( ( ord_less_nat @ J2 @ ( size_s1967422219310127201rm_a_b @ ( nth_li947802356562890492rm_a_b @ Xs @ I ) ) )
               => ( P @ ( nth_fs3949069584078227574rm_a_b @ ( nth_li947802356562890492rm_a_b @ Xs @ I ) @ J2 ) @ ( nth_fs3949069584078227574rm_a_b @ ( nth_li947802356562890492rm_a_b @ Ys2 @ I ) @ J2 ) ) ) )
         => ! [K3: nat] :
              ( ( ord_less_nat @ K3 @ ( size_s1967422219310127201rm_a_b @ ( concat8439565143799850800rm_a_b @ Xs ) ) )
             => ( P @ ( nth_fs3949069584078227574rm_a_b @ ( concat8439565143799850800rm_a_b @ Xs ) @ K3 ) @ ( nth_fs3949069584078227574rm_a_b @ ( concat8439565143799850800rm_a_b @ Ys2 ) @ K3 ) ) ) ) ) ) ).

% concat_all_nth
thf(fact_809_nth__concat__split,axiom,
    ! [I4: nat,Xs: list_l3960025731926578043rm_a_b] :
      ( ( ord_less_nat @ I4 @ ( size_s1967422219310127201rm_a_b @ ( concat8439565143799850800rm_a_b @ Xs ) ) )
     => ? [J2: nat,K2: nat] :
          ( ( ord_less_nat @ J2 @ ( size_s7600236048634125287rm_a_b @ Xs ) )
          & ( ord_less_nat @ K2 @ ( size_s1967422219310127201rm_a_b @ ( nth_li947802356562890492rm_a_b @ Xs @ J2 ) ) )
          & ( ( nth_fs3949069584078227574rm_a_b @ ( concat8439565143799850800rm_a_b @ Xs ) @ I4 )
            = ( nth_fs3949069584078227574rm_a_b @ ( nth_li947802356562890492rm_a_b @ Xs @ J2 ) @ K2 ) ) ) ) ).

% nth_concat_split
thf(fact_810_nth__concat__two__lists,axiom,
    ! [I4: nat,Xs: list_l3960025731926578043rm_a_b,Ys2: list_l3960025731926578043rm_a_b] :
      ( ( ord_less_nat @ I4 @ ( size_s1967422219310127201rm_a_b @ ( concat8439565143799850800rm_a_b @ Xs ) ) )
     => ( ( ( size_s7600236048634125287rm_a_b @ Ys2 )
          = ( size_s7600236048634125287rm_a_b @ Xs ) )
       => ( ! [I: nat] :
              ( ( ord_less_nat @ I @ ( size_s7600236048634125287rm_a_b @ Xs ) )
             => ( ( size_s1967422219310127201rm_a_b @ ( nth_li947802356562890492rm_a_b @ Ys2 @ I ) )
                = ( size_s1967422219310127201rm_a_b @ ( nth_li947802356562890492rm_a_b @ Xs @ I ) ) ) )
         => ? [J2: nat,K2: nat] :
              ( ( ord_less_nat @ J2 @ ( size_s7600236048634125287rm_a_b @ Xs ) )
              & ( ord_less_nat @ K2 @ ( size_s1967422219310127201rm_a_b @ ( nth_li947802356562890492rm_a_b @ Xs @ J2 ) ) )
              & ( ( nth_fs3949069584078227574rm_a_b @ ( concat8439565143799850800rm_a_b @ Xs ) @ I4 )
                = ( nth_fs3949069584078227574rm_a_b @ ( nth_li947802356562890492rm_a_b @ Xs @ J2 ) @ K2 ) )
              & ( ( nth_fs3949069584078227574rm_a_b @ ( concat8439565143799850800rm_a_b @ Ys2 ) @ I4 )
                = ( nth_fs3949069584078227574rm_a_b @ ( nth_li947802356562890492rm_a_b @ Ys2 @ J2 ) @ K2 ) ) ) ) ) ) ).

% nth_concat_two_lists
thf(fact_811_concat__nth__nthI,axiom,
    ! [Ss: list_l3960025731926578043rm_a_b,Ts: list_l3960025731926578043rm_a_b,P: fset_P3577670608877728751rm_a_b > fset_P3577670608877728751rm_a_b > $o] :
      ( ( ( size_s7600236048634125287rm_a_b @ Ss )
        = ( size_s7600236048634125287rm_a_b @ Ts ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s7600236048634125287rm_a_b @ Ts ) )
           => ( ( size_s1967422219310127201rm_a_b @ ( nth_li947802356562890492rm_a_b @ Ss @ I ) )
              = ( size_s1967422219310127201rm_a_b @ ( nth_li947802356562890492rm_a_b @ Ts @ I ) ) ) )
       => ( ! [I: nat] :
              ( ( ord_less_nat @ I @ ( size_s7600236048634125287rm_a_b @ Ts ) )
             => ! [J2: nat] :
                  ( ( ord_less_nat @ J2 @ ( size_s1967422219310127201rm_a_b @ ( nth_li947802356562890492rm_a_b @ Ts @ I ) ) )
                 => ( P @ ( nth_fs3949069584078227574rm_a_b @ ( nth_li947802356562890492rm_a_b @ Ss @ I ) @ J2 ) @ ( nth_fs3949069584078227574rm_a_b @ ( nth_li947802356562890492rm_a_b @ Ts @ I ) @ J2 ) ) ) )
         => ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_s1967422219310127201rm_a_b @ ( concat8439565143799850800rm_a_b @ Ts ) ) )
             => ( P @ ( nth_fs3949069584078227574rm_a_b @ ( concat8439565143799850800rm_a_b @ Ss ) @ I2 ) @ ( nth_fs3949069584078227574rm_a_b @ ( concat8439565143799850800rm_a_b @ Ts ) @ I2 ) ) ) ) ) ) ).

% concat_nth_nthI
thf(fact_812_product__lists_Osimps_I1_J,axiom,
    ( ( produc1542182858532876248rm_a_b @ nil_li1283984768393418997rm_a_b )
    = ( cons_l92702159317333157rm_a_b @ nil_se7334864872594423151rm_a_b @ nil_li1283984768393418997rm_a_b ) ) ).

% product_lists.simps(1)
thf(fact_813_product__lists_Osimps_I1_J,axiom,
    ( ( produc4235919412562105822rm_a_b @ nil_li2074289066168648443rm_a_b )
    = ( cons_l2079606646850324139rm_a_b @ nil_fs4020363321024022517rm_a_b @ nil_li2074289066168648443rm_a_b ) ) ).

% product_lists.simps(1)
thf(fact_814_concat__lists_Osimps_I1_J,axiom,
    ( ( missin4740983367183299007rm_a_b @ nil_li1283984768393418997rm_a_b )
    = ( cons_l92702159317333157rm_a_b @ nil_se7334864872594423151rm_a_b @ nil_li1283984768393418997rm_a_b ) ) ).

% concat_lists.simps(1)
thf(fact_815_concat__lists_Osimps_I1_J,axiom,
    ( ( missin837033523393336389rm_a_b @ nil_li2074289066168648443rm_a_b )
    = ( cons_l2079606646850324139rm_a_b @ nil_fs4020363321024022517rm_a_b @ nil_li2074289066168648443rm_a_b ) ) ).

% concat_lists.simps(1)
thf(fact_816_in__set__product__lists__length,axiom,
    ! [Xs: list_f7931864845379430901rm_a_b,Xss2: list_l3960025731926578043rm_a_b] :
      ( ( member3062333693255306910rm_a_b @ Xs @ ( set_li8657584070121143050rm_a_b @ ( produc4235919412562105822rm_a_b @ Xss2 ) ) )
     => ( ( size_s1967422219310127201rm_a_b @ Xs )
        = ( size_s7600236048634125287rm_a_b @ Xss2 ) ) ) ).

% in_set_product_lists_length
thf(fact_817_concat__lists__listset,axiom,
    ! [Xs: list_list_FOR_ftrs] :
      ( ( set_list_FOR_ftrs2 @ ( missin6446604053780320919R_ftrs @ Xs ) )
      = ( listset_FOR_ftrs @ ( map_li5604343147748686310R_ftrs @ set_FOR_ftrs2 @ Xs ) ) ) ).

% concat_lists_listset
thf(fact_818_nth__concat__diff,axiom,
    ! [I1: nat,Xs: list_l3960025731926578043rm_a_b,I22: nat] :
      ( ( ord_less_nat @ I1 @ ( size_s1967422219310127201rm_a_b @ ( concat8439565143799850800rm_a_b @ Xs ) ) )
     => ( ( ord_less_nat @ I22 @ ( size_s1967422219310127201rm_a_b @ ( concat8439565143799850800rm_a_b @ Xs ) ) )
       => ( ( I1 != I22 )
         => ? [J1: nat,K1: nat,J22: nat,K22: nat] :
              ( ( ( product_Pair_nat_nat @ J1 @ K1 )
               != ( product_Pair_nat_nat @ J22 @ K22 ) )
              & ( ord_less_nat @ J1 @ ( size_s7600236048634125287rm_a_b @ Xs ) )
              & ( ord_less_nat @ J22 @ ( size_s7600236048634125287rm_a_b @ Xs ) )
              & ( ord_less_nat @ K1 @ ( size_s1967422219310127201rm_a_b @ ( nth_li947802356562890492rm_a_b @ Xs @ J1 ) ) )
              & ( ord_less_nat @ K22 @ ( size_s1967422219310127201rm_a_b @ ( nth_li947802356562890492rm_a_b @ Xs @ J22 ) ) )
              & ( ( nth_fs3949069584078227574rm_a_b @ ( concat8439565143799850800rm_a_b @ Xs ) @ I1 )
                = ( nth_fs3949069584078227574rm_a_b @ ( nth_li947802356562890492rm_a_b @ Xs @ J1 ) @ K1 ) )
              & ( ( nth_fs3949069584078227574rm_a_b @ ( concat8439565143799850800rm_a_b @ Xs ) @ I22 )
                = ( nth_fs3949069584078227574rm_a_b @ ( nth_li947802356562890492rm_a_b @ Xs @ J22 ) @ K22 ) ) ) ) ) ) ).

% nth_concat_diff
thf(fact_819_nth__drop__0,axiom,
    ! [Ss: list_f7931864845379430901rm_a_b] :
      ( ( ord_less_nat @ zero_zero_nat @ ( size_s1967422219310127201rm_a_b @ Ss ) )
     => ( ( cons_f831858257744602021rm_a_b @ ( nth_fs3949069584078227574rm_a_b @ Ss @ zero_zero_nat ) @ ( drop_f1037359496611548469rm_a_b @ ( suc @ zero_zero_nat ) @ Ss ) )
        = Ss ) ) ).

% nth_drop_0
thf(fact_820_ImageI,axiom,
    ! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,R3: set_Pr2972776593051762503rm_a_b,A2: set_Pr4386577575007340137rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ R3 )
     => ( ( member5869715511025134514rm_a_b @ A @ A2 )
       => ( member5869715511025134514rm_a_b @ B @ ( image_871640889055226643rm_a_b @ R3 @ A2 ) ) ) ) ).

% ImageI
thf(fact_821_ImageI,axiom,
    ! [A: term_a_b,B: term_a_b,R3: set_Pr4386577575007340137rm_a_b,A2: set_term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R3 )
     => ( ( member_term_a_b @ A @ A2 )
       => ( member_term_a_b @ B @ ( image_6923787618150785661rm_a_b @ R3 @ A2 ) ) ) ) ).

% ImageI
thf(fact_822_ImageI,axiom,
    ! [A: nat > nat,B: nat,R3: set_Pr9093778441882193744at_nat,A2: set_nat_nat] :
      ( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ A @ B ) @ R3 )
     => ( ( member_nat_nat @ A @ A2 )
       => ( member_nat @ B @ ( image_nat_nat_nat @ R3 @ A2 ) ) ) ) ).

% ImageI
thf(fact_823_Id__onI,axiom,
    ! [A: term_a_b,A2: set_term_a_b] :
      ( ( member_term_a_b @ A @ A2 )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ A ) @ ( id_on_term_a_b @ A2 ) ) ) ).

% Id_onI
thf(fact_824_Id__onI,axiom,
    ! [A: produc357393685978478089rm_a_b,A2: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ A @ A2 )
     => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ A ) @ ( id_on_115818524817812774rm_a_b @ A2 ) ) ) ).

% Id_onI
thf(fact_825_drop__eq__Nil2,axiom,
    ! [N: nat,Xs: list_s5629610930313288943rm_a_b] :
      ( ( nil_se7334864872594423151rm_a_b
        = ( drop_s8295709470917206703rm_a_b @ N @ Xs ) )
      = ( ord_less_eq_nat @ ( size_s3663623692424790363rm_a_b @ Xs ) @ N ) ) ).

% drop_eq_Nil2
thf(fact_826_drop__eq__Nil2,axiom,
    ! [N: nat,Xs: list_f7931864845379430901rm_a_b] :
      ( ( nil_fs4020363321024022517rm_a_b
        = ( drop_f1037359496611548469rm_a_b @ N @ Xs ) )
      = ( ord_less_eq_nat @ ( size_s1967422219310127201rm_a_b @ Xs ) @ N ) ) ).

% drop_eq_Nil2
thf(fact_827_drop__eq__Nil,axiom,
    ! [N: nat,Xs: list_s5629610930313288943rm_a_b] :
      ( ( ( drop_s8295709470917206703rm_a_b @ N @ Xs )
        = nil_se7334864872594423151rm_a_b )
      = ( ord_less_eq_nat @ ( size_s3663623692424790363rm_a_b @ Xs ) @ N ) ) ).

% drop_eq_Nil
thf(fact_828_drop__eq__Nil,axiom,
    ! [N: nat,Xs: list_f7931864845379430901rm_a_b] :
      ( ( ( drop_f1037359496611548469rm_a_b @ N @ Xs )
        = nil_fs4020363321024022517rm_a_b )
      = ( ord_less_eq_nat @ ( size_s1967422219310127201rm_a_b @ Xs ) @ N ) ) ).

% drop_eq_Nil
thf(fact_829_drop__all,axiom,
    ! [Xs: list_s5629610930313288943rm_a_b,N: nat] :
      ( ( ord_less_eq_nat @ ( size_s3663623692424790363rm_a_b @ Xs ) @ N )
     => ( ( drop_s8295709470917206703rm_a_b @ N @ Xs )
        = nil_se7334864872594423151rm_a_b ) ) ).

% drop_all
thf(fact_830_drop__all,axiom,
    ! [Xs: list_f7931864845379430901rm_a_b,N: nat] :
      ( ( ord_less_eq_nat @ ( size_s1967422219310127201rm_a_b @ Xs ) @ N )
     => ( ( drop_f1037359496611548469rm_a_b @ N @ Xs )
        = nil_fs4020363321024022517rm_a_b ) ) ).

% drop_all
thf(fact_831_drop__map,axiom,
    ! [N: nat,F: fset_P3577670608877728751rm_a_b > set_Pr4386577575007340137rm_a_b,Xs: list_f7931864845379430901rm_a_b] :
      ( ( drop_s8295709470917206703rm_a_b @ N @ ( map_fs2147491833908496126rm_a_b @ F @ Xs ) )
      = ( map_fs2147491833908496126rm_a_b @ F @ ( drop_f1037359496611548469rm_a_b @ N @ Xs ) ) ) ).

% drop_map
thf(fact_832_in__set__dropD,axiom,
    ! [X: produc357393685978478089rm_a_b,N: nat,Xs: list_P8875379029341186191rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ X @ ( set_Pr2370880052973118494rm_a_b @ ( drop_P2146130716461001295rm_a_b @ N @ Xs ) ) )
     => ( member5869715511025134514rm_a_b @ X @ ( set_Pr2370880052973118494rm_a_b @ Xs ) ) ) ).

% in_set_dropD
thf(fact_833_in__set__dropD,axiom,
    ! [X: fOR_ftrs,N: nat,Xs: list_FOR_ftrs] :
      ( ( member_FOR_ftrs @ X @ ( set_FOR_ftrs2 @ ( drop_FOR_ftrs @ N @ Xs ) ) )
     => ( member_FOR_ftrs @ X @ ( set_FOR_ftrs2 @ Xs ) ) ) ).

% in_set_dropD
thf(fact_834_drop__Nil,axiom,
    ! [N: nat] :
      ( ( drop_s8295709470917206703rm_a_b @ N @ nil_se7334864872594423151rm_a_b )
      = nil_se7334864872594423151rm_a_b ) ).

% drop_Nil
thf(fact_835_drop__Nil,axiom,
    ! [N: nat] :
      ( ( drop_f1037359496611548469rm_a_b @ N @ nil_fs4020363321024022517rm_a_b )
      = nil_fs4020363321024022517rm_a_b ) ).

% drop_Nil
thf(fact_836_Relation_OImageE,axiom,
    ! [B: produc357393685978478089rm_a_b,R3: set_Pr2972776593051762503rm_a_b,A2: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ B @ ( image_871640889055226643rm_a_b @ R3 @ A2 ) )
     => ~ ! [X4: produc357393685978478089rm_a_b] :
            ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X4 @ B ) @ R3 )
           => ~ ( member5869715511025134514rm_a_b @ X4 @ A2 ) ) ) ).

% Relation.ImageE
thf(fact_837_Relation_OImageE,axiom,
    ! [B: term_a_b,R3: set_Pr4386577575007340137rm_a_b,A2: set_term_a_b] :
      ( ( member_term_a_b @ B @ ( image_6923787618150785661rm_a_b @ R3 @ A2 ) )
     => ~ ! [X4: term_a_b] :
            ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ B ) @ R3 )
           => ~ ( member_term_a_b @ X4 @ A2 ) ) ) ).

% Relation.ImageE
thf(fact_838_Relation_OImageE,axiom,
    ! [B: nat,R3: set_Pr9093778441882193744at_nat,A2: set_nat_nat] :
      ( ( member_nat @ B @ ( image_nat_nat_nat @ R3 @ A2 ) )
     => ~ ! [X4: nat > nat] :
            ( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ X4 @ B ) @ R3 )
           => ~ ( member_nat_nat @ X4 @ A2 ) ) ) ).

% Relation.ImageE
thf(fact_839_Image__iff,axiom,
    ! [B: produc357393685978478089rm_a_b,R3: set_Pr2972776593051762503rm_a_b,A2: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ B @ ( image_871640889055226643rm_a_b @ R3 @ A2 ) )
      = ( ? [X3: produc357393685978478089rm_a_b] :
            ( ( member5869715511025134514rm_a_b @ X3 @ A2 )
            & ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ B ) @ R3 ) ) ) ) ).

% Image_iff
thf(fact_840_Image__iff,axiom,
    ! [B: term_a_b,R3: set_Pr4386577575007340137rm_a_b,A2: set_term_a_b] :
      ( ( member_term_a_b @ B @ ( image_6923787618150785661rm_a_b @ R3 @ A2 ) )
      = ( ? [X3: term_a_b] :
            ( ( member_term_a_b @ X3 @ A2 )
            & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ B ) @ R3 ) ) ) ) ).

% Image_iff
thf(fact_841_Image__iff,axiom,
    ! [B: nat,R3: set_Pr9093778441882193744at_nat,A2: set_nat_nat] :
      ( ( member_nat @ B @ ( image_nat_nat_nat @ R3 @ A2 ) )
      = ( ? [X3: nat > nat] :
            ( ( member_nat_nat @ X3 @ A2 )
            & ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ X3 @ B ) @ R3 ) ) ) ) ).

% Image_iff
thf(fact_842_rev__ImageI,axiom,
    ! [A: produc357393685978478089rm_a_b,A2: set_Pr4386577575007340137rm_a_b,B: produc357393685978478089rm_a_b,R3: set_Pr2972776593051762503rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ A @ A2 )
     => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ R3 )
       => ( member5869715511025134514rm_a_b @ B @ ( image_871640889055226643rm_a_b @ R3 @ A2 ) ) ) ) ).

% rev_ImageI
thf(fact_843_rev__ImageI,axiom,
    ! [A: term_a_b,A2: set_term_a_b,B: term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
      ( ( member_term_a_b @ A @ A2 )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R3 )
       => ( member_term_a_b @ B @ ( image_6923787618150785661rm_a_b @ R3 @ A2 ) ) ) ) ).

% rev_ImageI
thf(fact_844_rev__ImageI,axiom,
    ! [A: nat > nat,A2: set_nat_nat,B: nat,R3: set_Pr9093778441882193744at_nat] :
      ( ( member_nat_nat @ A @ A2 )
     => ( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ A @ B ) @ R3 )
       => ( member_nat @ B @ ( image_nat_nat_nat @ R3 @ A2 ) ) ) ) ).

% rev_ImageI
thf(fact_845_Id__onE,axiom,
    ! [C2: produc357393685978478089rm_a_b,A2: set_term_a_b] :
      ( ( member5869715511025134514rm_a_b @ C2 @ ( id_on_term_a_b @ A2 ) )
     => ~ ! [X4: term_a_b] :
            ( ( member_term_a_b @ X4 @ A2 )
           => ( C2
             != ( produc7020197800436672577rm_a_b @ X4 @ X4 ) ) ) ) ).

% Id_onE
thf(fact_846_Id__onE,axiom,
    ! [C2: produc4523971788519308903rm_a_b,A2: set_Pr4386577575007340137rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ C2 @ ( id_on_115818524817812774rm_a_b @ A2 ) )
     => ~ ! [X4: produc357393685978478089rm_a_b] :
            ( ( member5869715511025134514rm_a_b @ X4 @ A2 )
           => ( C2
             != ( produc1763473618796451543rm_a_b @ X4 @ X4 ) ) ) ) ).

% Id_onE
thf(fact_847_Id__on__eqI,axiom,
    ! [A: term_a_b,B: term_a_b,A2: set_term_a_b] :
      ( ( A = B )
     => ( ( member_term_a_b @ A @ A2 )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( id_on_term_a_b @ A2 ) ) ) ) ).

% Id_on_eqI
thf(fact_848_Id__on__eqI,axiom,
    ! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,A2: set_Pr4386577575007340137rm_a_b] :
      ( ( A = B )
     => ( ( member5869715511025134514rm_a_b @ A @ A2 )
       => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( id_on_115818524817812774rm_a_b @ A2 ) ) ) ) ).

% Id_on_eqI
thf(fact_849_Id__on__iff,axiom,
    ! [X: term_a_b,Y: term_a_b,A2: set_term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( id_on_term_a_b @ A2 ) )
      = ( ( X = Y )
        & ( member_term_a_b @ X @ A2 ) ) ) ).

% Id_on_iff
thf(fact_850_Id__on__iff,axiom,
    ! [X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,A2: set_Pr4386577575007340137rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( id_on_115818524817812774rm_a_b @ A2 ) )
      = ( ( X = Y )
        & ( member5869715511025134514rm_a_b @ X @ A2 ) ) ) ).

% Id_on_iff
thf(fact_851_set__drop__subset,axiom,
    ! [N: nat,Xs: list_FOR_ftrs] : ( ord_le1454117246873690627R_ftrs @ ( set_FOR_ftrs2 @ ( drop_FOR_ftrs @ N @ Xs ) ) @ ( set_FOR_ftrs2 @ Xs ) ) ).

% set_drop_subset
thf(fact_852_set__drop__subset__set__drop,axiom,
    ! [N: nat,M: nat,Xs: list_FOR_ftrs] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ord_le1454117246873690627R_ftrs @ ( set_FOR_ftrs2 @ ( drop_FOR_ftrs @ M @ Xs ) ) @ ( set_FOR_ftrs2 @ ( drop_FOR_ftrs @ N @ Xs ) ) ) ) ).

% set_drop_subset_set_drop
thf(fact_853_Cons__nth__drop__Suc,axiom,
    ! [I4: nat,Xs: list_f7931864845379430901rm_a_b] :
      ( ( ord_less_nat @ I4 @ ( size_s1967422219310127201rm_a_b @ Xs ) )
     => ( ( cons_f831858257744602021rm_a_b @ ( nth_fs3949069584078227574rm_a_b @ Xs @ I4 ) @ ( drop_f1037359496611548469rm_a_b @ ( suc @ I4 ) @ Xs ) )
        = ( drop_f1037359496611548469rm_a_b @ I4 @ Xs ) ) ) ).

% Cons_nth_drop_Suc
thf(fact_854_less__prod__simp,axiom,
    ! [X1: nat > nat,Y1: nat,X22: nat > nat,Y22: nat] :
      ( ( ord_le6964573409913854876at_nat @ ( produc72220940542539688at_nat @ X1 @ Y1 ) @ ( produc72220940542539688at_nat @ X22 @ Y22 ) )
      = ( ( ord_less_nat_nat @ X1 @ X22 )
        | ( ( ord_less_eq_nat_nat @ X1 @ X22 )
          & ( ord_less_nat @ Y1 @ Y22 ) ) ) ) ).

% less_prod_simp
thf(fact_855_less__prod__simp,axiom,
    ! [X1: nat,Y1: nat,X22: nat,Y22: nat] :
      ( ( ord_le1203424502768444845at_nat @ ( product_Pair_nat_nat @ X1 @ Y1 ) @ ( product_Pair_nat_nat @ X22 @ Y22 ) )
      = ( ( ord_less_nat @ X1 @ X22 )
        | ( ( ord_less_eq_nat @ X1 @ X22 )
          & ( ord_less_nat @ Y1 @ Y22 ) ) ) ) ).

% less_prod_simp
thf(fact_856_less__prod__simp,axiom,
    ! [X1: nat,Y1: int,X22: nat,Y22: int] :
      ( ( ord_le6248945520114023945at_int @ ( product_Pair_nat_int @ X1 @ Y1 ) @ ( product_Pair_nat_int @ X22 @ Y22 ) )
      = ( ( ord_less_nat @ X1 @ X22 )
        | ( ( ord_less_eq_nat @ X1 @ X22 )
          & ( ord_less_int @ Y1 @ Y22 ) ) ) ) ).

% less_prod_simp
thf(fact_857_less__prod__simp,axiom,
    ! [X1: int,Y1: nat,X22: int,Y22: nat] :
      ( ( ord_le2202778214669499401nt_nat @ ( product_Pair_int_nat @ X1 @ Y1 ) @ ( product_Pair_int_nat @ X22 @ Y22 ) )
      = ( ( ord_less_int @ X1 @ X22 )
        | ( ( ord_less_eq_int @ X1 @ X22 )
          & ( ord_less_nat @ Y1 @ Y22 ) ) ) ) ).

% less_prod_simp
thf(fact_858_less__prod__simp,axiom,
    ! [X1: int,Y1: int,X22: int,Y22: int] :
      ( ( ord_le7248299232015078501nt_int @ ( product_Pair_int_int @ X1 @ Y1 ) @ ( product_Pair_int_int @ X22 @ Y22 ) )
      = ( ( ord_less_int @ X1 @ X22 )
        | ( ( ord_less_eq_int @ X1 @ X22 )
          & ( ord_less_int @ Y1 @ Y22 ) ) ) ) ).

% less_prod_simp
thf(fact_859_less__eq__prod__simp,axiom,
    ! [X1: nat > nat,Y1: nat,X22: nat > nat,Y22: nat] :
      ( ( ord_le2819838839419867280at_nat @ ( produc72220940542539688at_nat @ X1 @ Y1 ) @ ( produc72220940542539688at_nat @ X22 @ Y22 ) )
      = ( ( ord_less_nat_nat @ X1 @ X22 )
        | ( ( ord_less_eq_nat_nat @ X1 @ X22 )
          & ( ord_less_eq_nat @ Y1 @ Y22 ) ) ) ) ).

% less_eq_prod_simp
thf(fact_860_less__eq__prod__simp,axiom,
    ! [X1: nat,Y1: nat,X22: nat,Y22: nat] :
      ( ( ord_le8460144461188290721at_nat @ ( product_Pair_nat_nat @ X1 @ Y1 ) @ ( product_Pair_nat_nat @ X22 @ Y22 ) )
      = ( ( ord_less_nat @ X1 @ X22 )
        | ( ( ord_less_eq_nat @ X1 @ X22 )
          & ( ord_less_eq_nat @ Y1 @ Y22 ) ) ) ) ).

% less_eq_prod_simp
thf(fact_861_less__eq__prod__simp,axiom,
    ! [X1: nat,Y1: int,X22: nat,Y22: int] :
      ( ( ord_le4282293441679094013at_int @ ( product_Pair_nat_int @ X1 @ Y1 ) @ ( product_Pair_nat_int @ X22 @ Y22 ) )
      = ( ( ord_less_nat @ X1 @ X22 )
        | ( ( ord_less_eq_nat @ X1 @ X22 )
          & ( ord_less_eq_int @ Y1 @ Y22 ) ) ) ) ).

% less_eq_prod_simp
thf(fact_862_less__eq__prod__simp,axiom,
    ! [X1: int,Y1: nat,X22: int,Y22: nat] :
      ( ( ord_le236126136234569469nt_nat @ ( product_Pair_int_nat @ X1 @ Y1 ) @ ( product_Pair_int_nat @ X22 @ Y22 ) )
      = ( ( ord_less_int @ X1 @ X22 )
        | ( ( ord_less_eq_int @ X1 @ X22 )
          & ( ord_less_eq_nat @ Y1 @ Y22 ) ) ) ) ).

% less_eq_prod_simp
thf(fact_863_less__eq__prod__simp,axiom,
    ! [X1: int,Y1: int,X22: int,Y22: int] :
      ( ( ord_le5281647153580148569nt_int @ ( product_Pair_int_int @ X1 @ Y1 ) @ ( product_Pair_int_int @ X22 @ Y22 ) )
      = ( ( ord_less_int @ X1 @ X22 )
        | ( ( ord_less_eq_int @ X1 @ X22 )
          & ( ord_less_eq_int @ Y1 @ Y22 ) ) ) ) ).

% less_eq_prod_simp
thf(fact_864_divides__aux__eq,axiom,
    ! [Q3: nat,R3: nat] :
      ( ( unique5332122412489317741ux_nat @ ( product_Pair_nat_nat @ Q3 @ R3 ) )
      = ( R3 = zero_zero_nat ) ) ).

% divides_aux_eq
thf(fact_865_divides__aux__eq,axiom,
    ! [Q3: int,R3: int] :
      ( ( unique5329631941980267465ux_int @ ( product_Pair_int_int @ Q3 @ R3 ) )
      = ( R3 = zero_zero_int ) ) ).

% divides_aux_eq
thf(fact_866_filter2_Ocases,axiom,
    ! [X: produc6684372911298923872rm_a_b] :
      ( ! [P6: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b > $o,Uu2: list_s5629610930313288943rm_a_b] :
          ( X
         != ( produc6171443400230885072rm_a_b @ P6 @ ( produc510695698380873047rm_a_b @ nil_se7334864872594423151rm_a_b @ Uu2 ) ) )
     => ( ! [P6: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b > $o,V2: set_Pr4386577575007340137rm_a_b,Va: list_s5629610930313288943rm_a_b] :
            ( X
           != ( produc6171443400230885072rm_a_b @ P6 @ ( produc510695698380873047rm_a_b @ ( cons_s7847606222307733791rm_a_b @ V2 @ Va ) @ nil_se7334864872594423151rm_a_b ) ) )
       => ~ ! [P6: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b > $o,A7: set_Pr4386577575007340137rm_a_b,As2: list_s5629610930313288943rm_a_b,B5: set_Pr4386577575007340137rm_a_b,Bs2: list_s5629610930313288943rm_a_b] :
              ( X
             != ( produc6171443400230885072rm_a_b @ P6 @ ( produc510695698380873047rm_a_b @ ( cons_s7847606222307733791rm_a_b @ A7 @ As2 ) @ ( cons_s7847606222307733791rm_a_b @ B5 @ Bs2 ) ) ) ) ) ) ).

% filter2.cases
thf(fact_867_filter2_Ocases,axiom,
    ! [X: produc4507560098045251308rm_a_b] :
      ( ! [P6: set_Pr4386577575007340137rm_a_b > fset_P3577670608877728751rm_a_b > $o,Uu2: list_f7931864845379430901rm_a_b] :
          ( X
         != ( produc2305076824874716764rm_a_b @ P6 @ ( produc4120348330684003933rm_a_b @ nil_se7334864872594423151rm_a_b @ Uu2 ) ) )
     => ( ! [P6: set_Pr4386577575007340137rm_a_b > fset_P3577670608877728751rm_a_b > $o,V2: set_Pr4386577575007340137rm_a_b,Va: list_s5629610930313288943rm_a_b] :
            ( X
           != ( produc2305076824874716764rm_a_b @ P6 @ ( produc4120348330684003933rm_a_b @ ( cons_s7847606222307733791rm_a_b @ V2 @ Va ) @ nil_fs4020363321024022517rm_a_b ) ) )
       => ~ ! [P6: set_Pr4386577575007340137rm_a_b > fset_P3577670608877728751rm_a_b > $o,A7: set_Pr4386577575007340137rm_a_b,As2: list_s5629610930313288943rm_a_b,B5: fset_P3577670608877728751rm_a_b,Bs2: list_f7931864845379430901rm_a_b] :
              ( X
             != ( produc2305076824874716764rm_a_b @ P6 @ ( produc4120348330684003933rm_a_b @ ( cons_s7847606222307733791rm_a_b @ A7 @ As2 ) @ ( cons_f831858257744602021rm_a_b @ B5 @ Bs2 ) ) ) ) ) ) ).

% filter2.cases
thf(fact_868_filter2_Ocases,axiom,
    ! [X: produc6922486078077080660rm_a_b] :
      ( ! [P6: fset_P3577670608877728751rm_a_b > set_Pr4386577575007340137rm_a_b > $o,Uu2: list_s5629610930313288943rm_a_b] :
          ( X
         != ( produc6815238513020652996rm_a_b @ P6 @ ( produc5878835993175872977rm_a_b @ nil_fs4020363321024022517rm_a_b @ Uu2 ) ) )
     => ( ! [P6: fset_P3577670608877728751rm_a_b > set_Pr4386577575007340137rm_a_b > $o,V2: fset_P3577670608877728751rm_a_b,Va: list_f7931864845379430901rm_a_b] :
            ( X
           != ( produc6815238513020652996rm_a_b @ P6 @ ( produc5878835993175872977rm_a_b @ ( cons_f831858257744602021rm_a_b @ V2 @ Va ) @ nil_se7334864872594423151rm_a_b ) ) )
       => ~ ! [P6: fset_P3577670608877728751rm_a_b > set_Pr4386577575007340137rm_a_b > $o,A7: fset_P3577670608877728751rm_a_b,As2: list_f7931864845379430901rm_a_b,B5: set_Pr4386577575007340137rm_a_b,Bs2: list_s5629610930313288943rm_a_b] :
              ( X
             != ( produc6815238513020652996rm_a_b @ P6 @ ( produc5878835993175872977rm_a_b @ ( cons_f831858257744602021rm_a_b @ A7 @ As2 ) @ ( cons_s7847606222307733791rm_a_b @ B5 @ Bs2 ) ) ) ) ) ) ).

% filter2.cases
thf(fact_869_filter2_Ocases,axiom,
    ! [X: produc3511840931283393504rm_a_b] :
      ( ! [P6: fset_P3577670608877728751rm_a_b > fset_P3577670608877728751rm_a_b > $o,Uu2: list_f7931864845379430901rm_a_b] :
          ( X
         != ( produc7693669872013167440rm_a_b @ P6 @ ( produc8130127474510664151rm_a_b @ nil_fs4020363321024022517rm_a_b @ Uu2 ) ) )
     => ( ! [P6: fset_P3577670608877728751rm_a_b > fset_P3577670608877728751rm_a_b > $o,V2: fset_P3577670608877728751rm_a_b,Va: list_f7931864845379430901rm_a_b] :
            ( X
           != ( produc7693669872013167440rm_a_b @ P6 @ ( produc8130127474510664151rm_a_b @ ( cons_f831858257744602021rm_a_b @ V2 @ Va ) @ nil_fs4020363321024022517rm_a_b ) ) )
       => ~ ! [P6: fset_P3577670608877728751rm_a_b > fset_P3577670608877728751rm_a_b > $o,A7: fset_P3577670608877728751rm_a_b,As2: list_f7931864845379430901rm_a_b,B5: fset_P3577670608877728751rm_a_b,Bs2: list_f7931864845379430901rm_a_b] :
              ( X
             != ( produc7693669872013167440rm_a_b @ P6 @ ( produc8130127474510664151rm_a_b @ ( cons_f831858257744602021rm_a_b @ A7 @ As2 ) @ ( cons_f831858257744602021rm_a_b @ B5 @ Bs2 ) ) ) ) ) ) ).

% filter2.cases
thf(fact_870_collapse__state__list_Ocases,axiom,
    ! [X: produc8794552266035811103rm_a_b] :
      ( ! [Qn: fset_s6731944961250125903rm_a_b,Qs: fset_s6731944961250125903rm_a_b] :
          ( X
         != ( produc4052721450067169423rm_a_b @ Qn @ ( produc4958164796545313271rm_a_b @ Qs @ nil_se7334864872594423151rm_a_b ) ) )
     => ~ ! [Qn: fset_s6731944961250125903rm_a_b,Qs: fset_s6731944961250125903rm_a_b,Q4: set_Pr4386577575007340137rm_a_b,Qs2: list_s5629610930313288943rm_a_b] :
            ( X
           != ( produc4052721450067169423rm_a_b @ Qn @ ( produc4958164796545313271rm_a_b @ Qs @ ( cons_s7847606222307733791rm_a_b @ Q4 @ Qs2 ) ) ) ) ) ).

% collapse_state_list.cases
thf(fact_871_collapse__state__list_Ocases,axiom,
    ! [X: produc6270691069570780313rm_a_b] :
      ( ! [Qn: fset_f8829694286510474581rm_a_b,Qs: fset_f8829694286510474581rm_a_b] :
          ( X
         != ( produc8826034595809935241rm_a_b @ Qn @ ( produc5994328933639110263rm_a_b @ Qs @ nil_fs4020363321024022517rm_a_b ) ) )
     => ~ ! [Qn: fset_f8829694286510474581rm_a_b,Qs: fset_f8829694286510474581rm_a_b,Q4: fset_P3577670608877728751rm_a_b,Qs2: list_f7931864845379430901rm_a_b] :
            ( X
           != ( produc8826034595809935241rm_a_b @ Qn @ ( produc5994328933639110263rm_a_b @ Qs @ ( cons_f831858257744602021rm_a_b @ Q4 @ Qs2 ) ) ) ) ) ).

% collapse_state_list.cases
thf(fact_872_proper__intrvl_Oproper__interval__set__Compl__aux_Ocases,axiom,
    ! [X: produc7321815514089669226rm_a_b] :
      ( ! [Ao: option179594675238651567rm_a_b,N2: nat] :
          ( X
         != ( produc1802119990107635812rm_a_b @ Ao @ ( produc866814718296604094rm_a_b @ N2 @ ( produc510695698380873047rm_a_b @ nil_se7334864872594423151rm_a_b @ nil_se7334864872594423151rm_a_b ) ) ) )
     => ( ! [Ao: option179594675238651567rm_a_b,N2: nat,Y4: set_Pr4386577575007340137rm_a_b,Ys4: list_s5629610930313288943rm_a_b] :
            ( X
           != ( produc1802119990107635812rm_a_b @ Ao @ ( produc866814718296604094rm_a_b @ N2 @ ( produc510695698380873047rm_a_b @ nil_se7334864872594423151rm_a_b @ ( cons_s7847606222307733791rm_a_b @ Y4 @ Ys4 ) ) ) ) )
       => ( ! [Ao: option179594675238651567rm_a_b,N2: nat,X4: set_Pr4386577575007340137rm_a_b,Xs2: list_s5629610930313288943rm_a_b] :
              ( X
             != ( produc1802119990107635812rm_a_b @ Ao @ ( produc866814718296604094rm_a_b @ N2 @ ( produc510695698380873047rm_a_b @ ( cons_s7847606222307733791rm_a_b @ X4 @ Xs2 ) @ nil_se7334864872594423151rm_a_b ) ) ) )
         => ~ ! [Ao: option179594675238651567rm_a_b,N2: nat,X4: set_Pr4386577575007340137rm_a_b,Xs2: list_s5629610930313288943rm_a_b,Y4: set_Pr4386577575007340137rm_a_b,Ys4: list_s5629610930313288943rm_a_b] :
                ( X
               != ( produc1802119990107635812rm_a_b @ Ao @ ( produc866814718296604094rm_a_b @ N2 @ ( produc510695698380873047rm_a_b @ ( cons_s7847606222307733791rm_a_b @ X4 @ Xs2 ) @ ( cons_s7847606222307733791rm_a_b @ Y4 @ Ys4 ) ) ) ) ) ) ) ) ).

% proper_intrvl.proper_interval_set_Compl_aux.cases
thf(fact_873_proper__intrvl_Oproper__interval__set__Compl__aux_Ocases,axiom,
    ! [X: produc3501529727945571184rm_a_b] :
      ( ! [Ao: option8340779005562254261rm_a_b,N2: nat] :
          ( X
         != ( produc3174102978766136298rm_a_b @ Ao @ ( produc3703710239117002046rm_a_b @ N2 @ ( produc8130127474510664151rm_a_b @ nil_fs4020363321024022517rm_a_b @ nil_fs4020363321024022517rm_a_b ) ) ) )
     => ( ! [Ao: option8340779005562254261rm_a_b,N2: nat,Y4: fset_P3577670608877728751rm_a_b,Ys4: list_f7931864845379430901rm_a_b] :
            ( X
           != ( produc3174102978766136298rm_a_b @ Ao @ ( produc3703710239117002046rm_a_b @ N2 @ ( produc8130127474510664151rm_a_b @ nil_fs4020363321024022517rm_a_b @ ( cons_f831858257744602021rm_a_b @ Y4 @ Ys4 ) ) ) ) )
       => ( ! [Ao: option8340779005562254261rm_a_b,N2: nat,X4: fset_P3577670608877728751rm_a_b,Xs2: list_f7931864845379430901rm_a_b] :
              ( X
             != ( produc3174102978766136298rm_a_b @ Ao @ ( produc3703710239117002046rm_a_b @ N2 @ ( produc8130127474510664151rm_a_b @ ( cons_f831858257744602021rm_a_b @ X4 @ Xs2 ) @ nil_fs4020363321024022517rm_a_b ) ) ) )
         => ~ ! [Ao: option8340779005562254261rm_a_b,N2: nat,X4: fset_P3577670608877728751rm_a_b,Xs2: list_f7931864845379430901rm_a_b,Y4: fset_P3577670608877728751rm_a_b,Ys4: list_f7931864845379430901rm_a_b] :
                ( X
               != ( produc3174102978766136298rm_a_b @ Ao @ ( produc3703710239117002046rm_a_b @ N2 @ ( produc8130127474510664151rm_a_b @ ( cons_f831858257744602021rm_a_b @ X4 @ Xs2 ) @ ( cons_f831858257744602021rm_a_b @ Y4 @ Ys4 ) ) ) ) ) ) ) ) ).

% proper_intrvl.proper_interval_set_Compl_aux.cases
thf(fact_874_proper__intrvl_Oproper__interval__Compl__set__aux_Ocases,axiom,
    ! [X: produc4628417555811833631rm_a_b] :
      ( ! [Ao: option179594675238651567rm_a_b,X4: set_Pr4386577575007340137rm_a_b,Xs2: list_s5629610930313288943rm_a_b,Y4: set_Pr4386577575007340137rm_a_b,Ys4: list_s5629610930313288943rm_a_b] :
          ( X
         != ( produc7089420012340749711rm_a_b @ Ao @ ( produc510695698380873047rm_a_b @ ( cons_s7847606222307733791rm_a_b @ X4 @ Xs2 ) @ ( cons_s7847606222307733791rm_a_b @ Y4 @ Ys4 ) ) ) )
     => ( ! [Ao: option179594675238651567rm_a_b,Uv2: list_s5629610930313288943rm_a_b] :
            ( X
           != ( produc7089420012340749711rm_a_b @ Ao @ ( produc510695698380873047rm_a_b @ nil_se7334864872594423151rm_a_b @ Uv2 ) ) )
       => ~ ! [Ao: option179594675238651567rm_a_b,Uu2: list_s5629610930313288943rm_a_b] :
              ( X
             != ( produc7089420012340749711rm_a_b @ Ao @ ( produc510695698380873047rm_a_b @ Uu2 @ nil_se7334864872594423151rm_a_b ) ) ) ) ) ).

% proper_intrvl.proper_interval_Compl_set_aux.cases
thf(fact_875_proper__intrvl_Oproper__interval__Compl__set__aux_Ocases,axiom,
    ! [X: produc3924124187745943705rm_a_b] :
      ( ! [Ao: option8340779005562254261rm_a_b,X4: fset_P3577670608877728751rm_a_b,Xs2: list_f7931864845379430901rm_a_b,Y4: fset_P3577670608877728751rm_a_b,Ys4: list_f7931864845379430901rm_a_b] :
          ( X
         != ( produc1056333751507027081rm_a_b @ Ao @ ( produc8130127474510664151rm_a_b @ ( cons_f831858257744602021rm_a_b @ X4 @ Xs2 ) @ ( cons_f831858257744602021rm_a_b @ Y4 @ Ys4 ) ) ) )
     => ( ! [Ao: option8340779005562254261rm_a_b,Uv2: list_f7931864845379430901rm_a_b] :
            ( X
           != ( produc1056333751507027081rm_a_b @ Ao @ ( produc8130127474510664151rm_a_b @ nil_fs4020363321024022517rm_a_b @ Uv2 ) ) )
       => ~ ! [Ao: option8340779005562254261rm_a_b,Uu2: list_f7931864845379430901rm_a_b] :
              ( X
             != ( produc1056333751507027081rm_a_b @ Ao @ ( produc8130127474510664151rm_a_b @ Uu2 @ nil_fs4020363321024022517rm_a_b ) ) ) ) ) ).

% proper_intrvl.proper_interval_Compl_set_aux.cases
thf(fact_876_proper__intrvl_Oset__less__eq__aux__Compl_Ocases,axiom,
    ! [X: produc4628417555811833631rm_a_b] :
      ( ! [Ao: option179594675238651567rm_a_b,Ys4: list_s5629610930313288943rm_a_b] :
          ( X
         != ( produc7089420012340749711rm_a_b @ Ao @ ( produc510695698380873047rm_a_b @ nil_se7334864872594423151rm_a_b @ Ys4 ) ) )
     => ( ! [Ao: option179594675238651567rm_a_b,Xs2: list_s5629610930313288943rm_a_b] :
            ( X
           != ( produc7089420012340749711rm_a_b @ Ao @ ( produc510695698380873047rm_a_b @ Xs2 @ nil_se7334864872594423151rm_a_b ) ) )
       => ~ ! [Ao: option179594675238651567rm_a_b,X4: set_Pr4386577575007340137rm_a_b,Xs2: list_s5629610930313288943rm_a_b,Y4: set_Pr4386577575007340137rm_a_b,Ys4: list_s5629610930313288943rm_a_b] :
              ( X
             != ( produc7089420012340749711rm_a_b @ Ao @ ( produc510695698380873047rm_a_b @ ( cons_s7847606222307733791rm_a_b @ X4 @ Xs2 ) @ ( cons_s7847606222307733791rm_a_b @ Y4 @ Ys4 ) ) ) ) ) ) ).

% proper_intrvl.set_less_eq_aux_Compl.cases
thf(fact_877_proper__intrvl_Oset__less__eq__aux__Compl_Ocases,axiom,
    ! [X: produc3924124187745943705rm_a_b] :
      ( ! [Ao: option8340779005562254261rm_a_b,Ys4: list_f7931864845379430901rm_a_b] :
          ( X
         != ( produc1056333751507027081rm_a_b @ Ao @ ( produc8130127474510664151rm_a_b @ nil_fs4020363321024022517rm_a_b @ Ys4 ) ) )
     => ( ! [Ao: option8340779005562254261rm_a_b,Xs2: list_f7931864845379430901rm_a_b] :
            ( X
           != ( produc1056333751507027081rm_a_b @ Ao @ ( produc8130127474510664151rm_a_b @ Xs2 @ nil_fs4020363321024022517rm_a_b ) ) )
       => ~ ! [Ao: option8340779005562254261rm_a_b,X4: fset_P3577670608877728751rm_a_b,Xs2: list_f7931864845379430901rm_a_b,Y4: fset_P3577670608877728751rm_a_b,Ys4: list_f7931864845379430901rm_a_b] :
              ( X
             != ( produc1056333751507027081rm_a_b @ Ao @ ( produc8130127474510664151rm_a_b @ ( cons_f831858257744602021rm_a_b @ X4 @ Xs2 ) @ ( cons_f831858257744602021rm_a_b @ Y4 @ Ys4 ) ) ) ) ) ) ).

% proper_intrvl.set_less_eq_aux_Compl.cases
thf(fact_878_proper__intrvl_OCompl__set__less__eq__aux_Ocases,axiom,
    ! [X: produc4628417555811833631rm_a_b] :
      ( ! [Ao: option179594675238651567rm_a_b] :
          ( X
         != ( produc7089420012340749711rm_a_b @ Ao @ ( produc510695698380873047rm_a_b @ nil_se7334864872594423151rm_a_b @ nil_se7334864872594423151rm_a_b ) ) )
     => ( ! [Ao: option179594675238651567rm_a_b,Y4: set_Pr4386577575007340137rm_a_b,Ys4: list_s5629610930313288943rm_a_b] :
            ( X
           != ( produc7089420012340749711rm_a_b @ Ao @ ( produc510695698380873047rm_a_b @ nil_se7334864872594423151rm_a_b @ ( cons_s7847606222307733791rm_a_b @ Y4 @ Ys4 ) ) ) )
       => ( ! [Ao: option179594675238651567rm_a_b,X4: set_Pr4386577575007340137rm_a_b,Xs2: list_s5629610930313288943rm_a_b] :
              ( X
             != ( produc7089420012340749711rm_a_b @ Ao @ ( produc510695698380873047rm_a_b @ ( cons_s7847606222307733791rm_a_b @ X4 @ Xs2 ) @ nil_se7334864872594423151rm_a_b ) ) )
         => ~ ! [Ao: option179594675238651567rm_a_b,X4: set_Pr4386577575007340137rm_a_b,Xs2: list_s5629610930313288943rm_a_b,Y4: set_Pr4386577575007340137rm_a_b,Ys4: list_s5629610930313288943rm_a_b] :
                ( X
               != ( produc7089420012340749711rm_a_b @ Ao @ ( produc510695698380873047rm_a_b @ ( cons_s7847606222307733791rm_a_b @ X4 @ Xs2 ) @ ( cons_s7847606222307733791rm_a_b @ Y4 @ Ys4 ) ) ) ) ) ) ) ).

% proper_intrvl.Compl_set_less_eq_aux.cases
thf(fact_879_proper__intrvl_OCompl__set__less__eq__aux_Ocases,axiom,
    ! [X: produc3924124187745943705rm_a_b] :
      ( ! [Ao: option8340779005562254261rm_a_b] :
          ( X
         != ( produc1056333751507027081rm_a_b @ Ao @ ( produc8130127474510664151rm_a_b @ nil_fs4020363321024022517rm_a_b @ nil_fs4020363321024022517rm_a_b ) ) )
     => ( ! [Ao: option8340779005562254261rm_a_b,Y4: fset_P3577670608877728751rm_a_b,Ys4: list_f7931864845379430901rm_a_b] :
            ( X
           != ( produc1056333751507027081rm_a_b @ Ao @ ( produc8130127474510664151rm_a_b @ nil_fs4020363321024022517rm_a_b @ ( cons_f831858257744602021rm_a_b @ Y4 @ Ys4 ) ) ) )
       => ( ! [Ao: option8340779005562254261rm_a_b,X4: fset_P3577670608877728751rm_a_b,Xs2: list_f7931864845379430901rm_a_b] :
              ( X
             != ( produc1056333751507027081rm_a_b @ Ao @ ( produc8130127474510664151rm_a_b @ ( cons_f831858257744602021rm_a_b @ X4 @ Xs2 ) @ nil_fs4020363321024022517rm_a_b ) ) )
         => ~ ! [Ao: option8340779005562254261rm_a_b,X4: fset_P3577670608877728751rm_a_b,Xs2: list_f7931864845379430901rm_a_b,Y4: fset_P3577670608877728751rm_a_b,Ys4: list_f7931864845379430901rm_a_b] :
                ( X
               != ( produc1056333751507027081rm_a_b @ Ao @ ( produc8130127474510664151rm_a_b @ ( cons_f831858257744602021rm_a_b @ X4 @ Xs2 ) @ ( cons_f831858257744602021rm_a_b @ Y4 @ Ys4 ) ) ) ) ) ) ) ).

% proper_intrvl.Compl_set_less_eq_aux.cases
thf(fact_880_ord__class_Osorted__list__subset_Ocases,axiom,
    ! [X: produc6684372911298923872rm_a_b] :
      ( ! [Eq: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b > $o,Ys4: list_s5629610930313288943rm_a_b] :
          ( X
         != ( produc6171443400230885072rm_a_b @ Eq @ ( produc510695698380873047rm_a_b @ nil_se7334864872594423151rm_a_b @ Ys4 ) ) )
     => ( ! [Eq: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b > $o,X4: set_Pr4386577575007340137rm_a_b,Xs2: list_s5629610930313288943rm_a_b] :
            ( X
           != ( produc6171443400230885072rm_a_b @ Eq @ ( produc510695698380873047rm_a_b @ ( cons_s7847606222307733791rm_a_b @ X4 @ Xs2 ) @ nil_se7334864872594423151rm_a_b ) ) )
       => ~ ! [Eq: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b > $o,X4: set_Pr4386577575007340137rm_a_b,Xs2: list_s5629610930313288943rm_a_b,Y4: set_Pr4386577575007340137rm_a_b,Ys4: list_s5629610930313288943rm_a_b] :
              ( X
             != ( produc6171443400230885072rm_a_b @ Eq @ ( produc510695698380873047rm_a_b @ ( cons_s7847606222307733791rm_a_b @ X4 @ Xs2 ) @ ( cons_s7847606222307733791rm_a_b @ Y4 @ Ys4 ) ) ) ) ) ) ).

% ord_class.sorted_list_subset.cases
thf(fact_881_ord__class_Osorted__list__subset_Ocases,axiom,
    ! [X: produc3511840931283393504rm_a_b] :
      ( ! [Eq: fset_P3577670608877728751rm_a_b > fset_P3577670608877728751rm_a_b > $o,Ys4: list_f7931864845379430901rm_a_b] :
          ( X
         != ( produc7693669872013167440rm_a_b @ Eq @ ( produc8130127474510664151rm_a_b @ nil_fs4020363321024022517rm_a_b @ Ys4 ) ) )
     => ( ! [Eq: fset_P3577670608877728751rm_a_b > fset_P3577670608877728751rm_a_b > $o,X4: fset_P3577670608877728751rm_a_b,Xs2: list_f7931864845379430901rm_a_b] :
            ( X
           != ( produc7693669872013167440rm_a_b @ Eq @ ( produc8130127474510664151rm_a_b @ ( cons_f831858257744602021rm_a_b @ X4 @ Xs2 ) @ nil_fs4020363321024022517rm_a_b ) ) )
       => ~ ! [Eq: fset_P3577670608877728751rm_a_b > fset_P3577670608877728751rm_a_b > $o,X4: fset_P3577670608877728751rm_a_b,Xs2: list_f7931864845379430901rm_a_b,Y4: fset_P3577670608877728751rm_a_b,Ys4: list_f7931864845379430901rm_a_b] :
              ( X
             != ( produc7693669872013167440rm_a_b @ Eq @ ( produc8130127474510664151rm_a_b @ ( cons_f831858257744602021rm_a_b @ X4 @ Xs2 ) @ ( cons_f831858257744602021rm_a_b @ Y4 @ Ys4 ) ) ) ) ) ) ).

% ord_class.sorted_list_subset.cases
thf(fact_882_quicksort__part_Ocases,axiom,
    ! [X: produc787269501071769549rm_a_b] :
      ( ! [Ac: list_s5629610930313288943rm_a_b,X4: set_Pr4386577575007340137rm_a_b,Lts: list_s5629610930313288943rm_a_b,Eqs: list_s5629610930313288943rm_a_b,Gts: list_s5629610930313288943rm_a_b] :
          ( X
         != ( produc7700493563723880125rm_a_b @ Ac @ ( produc7491607707020564805rm_a_b @ X4 @ ( produc8860274016745896519rm_a_b @ Lts @ ( produc1110353298344892239rm_a_b @ Eqs @ ( produc510695698380873047rm_a_b @ Gts @ nil_se7334864872594423151rm_a_b ) ) ) ) ) )
     => ~ ! [Ac: list_s5629610930313288943rm_a_b,X4: set_Pr4386577575007340137rm_a_b,Lts: list_s5629610930313288943rm_a_b,Eqs: list_s5629610930313288943rm_a_b,Gts: list_s5629610930313288943rm_a_b,Z: set_Pr4386577575007340137rm_a_b,Zs: list_s5629610930313288943rm_a_b] :
            ( X
           != ( produc7700493563723880125rm_a_b @ Ac @ ( produc7491607707020564805rm_a_b @ X4 @ ( produc8860274016745896519rm_a_b @ Lts @ ( produc1110353298344892239rm_a_b @ Eqs @ ( produc510695698380873047rm_a_b @ Gts @ ( cons_s7847606222307733791rm_a_b @ Z @ Zs ) ) ) ) ) ) ) ) ).

% quicksort_part.cases
thf(fact_883_quicksort__part_Ocases,axiom,
    ! [X: produc4094060736509213621rm_a_b] :
      ( ! [Ac: list_f7931864845379430901rm_a_b,X4: fset_P3577670608877728751rm_a_b,Lts: list_f7931864845379430901rm_a_b,Eqs: list_f7931864845379430901rm_a_b,Gts: list_f7931864845379430901rm_a_b] :
          ( X
         != ( produc307254047110923941rm_a_b @ Ac @ ( produc1546334010041838899rm_a_b @ X4 @ ( produc615127835018616123rm_a_b @ Lts @ ( produc3206704591963461193rm_a_b @ Eqs @ ( produc8130127474510664151rm_a_b @ Gts @ nil_fs4020363321024022517rm_a_b ) ) ) ) ) )
     => ~ ! [Ac: list_f7931864845379430901rm_a_b,X4: fset_P3577670608877728751rm_a_b,Lts: list_f7931864845379430901rm_a_b,Eqs: list_f7931864845379430901rm_a_b,Gts: list_f7931864845379430901rm_a_b,Z: fset_P3577670608877728751rm_a_b,Zs: list_f7931864845379430901rm_a_b] :
            ( X
           != ( produc307254047110923941rm_a_b @ Ac @ ( produc1546334010041838899rm_a_b @ X4 @ ( produc615127835018616123rm_a_b @ Lts @ ( produc3206704591963461193rm_a_b @ Eqs @ ( produc8130127474510664151rm_a_b @ Gts @ ( cons_f831858257744602021rm_a_b @ Z @ Zs ) ) ) ) ) ) ) ) ).

% quicksort_part.cases
thf(fact_884_ord_Osorted__list__subset_Ocases,axiom,
    ! [X: produc6684372911298923872rm_a_b] :
      ( ! [Eq: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b > $o,Ys4: list_s5629610930313288943rm_a_b] :
          ( X
         != ( produc6171443400230885072rm_a_b @ Eq @ ( produc510695698380873047rm_a_b @ nil_se7334864872594423151rm_a_b @ Ys4 ) ) )
     => ( ! [Eq: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b > $o,X4: set_Pr4386577575007340137rm_a_b,Xs2: list_s5629610930313288943rm_a_b] :
            ( X
           != ( produc6171443400230885072rm_a_b @ Eq @ ( produc510695698380873047rm_a_b @ ( cons_s7847606222307733791rm_a_b @ X4 @ Xs2 ) @ nil_se7334864872594423151rm_a_b ) ) )
       => ~ ! [Eq: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b > $o,X4: set_Pr4386577575007340137rm_a_b,Xs2: list_s5629610930313288943rm_a_b,Y4: set_Pr4386577575007340137rm_a_b,Ys4: list_s5629610930313288943rm_a_b] :
              ( X
             != ( produc6171443400230885072rm_a_b @ Eq @ ( produc510695698380873047rm_a_b @ ( cons_s7847606222307733791rm_a_b @ X4 @ Xs2 ) @ ( cons_s7847606222307733791rm_a_b @ Y4 @ Ys4 ) ) ) ) ) ) ).

% ord.sorted_list_subset.cases
thf(fact_885_ord_Osorted__list__subset_Ocases,axiom,
    ! [X: produc3511840931283393504rm_a_b] :
      ( ! [Eq: fset_P3577670608877728751rm_a_b > fset_P3577670608877728751rm_a_b > $o,Ys4: list_f7931864845379430901rm_a_b] :
          ( X
         != ( produc7693669872013167440rm_a_b @ Eq @ ( produc8130127474510664151rm_a_b @ nil_fs4020363321024022517rm_a_b @ Ys4 ) ) )
     => ( ! [Eq: fset_P3577670608877728751rm_a_b > fset_P3577670608877728751rm_a_b > $o,X4: fset_P3577670608877728751rm_a_b,Xs2: list_f7931864845379430901rm_a_b] :
            ( X
           != ( produc7693669872013167440rm_a_b @ Eq @ ( produc8130127474510664151rm_a_b @ ( cons_f831858257744602021rm_a_b @ X4 @ Xs2 ) @ nil_fs4020363321024022517rm_a_b ) ) )
       => ~ ! [Eq: fset_P3577670608877728751rm_a_b > fset_P3577670608877728751rm_a_b > $o,X4: fset_P3577670608877728751rm_a_b,Xs2: list_f7931864845379430901rm_a_b,Y4: fset_P3577670608877728751rm_a_b,Ys4: list_f7931864845379430901rm_a_b] :
              ( X
             != ( produc7693669872013167440rm_a_b @ Eq @ ( produc8130127474510664151rm_a_b @ ( cons_f831858257744602021rm_a_b @ X4 @ Xs2 ) @ ( cons_f831858257744602021rm_a_b @ Y4 @ Ys4 ) ) ) ) ) ) ).

% ord.sorted_list_subset.cases
thf(fact_886_ord_Oquicksort__part_Ocases,axiom,
    ! [X: produc787269501071769549rm_a_b] :
      ( ! [Ac: list_s5629610930313288943rm_a_b,X4: set_Pr4386577575007340137rm_a_b,Lts: list_s5629610930313288943rm_a_b,Eqs: list_s5629610930313288943rm_a_b,Gts: list_s5629610930313288943rm_a_b] :
          ( X
         != ( produc7700493563723880125rm_a_b @ Ac @ ( produc7491607707020564805rm_a_b @ X4 @ ( produc8860274016745896519rm_a_b @ Lts @ ( produc1110353298344892239rm_a_b @ Eqs @ ( produc510695698380873047rm_a_b @ Gts @ nil_se7334864872594423151rm_a_b ) ) ) ) ) )
     => ~ ! [Ac: list_s5629610930313288943rm_a_b,X4: set_Pr4386577575007340137rm_a_b,Lts: list_s5629610930313288943rm_a_b,Eqs: list_s5629610930313288943rm_a_b,Gts: list_s5629610930313288943rm_a_b,Z: set_Pr4386577575007340137rm_a_b,Zs: list_s5629610930313288943rm_a_b] :
            ( X
           != ( produc7700493563723880125rm_a_b @ Ac @ ( produc7491607707020564805rm_a_b @ X4 @ ( produc8860274016745896519rm_a_b @ Lts @ ( produc1110353298344892239rm_a_b @ Eqs @ ( produc510695698380873047rm_a_b @ Gts @ ( cons_s7847606222307733791rm_a_b @ Z @ Zs ) ) ) ) ) ) ) ) ).

% ord.quicksort_part.cases
thf(fact_887_ord_Oquicksort__part_Ocases,axiom,
    ! [X: produc4094060736509213621rm_a_b] :
      ( ! [Ac: list_f7931864845379430901rm_a_b,X4: fset_P3577670608877728751rm_a_b,Lts: list_f7931864845379430901rm_a_b,Eqs: list_f7931864845379430901rm_a_b,Gts: list_f7931864845379430901rm_a_b] :
          ( X
         != ( produc307254047110923941rm_a_b @ Ac @ ( produc1546334010041838899rm_a_b @ X4 @ ( produc615127835018616123rm_a_b @ Lts @ ( produc3206704591963461193rm_a_b @ Eqs @ ( produc8130127474510664151rm_a_b @ Gts @ nil_fs4020363321024022517rm_a_b ) ) ) ) ) )
     => ~ ! [Ac: list_f7931864845379430901rm_a_b,X4: fset_P3577670608877728751rm_a_b,Lts: list_f7931864845379430901rm_a_b,Eqs: list_f7931864845379430901rm_a_b,Gts: list_f7931864845379430901rm_a_b,Z: fset_P3577670608877728751rm_a_b,Zs: list_f7931864845379430901rm_a_b] :
            ( X
           != ( produc307254047110923941rm_a_b @ Ac @ ( produc1546334010041838899rm_a_b @ X4 @ ( produc615127835018616123rm_a_b @ Lts @ ( produc3206704591963461193rm_a_b @ Eqs @ ( produc8130127474510664151rm_a_b @ Gts @ ( cons_f831858257744602021rm_a_b @ Z @ Zs ) ) ) ) ) ) ) ) ).

% ord.quicksort_part.cases
thf(fact_888_closure__impl_Ocases,axiom,
    ! [X: produc2719214607420647895rm_a_b] :
      ( ! [F2: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b] :
          ( X
         != ( produc5584162508467506759rm_a_b @ F2 @ nil_se7334864872594423151rm_a_b ) )
     => ~ ! [F2: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b,X4: set_Pr4386577575007340137rm_a_b,S3: list_s5629610930313288943rm_a_b] :
            ( X
           != ( produc5584162508467506759rm_a_b @ F2 @ ( cons_s7847606222307733791rm_a_b @ X4 @ S3 ) ) ) ) ).

% closure_impl.cases
thf(fact_889_closure__impl_Ocases,axiom,
    ! [X: produc7679025466015609815rm_a_b] :
      ( ! [F2: fset_P3577670608877728751rm_a_b > fset_P3577670608877728751rm_a_b > fset_P3577670608877728751rm_a_b] :
          ( X
         != ( produc8208750159043342151rm_a_b @ F2 @ nil_fs4020363321024022517rm_a_b ) )
     => ~ ! [F2: fset_P3577670608877728751rm_a_b > fset_P3577670608877728751rm_a_b > fset_P3577670608877728751rm_a_b,X4: fset_P3577670608877728751rm_a_b,S3: list_f7931864845379430901rm_a_b] :
            ( X
           != ( produc8208750159043342151rm_a_b @ F2 @ ( cons_f831858257744602021rm_a_b @ X4 @ S3 ) ) ) ) ).

% closure_impl.cases
thf(fact_890_span_Ocases,axiom,
    ! [X: produc2965730177405805456rm_a_b] :
      ( ! [P6: set_Pr4386577575007340137rm_a_b > $o,X4: set_Pr4386577575007340137rm_a_b,Xs2: list_s5629610930313288943rm_a_b] :
          ( X
         != ( produc7007422553166974018rm_a_b @ P6 @ ( cons_s7847606222307733791rm_a_b @ X4 @ Xs2 ) ) )
     => ~ ! [Uu2: set_Pr4386577575007340137rm_a_b > $o] :
            ( X
           != ( produc7007422553166974018rm_a_b @ Uu2 @ nil_se7334864872594423151rm_a_b ) ) ) ).

% span.cases
thf(fact_891_span_Ocases,axiom,
    ! [X: produc4499360472861772060rm_a_b] :
      ( ! [P6: fset_P3577670608877728751rm_a_b > $o,X4: fset_P3577670608877728751rm_a_b,Xs2: list_f7931864845379430901rm_a_b] :
          ( X
         != ( produc3510664975902516686rm_a_b @ P6 @ ( cons_f831858257744602021rm_a_b @ X4 @ Xs2 ) ) )
     => ~ ! [Uu2: fset_P3577670608877728751rm_a_b > $o] :
            ( X
           != ( produc3510664975902516686rm_a_b @ Uu2 @ nil_fs4020363321024022517rm_a_b ) ) ) ).

% span.cases
thf(fact_892_distinct__eq_Ocases,axiom,
    ! [X: produc3707020311488567656rm_a_b] :
      ( ! [Uu2: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b > $o] :
          ( X
         != ( produc7612248553489884120rm_a_b @ Uu2 @ nil_se7334864872594423151rm_a_b ) )
     => ~ ! [Eq: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b > $o,X4: set_Pr4386577575007340137rm_a_b,Xs2: list_s5629610930313288943rm_a_b] :
            ( X
           != ( produc7612248553489884120rm_a_b @ Eq @ ( cons_s7847606222307733791rm_a_b @ X4 @ Xs2 ) ) ) ) ).

% distinct_eq.cases
thf(fact_893_distinct__eq_Ocases,axiom,
    ! [X: produc7748183047977949806rm_a_b] :
      ( ! [Uu2: fset_P3577670608877728751rm_a_b > fset_P3577670608877728751rm_a_b > $o] :
          ( X
         != ( produc6659827884187831134rm_a_b @ Uu2 @ nil_fs4020363321024022517rm_a_b ) )
     => ~ ! [Eq: fset_P3577670608877728751rm_a_b > fset_P3577670608877728751rm_a_b > $o,X4: fset_P3577670608877728751rm_a_b,Xs2: list_f7931864845379430901rm_a_b] :
            ( X
           != ( produc6659827884187831134rm_a_b @ Eq @ ( cons_f831858257744602021rm_a_b @ X4 @ Xs2 ) ) ) ) ).

% distinct_eq.cases
thf(fact_894_successively_Ocases,axiom,
    ! [X: produc3707020311488567656rm_a_b] :
      ( ! [P6: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b > $o] :
          ( X
         != ( produc7612248553489884120rm_a_b @ P6 @ nil_se7334864872594423151rm_a_b ) )
     => ( ! [P6: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b > $o,X4: set_Pr4386577575007340137rm_a_b] :
            ( X
           != ( produc7612248553489884120rm_a_b @ P6 @ ( cons_s7847606222307733791rm_a_b @ X4 @ nil_se7334864872594423151rm_a_b ) ) )
       => ~ ! [P6: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b > $o,X4: set_Pr4386577575007340137rm_a_b,Y4: set_Pr4386577575007340137rm_a_b,Xs2: list_s5629610930313288943rm_a_b] :
              ( X
             != ( produc7612248553489884120rm_a_b @ P6 @ ( cons_s7847606222307733791rm_a_b @ X4 @ ( cons_s7847606222307733791rm_a_b @ Y4 @ Xs2 ) ) ) ) ) ) ).

% successively.cases
thf(fact_895_successively_Ocases,axiom,
    ! [X: produc7748183047977949806rm_a_b] :
      ( ! [P6: fset_P3577670608877728751rm_a_b > fset_P3577670608877728751rm_a_b > $o] :
          ( X
         != ( produc6659827884187831134rm_a_b @ P6 @ nil_fs4020363321024022517rm_a_b ) )
     => ( ! [P6: fset_P3577670608877728751rm_a_b > fset_P3577670608877728751rm_a_b > $o,X4: fset_P3577670608877728751rm_a_b] :
            ( X
           != ( produc6659827884187831134rm_a_b @ P6 @ ( cons_f831858257744602021rm_a_b @ X4 @ nil_fs4020363321024022517rm_a_b ) ) )
       => ~ ! [P6: fset_P3577670608877728751rm_a_b > fset_P3577670608877728751rm_a_b > $o,X4: fset_P3577670608877728751rm_a_b,Y4: fset_P3577670608877728751rm_a_b,Xs2: list_f7931864845379430901rm_a_b] :
              ( X
             != ( produc6659827884187831134rm_a_b @ P6 @ ( cons_f831858257744602021rm_a_b @ X4 @ ( cons_f831858257744602021rm_a_b @ Y4 @ Xs2 ) ) ) ) ) ) ).

% successively.cases
thf(fact_896_inf__concat__simple_Ocases,axiom,
    ! [X: produc8199716216217303280at_nat] :
      ( ! [F2: nat > nat] :
          ( X
         != ( produc72220940542539688at_nat @ F2 @ zero_zero_nat ) )
     => ~ ! [F2: nat > nat,N2: nat] :
            ( X
           != ( produc72220940542539688at_nat @ F2 @ ( suc @ N2 ) ) ) ) ).

% inf_concat_simple.cases
thf(fact_897_proper__intrvl_Oexhaustive__above_Ocases,axiom,
    ! [X: produc3431338860107707501rm_a_b] :
      ( ! [X4: set_Pr4386577575007340137rm_a_b] :
          ( X
         != ( produc6484470120179297885rm_a_b @ X4 @ nil_se7334864872594423151rm_a_b ) )
     => ~ ! [X4: set_Pr4386577575007340137rm_a_b,Y4: set_Pr4386577575007340137rm_a_b,Ys4: list_s5629610930313288943rm_a_b] :
            ( X
           != ( produc6484470120179297885rm_a_b @ X4 @ ( cons_s7847606222307733791rm_a_b @ Y4 @ Ys4 ) ) ) ) ).

% proper_intrvl.exhaustive_above.cases
thf(fact_898_proper__intrvl_Oexhaustive__above_Ocases,axiom,
    ! [X: produc8900648260557066093rm_a_b] :
      ( ! [X4: fset_P3577670608877728751rm_a_b] :
          ( X
         != ( produc7398763362198749021rm_a_b @ X4 @ nil_fs4020363321024022517rm_a_b ) )
     => ~ ! [X4: fset_P3577670608877728751rm_a_b,Y4: fset_P3577670608877728751rm_a_b,Ys4: list_f7931864845379430901rm_a_b] :
            ( X
           != ( produc7398763362198749021rm_a_b @ X4 @ ( cons_f831858257744602021rm_a_b @ Y4 @ Ys4 ) ) ) ) ).

% proper_intrvl.exhaustive_above.cases
thf(fact_899_remove__prefix_Ocases,axiom,
    ! [X: produc8638778270066257639rm_a_b] :
      ( ! [X4: set_Pr4386577575007340137rm_a_b,Xs2: list_s5629610930313288943rm_a_b,Y4: set_Pr4386577575007340137rm_a_b,Ys4: list_s5629610930313288943rm_a_b] :
          ( X
         != ( produc510695698380873047rm_a_b @ ( cons_s7847606222307733791rm_a_b @ X4 @ Xs2 ) @ ( cons_s7847606222307733791rm_a_b @ Y4 @ Ys4 ) ) )
     => ( ! [Ys4: list_s5629610930313288943rm_a_b] :
            ( X
           != ( produc510695698380873047rm_a_b @ nil_se7334864872594423151rm_a_b @ Ys4 ) )
       => ~ ! [V2: set_Pr4386577575007340137rm_a_b,Va: list_s5629610930313288943rm_a_b] :
              ( X
             != ( produc510695698380873047rm_a_b @ ( cons_s7847606222307733791rm_a_b @ V2 @ Va ) @ nil_se7334864872594423151rm_a_b ) ) ) ) ).

% remove_prefix.cases
thf(fact_900_remove__prefix_Ocases,axiom,
    ! [X: produc4312625336439893351rm_a_b] :
      ( ! [X4: fset_P3577670608877728751rm_a_b,Xs2: list_f7931864845379430901rm_a_b,Y4: fset_P3577670608877728751rm_a_b,Ys4: list_f7931864845379430901rm_a_b] :
          ( X
         != ( produc8130127474510664151rm_a_b @ ( cons_f831858257744602021rm_a_b @ X4 @ Xs2 ) @ ( cons_f831858257744602021rm_a_b @ Y4 @ Ys4 ) ) )
     => ( ! [Ys4: list_f7931864845379430901rm_a_b] :
            ( X
           != ( produc8130127474510664151rm_a_b @ nil_fs4020363321024022517rm_a_b @ Ys4 ) )
       => ~ ! [V2: fset_P3577670608877728751rm_a_b,Va: list_f7931864845379430901rm_a_b] :
              ( X
             != ( produc8130127474510664151rm_a_b @ ( cons_f831858257744602021rm_a_b @ V2 @ Va ) @ nil_fs4020363321024022517rm_a_b ) ) ) ) ).

% remove_prefix.cases
thf(fact_901_list__inter_Ocases,axiom,
    ! [X: produc8638778270066257639rm_a_b] :
      ( ! [Bs2: list_s5629610930313288943rm_a_b] :
          ( X
         != ( produc510695698380873047rm_a_b @ nil_se7334864872594423151rm_a_b @ Bs2 ) )
     => ~ ! [A7: set_Pr4386577575007340137rm_a_b,As2: list_s5629610930313288943rm_a_b,Bs2: list_s5629610930313288943rm_a_b] :
            ( X
           != ( produc510695698380873047rm_a_b @ ( cons_s7847606222307733791rm_a_b @ A7 @ As2 ) @ Bs2 ) ) ) ).

% list_inter.cases
thf(fact_902_list__inter_Ocases,axiom,
    ! [X: produc4312625336439893351rm_a_b] :
      ( ! [Bs2: list_f7931864845379430901rm_a_b] :
          ( X
         != ( produc8130127474510664151rm_a_b @ nil_fs4020363321024022517rm_a_b @ Bs2 ) )
     => ~ ! [A7: fset_P3577670608877728751rm_a_b,As2: list_f7931864845379430901rm_a_b,Bs2: list_f7931864845379430901rm_a_b] :
            ( X
           != ( produc8130127474510664151rm_a_b @ ( cons_f831858257744602021rm_a_b @ A7 @ As2 ) @ Bs2 ) ) ) ).

% list_inter.cases
thf(fact_903_union__list__sorted_Ocases,axiom,
    ! [X: produc8638778270066257639rm_a_b] :
      ( ! [X4: set_Pr4386577575007340137rm_a_b,Xs2: list_s5629610930313288943rm_a_b,Y4: set_Pr4386577575007340137rm_a_b,Ys4: list_s5629610930313288943rm_a_b] :
          ( X
         != ( produc510695698380873047rm_a_b @ ( cons_s7847606222307733791rm_a_b @ X4 @ Xs2 ) @ ( cons_s7847606222307733791rm_a_b @ Y4 @ Ys4 ) ) )
     => ( ! [Ys4: list_s5629610930313288943rm_a_b] :
            ( X
           != ( produc510695698380873047rm_a_b @ nil_se7334864872594423151rm_a_b @ Ys4 ) )
       => ~ ! [V2: set_Pr4386577575007340137rm_a_b,Va: list_s5629610930313288943rm_a_b] :
              ( X
             != ( produc510695698380873047rm_a_b @ ( cons_s7847606222307733791rm_a_b @ V2 @ Va ) @ nil_se7334864872594423151rm_a_b ) ) ) ) ).

% union_list_sorted.cases
thf(fact_904_union__list__sorted_Ocases,axiom,
    ! [X: produc4312625336439893351rm_a_b] :
      ( ! [X4: fset_P3577670608877728751rm_a_b,Xs2: list_f7931864845379430901rm_a_b,Y4: fset_P3577670608877728751rm_a_b,Ys4: list_f7931864845379430901rm_a_b] :
          ( X
         != ( produc8130127474510664151rm_a_b @ ( cons_f831858257744602021rm_a_b @ X4 @ Xs2 ) @ ( cons_f831858257744602021rm_a_b @ Y4 @ Ys4 ) ) )
     => ( ! [Ys4: list_f7931864845379430901rm_a_b] :
            ( X
           != ( produc8130127474510664151rm_a_b @ nil_fs4020363321024022517rm_a_b @ Ys4 ) )
       => ~ ! [V2: fset_P3577670608877728751rm_a_b,Va: list_f7931864845379430901rm_a_b] :
              ( X
             != ( produc8130127474510664151rm_a_b @ ( cons_f831858257744602021rm_a_b @ V2 @ Va ) @ nil_fs4020363321024022517rm_a_b ) ) ) ) ).

% union_list_sorted.cases
thf(fact_905_shuffles_Ocases,axiom,
    ! [X: produc8638778270066257639rm_a_b] :
      ( ! [Ys4: list_s5629610930313288943rm_a_b] :
          ( X
         != ( produc510695698380873047rm_a_b @ nil_se7334864872594423151rm_a_b @ Ys4 ) )
     => ( ! [Xs2: list_s5629610930313288943rm_a_b] :
            ( X
           != ( produc510695698380873047rm_a_b @ Xs2 @ nil_se7334864872594423151rm_a_b ) )
       => ~ ! [X4: set_Pr4386577575007340137rm_a_b,Xs2: list_s5629610930313288943rm_a_b,Y4: set_Pr4386577575007340137rm_a_b,Ys4: list_s5629610930313288943rm_a_b] :
              ( X
             != ( produc510695698380873047rm_a_b @ ( cons_s7847606222307733791rm_a_b @ X4 @ Xs2 ) @ ( cons_s7847606222307733791rm_a_b @ Y4 @ Ys4 ) ) ) ) ) ).

% shuffles.cases
thf(fact_906_shuffles_Ocases,axiom,
    ! [X: produc4312625336439893351rm_a_b] :
      ( ! [Ys4: list_f7931864845379430901rm_a_b] :
          ( X
         != ( produc8130127474510664151rm_a_b @ nil_fs4020363321024022517rm_a_b @ Ys4 ) )
     => ( ! [Xs2: list_f7931864845379430901rm_a_b] :
            ( X
           != ( produc8130127474510664151rm_a_b @ Xs2 @ nil_fs4020363321024022517rm_a_b ) )
       => ~ ! [X4: fset_P3577670608877728751rm_a_b,Xs2: list_f7931864845379430901rm_a_b,Y4: fset_P3577670608877728751rm_a_b,Ys4: list_f7931864845379430901rm_a_b] :
              ( X
             != ( produc8130127474510664151rm_a_b @ ( cons_f831858257744602021rm_a_b @ X4 @ Xs2 ) @ ( cons_f831858257744602021rm_a_b @ Y4 @ Ys4 ) ) ) ) ) ).

% shuffles.cases
thf(fact_907_ord_Oquicksort__acc_Ocases,axiom,
    ! [X: produc8638778270066257639rm_a_b] :
      ( ! [Ac: list_s5629610930313288943rm_a_b] :
          ( X
         != ( produc510695698380873047rm_a_b @ Ac @ nil_se7334864872594423151rm_a_b ) )
     => ( ! [Ac: list_s5629610930313288943rm_a_b,X4: set_Pr4386577575007340137rm_a_b] :
            ( X
           != ( produc510695698380873047rm_a_b @ Ac @ ( cons_s7847606222307733791rm_a_b @ X4 @ nil_se7334864872594423151rm_a_b ) ) )
       => ~ ! [Ac: list_s5629610930313288943rm_a_b,X4: set_Pr4386577575007340137rm_a_b,V2: set_Pr4386577575007340137rm_a_b,Va: list_s5629610930313288943rm_a_b] :
              ( X
             != ( produc510695698380873047rm_a_b @ Ac @ ( cons_s7847606222307733791rm_a_b @ X4 @ ( cons_s7847606222307733791rm_a_b @ V2 @ Va ) ) ) ) ) ) ).

% ord.quicksort_acc.cases
thf(fact_908_ord_Oquicksort__acc_Ocases,axiom,
    ! [X: produc4312625336439893351rm_a_b] :
      ( ! [Ac: list_f7931864845379430901rm_a_b] :
          ( X
         != ( produc8130127474510664151rm_a_b @ Ac @ nil_fs4020363321024022517rm_a_b ) )
     => ( ! [Ac: list_f7931864845379430901rm_a_b,X4: fset_P3577670608877728751rm_a_b] :
            ( X
           != ( produc8130127474510664151rm_a_b @ Ac @ ( cons_f831858257744602021rm_a_b @ X4 @ nil_fs4020363321024022517rm_a_b ) ) )
       => ~ ! [Ac: list_f7931864845379430901rm_a_b,X4: fset_P3577670608877728751rm_a_b,V2: fset_P3577670608877728751rm_a_b,Va: list_f7931864845379430901rm_a_b] :
              ( X
             != ( produc8130127474510664151rm_a_b @ Ac @ ( cons_f831858257744602021rm_a_b @ X4 @ ( cons_f831858257744602021rm_a_b @ V2 @ Va ) ) ) ) ) ) ).

% ord.quicksort_acc.cases
thf(fact_909_quicksort__acc_Ocases,axiom,
    ! [X: produc8638778270066257639rm_a_b] :
      ( ! [Ac: list_s5629610930313288943rm_a_b] :
          ( X
         != ( produc510695698380873047rm_a_b @ Ac @ nil_se7334864872594423151rm_a_b ) )
     => ( ! [Ac: list_s5629610930313288943rm_a_b,X4: set_Pr4386577575007340137rm_a_b] :
            ( X
           != ( produc510695698380873047rm_a_b @ Ac @ ( cons_s7847606222307733791rm_a_b @ X4 @ nil_se7334864872594423151rm_a_b ) ) )
       => ~ ! [Ac: list_s5629610930313288943rm_a_b,X4: set_Pr4386577575007340137rm_a_b,V2: set_Pr4386577575007340137rm_a_b,Va: list_s5629610930313288943rm_a_b] :
              ( X
             != ( produc510695698380873047rm_a_b @ Ac @ ( cons_s7847606222307733791rm_a_b @ X4 @ ( cons_s7847606222307733791rm_a_b @ V2 @ Va ) ) ) ) ) ) ).

% quicksort_acc.cases
thf(fact_910_quicksort__acc_Ocases,axiom,
    ! [X: produc4312625336439893351rm_a_b] :
      ( ! [Ac: list_f7931864845379430901rm_a_b] :
          ( X
         != ( produc8130127474510664151rm_a_b @ Ac @ nil_fs4020363321024022517rm_a_b ) )
     => ( ! [Ac: list_f7931864845379430901rm_a_b,X4: fset_P3577670608877728751rm_a_b] :
            ( X
           != ( produc8130127474510664151rm_a_b @ Ac @ ( cons_f831858257744602021rm_a_b @ X4 @ nil_fs4020363321024022517rm_a_b ) ) )
       => ~ ! [Ac: list_f7931864845379430901rm_a_b,X4: fset_P3577670608877728751rm_a_b,V2: fset_P3577670608877728751rm_a_b,Va: list_f7931864845379430901rm_a_b] :
              ( X
             != ( produc8130127474510664151rm_a_b @ Ac @ ( cons_f831858257744602021rm_a_b @ X4 @ ( cons_f831858257744602021rm_a_b @ V2 @ Va ) ) ) ) ) ) ).

% quicksort_acc.cases
thf(fact_911_proper__intrvl_Oproper__interval__set__aux_Ocases,axiom,
    ! [X: produc8638778270066257639rm_a_b] :
      ( ! [Xs2: list_s5629610930313288943rm_a_b] :
          ( X
         != ( produc510695698380873047rm_a_b @ Xs2 @ nil_se7334864872594423151rm_a_b ) )
     => ( ! [Y4: set_Pr4386577575007340137rm_a_b,Ys4: list_s5629610930313288943rm_a_b] :
            ( X
           != ( produc510695698380873047rm_a_b @ nil_se7334864872594423151rm_a_b @ ( cons_s7847606222307733791rm_a_b @ Y4 @ Ys4 ) ) )
       => ~ ! [X4: set_Pr4386577575007340137rm_a_b,Xs2: list_s5629610930313288943rm_a_b,Y4: set_Pr4386577575007340137rm_a_b,Ys4: list_s5629610930313288943rm_a_b] :
              ( X
             != ( produc510695698380873047rm_a_b @ ( cons_s7847606222307733791rm_a_b @ X4 @ Xs2 ) @ ( cons_s7847606222307733791rm_a_b @ Y4 @ Ys4 ) ) ) ) ) ).

% proper_intrvl.proper_interval_set_aux.cases
thf(fact_912_proper__intrvl_Oproper__interval__set__aux_Ocases,axiom,
    ! [X: produc4312625336439893351rm_a_b] :
      ( ! [Xs2: list_f7931864845379430901rm_a_b] :
          ( X
         != ( produc8130127474510664151rm_a_b @ Xs2 @ nil_fs4020363321024022517rm_a_b ) )
     => ( ! [Y4: fset_P3577670608877728751rm_a_b,Ys4: list_f7931864845379430901rm_a_b] :
            ( X
           != ( produc8130127474510664151rm_a_b @ nil_fs4020363321024022517rm_a_b @ ( cons_f831858257744602021rm_a_b @ Y4 @ Ys4 ) ) )
       => ~ ! [X4: fset_P3577670608877728751rm_a_b,Xs2: list_f7931864845379430901rm_a_b,Y4: fset_P3577670608877728751rm_a_b,Ys4: list_f7931864845379430901rm_a_b] :
              ( X
             != ( produc8130127474510664151rm_a_b @ ( cons_f831858257744602021rm_a_b @ X4 @ Xs2 ) @ ( cons_f831858257744602021rm_a_b @ Y4 @ Ys4 ) ) ) ) ) ).

% proper_intrvl.proper_interval_set_aux.cases
thf(fact_913_zip__option_Ocases,axiom,
    ! [X: produc8638778270066257639rm_a_b] :
      ( ( X
       != ( produc510695698380873047rm_a_b @ nil_se7334864872594423151rm_a_b @ nil_se7334864872594423151rm_a_b ) )
     => ( ! [X4: set_Pr4386577575007340137rm_a_b,Xs2: list_s5629610930313288943rm_a_b,Y4: set_Pr4386577575007340137rm_a_b,Ys4: list_s5629610930313288943rm_a_b] :
            ( X
           != ( produc510695698380873047rm_a_b @ ( cons_s7847606222307733791rm_a_b @ X4 @ Xs2 ) @ ( cons_s7847606222307733791rm_a_b @ Y4 @ Ys4 ) ) )
       => ( ! [X4: set_Pr4386577575007340137rm_a_b,Xs2: list_s5629610930313288943rm_a_b] :
              ( X
             != ( produc510695698380873047rm_a_b @ ( cons_s7847606222307733791rm_a_b @ X4 @ Xs2 ) @ nil_se7334864872594423151rm_a_b ) )
         => ~ ! [Y4: set_Pr4386577575007340137rm_a_b,Ys4: list_s5629610930313288943rm_a_b] :
                ( X
               != ( produc510695698380873047rm_a_b @ nil_se7334864872594423151rm_a_b @ ( cons_s7847606222307733791rm_a_b @ Y4 @ Ys4 ) ) ) ) ) ) ).

% zip_option.cases
thf(fact_914_zip__option_Ocases,axiom,
    ! [X: produc7246826262118695533rm_a_b] :
      ( ( X
       != ( produc4120348330684003933rm_a_b @ nil_se7334864872594423151rm_a_b @ nil_fs4020363321024022517rm_a_b ) )
     => ( ! [X4: set_Pr4386577575007340137rm_a_b,Xs2: list_s5629610930313288943rm_a_b,Y4: fset_P3577670608877728751rm_a_b,Ys4: list_f7931864845379430901rm_a_b] :
            ( X
           != ( produc4120348330684003933rm_a_b @ ( cons_s7847606222307733791rm_a_b @ X4 @ Xs2 ) @ ( cons_f831858257744602021rm_a_b @ Y4 @ Ys4 ) ) )
       => ( ! [X4: set_Pr4386577575007340137rm_a_b,Xs2: list_s5629610930313288943rm_a_b] :
              ( X
             != ( produc4120348330684003933rm_a_b @ ( cons_s7847606222307733791rm_a_b @ X4 @ Xs2 ) @ nil_fs4020363321024022517rm_a_b ) )
         => ~ ! [Y4: fset_P3577670608877728751rm_a_b,Ys4: list_f7931864845379430901rm_a_b] :
                ( X
               != ( produc4120348330684003933rm_a_b @ nil_se7334864872594423151rm_a_b @ ( cons_f831858257744602021rm_a_b @ Y4 @ Ys4 ) ) ) ) ) ) ).

% zip_option.cases
thf(fact_915_zip__option_Ocases,axiom,
    ! [X: produc6634920076183926497rm_a_b] :
      ( ( X
       != ( produc5878835993175872977rm_a_b @ nil_fs4020363321024022517rm_a_b @ nil_se7334864872594423151rm_a_b ) )
     => ( ! [X4: fset_P3577670608877728751rm_a_b,Xs2: list_f7931864845379430901rm_a_b,Y4: set_Pr4386577575007340137rm_a_b,Ys4: list_s5629610930313288943rm_a_b] :
            ( X
           != ( produc5878835993175872977rm_a_b @ ( cons_f831858257744602021rm_a_b @ X4 @ Xs2 ) @ ( cons_s7847606222307733791rm_a_b @ Y4 @ Ys4 ) ) )
       => ( ! [X4: fset_P3577670608877728751rm_a_b,Xs2: list_f7931864845379430901rm_a_b] :
              ( X
             != ( produc5878835993175872977rm_a_b @ ( cons_f831858257744602021rm_a_b @ X4 @ Xs2 ) @ nil_se7334864872594423151rm_a_b ) )
         => ~ ! [Y4: set_Pr4386577575007340137rm_a_b,Ys4: list_s5629610930313288943rm_a_b] :
                ( X
               != ( produc5878835993175872977rm_a_b @ nil_fs4020363321024022517rm_a_b @ ( cons_s7847606222307733791rm_a_b @ Y4 @ Ys4 ) ) ) ) ) ) ).

% zip_option.cases
thf(fact_916_zip__option_Ocases,axiom,
    ! [X: produc4312625336439893351rm_a_b] :
      ( ( X
       != ( produc8130127474510664151rm_a_b @ nil_fs4020363321024022517rm_a_b @ nil_fs4020363321024022517rm_a_b ) )
     => ( ! [X4: fset_P3577670608877728751rm_a_b,Xs2: list_f7931864845379430901rm_a_b,Y4: fset_P3577670608877728751rm_a_b,Ys4: list_f7931864845379430901rm_a_b] :
            ( X
           != ( produc8130127474510664151rm_a_b @ ( cons_f831858257744602021rm_a_b @ X4 @ Xs2 ) @ ( cons_f831858257744602021rm_a_b @ Y4 @ Ys4 ) ) )
       => ( ! [X4: fset_P3577670608877728751rm_a_b,Xs2: list_f7931864845379430901rm_a_b] :
              ( X
             != ( produc8130127474510664151rm_a_b @ ( cons_f831858257744602021rm_a_b @ X4 @ Xs2 ) @ nil_fs4020363321024022517rm_a_b ) )
         => ~ ! [Y4: fset_P3577670608877728751rm_a_b,Ys4: list_f7931864845379430901rm_a_b] :
                ( X
               != ( produc8130127474510664151rm_a_b @ nil_fs4020363321024022517rm_a_b @ ( cons_f831858257744602021rm_a_b @ Y4 @ Ys4 ) ) ) ) ) ) ).

% zip_option.cases
thf(fact_917_list__of__permutation__element__n_Opelims,axiom,
    ! [X: set_Pr4386577575007340137rm_a_b,Xa: nat,Xb: list_s5629610930313288943rm_a_b,Y: list_l6192929068897674997rm_a_b] :
      ( ( ( basic_1071435344846903220rm_a_b @ X @ Xa @ Xb )
        = Y )
     => ( ( accp_P8087403281110482147rm_a_b @ basic_3690947911968671523rm_a_b @ ( produc5324509014790280742rm_a_b @ X @ ( produc2544545499933428934rm_a_b @ Xa @ Xb ) ) )
       => ( ( ( Xa = zero_zero_nat )
           => ( ( Y
                = ( cons_l92702159317333157rm_a_b @ nil_se7334864872594423151rm_a_b @ nil_li1283984768393418997rm_a_b ) )
             => ~ ( accp_P8087403281110482147rm_a_b @ basic_3690947911968671523rm_a_b @ ( produc5324509014790280742rm_a_b @ X @ ( produc2544545499933428934rm_a_b @ zero_zero_nat @ Xb ) ) ) ) )
         => ~ ! [N2: nat] :
                ( ( Xa
                  = ( suc @ N2 ) )
               => ( ( Y
                    = ( concat3770752595006032944rm_a_b @ ( map_li5449371397125115274rm_a_b @ ( basic_6852907471535255442rm_a_b @ X ) @ ( n_list6731552439858285943rm_a_b @ N2 @ Xb ) ) ) )
                 => ~ ( accp_P8087403281110482147rm_a_b @ basic_3690947911968671523rm_a_b @ ( produc5324509014790280742rm_a_b @ X @ ( produc2544545499933428934rm_a_b @ ( suc @ N2 ) @ Xb ) ) ) ) ) ) ) ) ).

% list_of_permutation_element_n.pelims
thf(fact_918_list__of__permutation__element__n_Opelims,axiom,
    ! [X: fset_P3577670608877728751rm_a_b,Xa: nat,Xb: list_f7931864845379430901rm_a_b,Y: list_l3960025731926578043rm_a_b] :
      ( ( ( basic_205589215891509178rm_a_b @ X @ Xa @ Xb )
        = Y )
     => ( ( accp_P3725179739350892143rm_a_b @ basic_6122076941655046057rm_a_b @ ( produc7634437889667727026rm_a_b @ X @ ( produc2848911566537383500rm_a_b @ Xa @ Xb ) ) )
       => ( ( ( Xa = zero_zero_nat )
           => ( ( Y
                = ( cons_l2079606646850324139rm_a_b @ nil_fs4020363321024022517rm_a_b @ nil_li2074289066168648443rm_a_b ) )
             => ~ ( accp_P3725179739350892143rm_a_b @ basic_6122076941655046057rm_a_b @ ( produc7634437889667727026rm_a_b @ X @ ( produc2848911566537383500rm_a_b @ zero_zero_nat @ Xb ) ) ) ) )
         => ~ ! [N2: nat] :
                ( ( Xa
                  = ( suc @ N2 ) )
               => ( ( Y
                    = ( concat7128574750897932470rm_a_b @ ( map_li8124276902986747786rm_a_b @ ( basic_5157831931618400408rm_a_b @ X ) @ ( n_list7988791480787244029rm_a_b @ N2 @ Xb ) ) ) )
                 => ~ ( accp_P3725179739350892143rm_a_b @ basic_6122076941655046057rm_a_b @ ( produc7634437889667727026rm_a_b @ X @ ( produc2848911566537383500rm_a_b @ ( suc @ N2 ) @ Xb ) ) ) ) ) ) ) ) ).

% list_of_permutation_element_n.pelims
thf(fact_919_concat__index__split__less__length__concat_I4_J,axiom,
    ! [I4: nat,Xs: list_l3960025731926578043rm_a_b,M: nat,N: nat] :
      ( ( ord_less_nat @ I4 @ ( size_s1967422219310127201rm_a_b @ ( concat8439565143799850800rm_a_b @ Xs ) ) )
     => ( ( ( concat3863192905831170157rm_a_b @ ( product_Pair_nat_nat @ zero_zero_nat @ I4 ) @ Xs )
          = ( product_Pair_nat_nat @ M @ N ) )
       => ( ( nth_fs3949069584078227574rm_a_b @ ( concat8439565143799850800rm_a_b @ Xs ) @ I4 )
          = ( nth_fs3949069584078227574rm_a_b @ ( nth_li947802356562890492rm_a_b @ Xs @ M ) @ N ) ) ) ) ).

% concat_index_split_less_length_concat(4)
thf(fact_920_concat__index__split__unique,axiom,
    ! [I4: nat,Xs: list_l3960025731926578043rm_a_b,Ys2: list_l3960025731926578043rm_a_b,N: nat] :
      ( ( ord_less_nat @ I4 @ ( size_s1967422219310127201rm_a_b @ ( concat8439565143799850800rm_a_b @ Xs ) ) )
     => ( ( ( size_s7600236048634125287rm_a_b @ Xs )
          = ( size_s7600236048634125287rm_a_b @ Ys2 ) )
       => ( ! [I: nat] :
              ( ( ord_less_nat @ I @ ( size_s7600236048634125287rm_a_b @ Xs ) )
             => ( ( size_s1967422219310127201rm_a_b @ ( nth_li947802356562890492rm_a_b @ Xs @ I ) )
                = ( size_s1967422219310127201rm_a_b @ ( nth_li947802356562890492rm_a_b @ Ys2 @ I ) ) ) )
         => ( ( concat3863192905831170157rm_a_b @ ( product_Pair_nat_nat @ N @ I4 ) @ Xs )
            = ( concat3863192905831170157rm_a_b @ ( product_Pair_nat_nat @ N @ I4 ) @ Ys2 ) ) ) ) ) ).

% concat_index_split_unique
thf(fact_921_concat__index__split__less__length__concat_I3_J,axiom,
    ! [I4: nat,Xs: list_l3960025731926578043rm_a_b,M: nat,N: nat] :
      ( ( ord_less_nat @ I4 @ ( size_s1967422219310127201rm_a_b @ ( concat8439565143799850800rm_a_b @ Xs ) ) )
     => ( ( ( concat3863192905831170157rm_a_b @ ( product_Pair_nat_nat @ zero_zero_nat @ I4 ) @ Xs )
          = ( product_Pair_nat_nat @ M @ N ) )
       => ( ord_less_nat @ N @ ( size_s1967422219310127201rm_a_b @ ( nth_li947802356562890492rm_a_b @ Xs @ M ) ) ) ) ) ).

% concat_index_split_less_length_concat(3)
thf(fact_922_concat__index__split__less__length__concat_I2_J,axiom,
    ! [I4: nat,Xs: list_l3960025731926578043rm_a_b,M: nat,N: nat] :
      ( ( ord_less_nat @ I4 @ ( size_s1967422219310127201rm_a_b @ ( concat8439565143799850800rm_a_b @ Xs ) ) )
     => ( ( ( concat3863192905831170157rm_a_b @ ( product_Pair_nat_nat @ zero_zero_nat @ I4 ) @ Xs )
          = ( product_Pair_nat_nat @ M @ N ) )
       => ( ord_less_nat @ M @ ( size_s7600236048634125287rm_a_b @ Xs ) ) ) ) ).

% concat_index_split_less_length_concat(2)
thf(fact_923_Utils_Otrancl__list_Olist__trancl,axiom,
    ! [Xs: list_term_a_b,Ys2: list_term_a_b,R: set_Pr4386577575007340137rm_a_b,I4: nat,Z4: term_a_b] :
      ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs @ Ys2 ) @ ( trancl_list_term_a_b @ R ) )
     => ( ( ord_less_nat @ I4 @ ( size_s8906293707977694520rm_a_b @ Ys2 ) )
       => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( nth_term_a_b @ Ys2 @ I4 ) @ Z4 ) @ R )
         => ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs @ ( list_update_term_a_b @ Ys2 @ I4 @ Z4 ) ) @ ( trancl_list_term_a_b @ R ) ) ) ) ) ).

% Utils.trancl_list.list_trancl
thf(fact_924_Utils_Otrancl__list_Olist__trancl,axiom,
    ! [Xs: list_f7931864845379430901rm_a_b,Ys2: list_f7931864845379430901rm_a_b,R: set_Pr3641583360614959815rm_a_b,I4: nat,Z4: fset_P3577670608877728751rm_a_b] :
      ( ( member6557215607983664784rm_a_b @ ( produc8130127474510664151rm_a_b @ Xs @ Ys2 ) @ ( trancl6568930836738952055rm_a_b @ R ) )
     => ( ( ord_less_nat @ I4 @ ( size_s1967422219310127201rm_a_b @ Ys2 ) )
       => ( ( member2354150401426528528rm_a_b @ ( produc6898453836806167895rm_a_b @ ( nth_fs3949069584078227574rm_a_b @ Ys2 @ I4 ) @ Z4 ) @ R )
         => ( member6557215607983664784rm_a_b @ ( produc8130127474510664151rm_a_b @ Xs @ ( list_u7212055091542668893rm_a_b @ Ys2 @ I4 @ Z4 ) ) @ ( trancl6568930836738952055rm_a_b @ R ) ) ) ) ) ).

% Utils.trancl_list.list_trancl
thf(fact_925_Utils_Otrancl__list_Osimps,axiom,
    ! [A1: list_term_a_b,A22: list_term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ A1 @ A22 ) @ ( trancl_list_term_a_b @ R ) )
      = ( ? [Xs3: list_term_a_b,Ys3: list_term_a_b] :
            ( ( A1 = Xs3 )
            & ( A22 = Ys3 )
            & ( ( size_s8906293707977694520rm_a_b @ Xs3 )
              = ( size_s8906293707977694520rm_a_b @ Ys3 ) )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ ( size_s8906293707977694520rm_a_b @ Ys3 ) )
               => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( nth_term_a_b @ Xs3 @ I3 ) @ ( nth_term_a_b @ Ys3 @ I3 ) ) @ R ) ) )
        | ? [Xs3: list_term_a_b,Ys3: list_term_a_b,I3: nat,Z3: term_a_b] :
            ( ( A1 = Xs3 )
            & ( A22
              = ( list_update_term_a_b @ Ys3 @ I3 @ Z3 ) )
            & ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs3 @ Ys3 ) @ ( trancl_list_term_a_b @ R ) )
            & ( ord_less_nat @ I3 @ ( size_s8906293707977694520rm_a_b @ Ys3 ) )
            & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( nth_term_a_b @ Ys3 @ I3 ) @ Z3 ) @ R ) ) ) ) ).

% Utils.trancl_list.simps
thf(fact_926_Utils_Otrancl__list_Osimps,axiom,
    ! [A1: list_f7931864845379430901rm_a_b,A22: list_f7931864845379430901rm_a_b,R: set_Pr3641583360614959815rm_a_b] :
      ( ( member6557215607983664784rm_a_b @ ( produc8130127474510664151rm_a_b @ A1 @ A22 ) @ ( trancl6568930836738952055rm_a_b @ R ) )
      = ( ? [Xs3: list_f7931864845379430901rm_a_b,Ys3: list_f7931864845379430901rm_a_b] :
            ( ( A1 = Xs3 )
            & ( A22 = Ys3 )
            & ( ( size_s1967422219310127201rm_a_b @ Xs3 )
              = ( size_s1967422219310127201rm_a_b @ Ys3 ) )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ ( size_s1967422219310127201rm_a_b @ Ys3 ) )
               => ( member2354150401426528528rm_a_b @ ( produc6898453836806167895rm_a_b @ ( nth_fs3949069584078227574rm_a_b @ Xs3 @ I3 ) @ ( nth_fs3949069584078227574rm_a_b @ Ys3 @ I3 ) ) @ R ) ) )
        | ? [Xs3: list_f7931864845379430901rm_a_b,Ys3: list_f7931864845379430901rm_a_b,I3: nat,Z3: fset_P3577670608877728751rm_a_b] :
            ( ( A1 = Xs3 )
            & ( A22
              = ( list_u7212055091542668893rm_a_b @ Ys3 @ I3 @ Z3 ) )
            & ( member6557215607983664784rm_a_b @ ( produc8130127474510664151rm_a_b @ Xs3 @ Ys3 ) @ ( trancl6568930836738952055rm_a_b @ R ) )
            & ( ord_less_nat @ I3 @ ( size_s1967422219310127201rm_a_b @ Ys3 ) )
            & ( member2354150401426528528rm_a_b @ ( produc6898453836806167895rm_a_b @ ( nth_fs3949069584078227574rm_a_b @ Ys3 @ I3 ) @ Z3 ) @ R ) ) ) ) ).

% Utils.trancl_list.simps
thf(fact_927_Utils_Otrancl__list__appendI,axiom,
    ! [Xs: list_term_a_b,Ys2: list_term_a_b,R: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b] :
      ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs @ Ys2 ) @ ( trancl_list_term_a_b @ R ) )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R )
       => ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ ( cons_term_a_b @ X @ Xs ) @ ( cons_term_a_b @ Y @ Ys2 ) ) @ ( trancl_list_term_a_b @ R ) ) ) ) ).

% Utils.trancl_list_appendI
thf(fact_928_Utils_Otrancl__list_Obase,axiom,
    ! [Xs: list_term_a_b,Ys2: list_term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( ( size_s8906293707977694520rm_a_b @ Xs )
        = ( size_s8906293707977694520rm_a_b @ Ys2 ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s8906293707977694520rm_a_b @ Ys2 ) )
           => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( nth_term_a_b @ Xs @ I ) @ ( nth_term_a_b @ Ys2 @ I ) ) @ R ) )
       => ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs @ Ys2 ) @ ( trancl_list_term_a_b @ R ) ) ) ) ).

% Utils.trancl_list.base
thf(fact_929_Utils_Otrancl__list_Obase,axiom,
    ! [Xs: list_f7931864845379430901rm_a_b,Ys2: list_f7931864845379430901rm_a_b,R: set_Pr3641583360614959815rm_a_b] :
      ( ( ( size_s1967422219310127201rm_a_b @ Xs )
        = ( size_s1967422219310127201rm_a_b @ Ys2 ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s1967422219310127201rm_a_b @ Ys2 ) )
           => ( member2354150401426528528rm_a_b @ ( produc6898453836806167895rm_a_b @ ( nth_fs3949069584078227574rm_a_b @ Xs @ I ) @ ( nth_fs3949069584078227574rm_a_b @ Ys2 @ I ) ) @ R ) )
       => ( member6557215607983664784rm_a_b @ ( produc8130127474510664151rm_a_b @ Xs @ Ys2 ) @ ( trancl6568930836738952055rm_a_b @ R ) ) ) ) ).

% Utils.trancl_list.base
thf(fact_930_Utils_Otrancl__list_Ocases,axiom,
    ! [A1: list_term_a_b,A22: list_term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ A1 @ A22 ) @ ( trancl_list_term_a_b @ R ) )
     => ( ( ( ( size_s8906293707977694520rm_a_b @ A1 )
            = ( size_s8906293707977694520rm_a_b @ A22 ) )
         => ~ ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ ( size_s8906293707977694520rm_a_b @ A22 ) )
               => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( nth_term_a_b @ A1 @ I2 ) @ ( nth_term_a_b @ A22 @ I2 ) ) @ R ) ) )
       => ~ ! [Ys4: list_term_a_b,I: nat,Z: term_a_b] :
              ( ( A22
                = ( list_update_term_a_b @ Ys4 @ I @ Z ) )
             => ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ A1 @ Ys4 ) @ ( trancl_list_term_a_b @ R ) )
               => ( ( ord_less_nat @ I @ ( size_s8906293707977694520rm_a_b @ Ys4 ) )
                 => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( nth_term_a_b @ Ys4 @ I ) @ Z ) @ R ) ) ) ) ) ) ).

% Utils.trancl_list.cases
thf(fact_931_Utils_Otrancl__list_Ocases,axiom,
    ! [A1: list_f7931864845379430901rm_a_b,A22: list_f7931864845379430901rm_a_b,R: set_Pr3641583360614959815rm_a_b] :
      ( ( member6557215607983664784rm_a_b @ ( produc8130127474510664151rm_a_b @ A1 @ A22 ) @ ( trancl6568930836738952055rm_a_b @ R ) )
     => ( ( ( ( size_s1967422219310127201rm_a_b @ A1 )
            = ( size_s1967422219310127201rm_a_b @ A22 ) )
         => ~ ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ ( size_s1967422219310127201rm_a_b @ A22 ) )
               => ( member2354150401426528528rm_a_b @ ( produc6898453836806167895rm_a_b @ ( nth_fs3949069584078227574rm_a_b @ A1 @ I2 ) @ ( nth_fs3949069584078227574rm_a_b @ A22 @ I2 ) ) @ R ) ) )
       => ~ ! [Ys4: list_f7931864845379430901rm_a_b,I: nat,Z: fset_P3577670608877728751rm_a_b] :
              ( ( A22
                = ( list_u7212055091542668893rm_a_b @ Ys4 @ I @ Z ) )
             => ( ( member6557215607983664784rm_a_b @ ( produc8130127474510664151rm_a_b @ A1 @ Ys4 ) @ ( trancl6568930836738952055rm_a_b @ R ) )
               => ( ( ord_less_nat @ I @ ( size_s1967422219310127201rm_a_b @ Ys4 ) )
                 => ~ ( member2354150401426528528rm_a_b @ ( produc6898453836806167895rm_a_b @ ( nth_fs3949069584078227574rm_a_b @ Ys4 @ I ) @ Z ) @ R ) ) ) ) ) ) ).

% Utils.trancl_list.cases
thf(fact_932_concat__index__split__sound__fst__arg,axiom,
    ! [I4: nat,Xs: list_l3960025731926578043rm_a_b] :
      ( ( ord_less_nat @ I4 @ ( size_s1967422219310127201rm_a_b @ ( concat8439565143799850800rm_a_b @ Xs ) ) )
     => ( ord_less_nat @ ( product_fst_nat_nat @ ( concat3863192905831170157rm_a_b @ ( product_Pair_nat_nat @ zero_zero_nat @ I4 ) @ Xs ) ) @ ( size_s7600236048634125287rm_a_b @ Xs ) ) ) ).

% concat_index_split_sound_fst_arg
thf(fact_933_Cons__lenlex__iff,axiom,
    ! [M: term_a_b,Ms: list_term_a_b,N: term_a_b,Ns: list_term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
      ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ ( cons_term_a_b @ M @ Ms ) @ ( cons_term_a_b @ N @ Ns ) ) @ ( lenlex_term_a_b @ R3 ) )
      = ( ( ord_less_nat @ ( size_s8906293707977694520rm_a_b @ Ms ) @ ( size_s8906293707977694520rm_a_b @ Ns ) )
        | ( ( ( size_s8906293707977694520rm_a_b @ Ms )
            = ( size_s8906293707977694520rm_a_b @ Ns ) )
          & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ M @ N ) @ R3 ) )
        | ( ( M = N )
          & ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Ms @ Ns ) @ ( lenlex_term_a_b @ R3 ) ) ) ) ) ).

% Cons_lenlex_iff
thf(fact_934_Cons__lenlex__iff,axiom,
    ! [M: fset_P3577670608877728751rm_a_b,Ms: list_f7931864845379430901rm_a_b,N: fset_P3577670608877728751rm_a_b,Ns: list_f7931864845379430901rm_a_b,R3: set_Pr3641583360614959815rm_a_b] :
      ( ( member6557215607983664784rm_a_b @ ( produc8130127474510664151rm_a_b @ ( cons_f831858257744602021rm_a_b @ M @ Ms ) @ ( cons_f831858257744602021rm_a_b @ N @ Ns ) ) @ ( lenlex227106943944653562rm_a_b @ R3 ) )
      = ( ( ord_less_nat @ ( size_s1967422219310127201rm_a_b @ Ms ) @ ( size_s1967422219310127201rm_a_b @ Ns ) )
        | ( ( ( size_s1967422219310127201rm_a_b @ Ms )
            = ( size_s1967422219310127201rm_a_b @ Ns ) )
          & ( member2354150401426528528rm_a_b @ ( produc6898453836806167895rm_a_b @ M @ N ) @ R3 ) )
        | ( ( M = N )
          & ( member6557215607983664784rm_a_b @ ( produc8130127474510664151rm_a_b @ Ms @ Ns ) @ ( lenlex227106943944653562rm_a_b @ R3 ) ) ) ) ) ).

% Cons_lenlex_iff
thf(fact_935_Cons__in__lex,axiom,
    ! [X: term_a_b,Xs: list_term_a_b,Y: term_a_b,Ys2: list_term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
      ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ ( cons_term_a_b @ X @ Xs ) @ ( cons_term_a_b @ Y @ Ys2 ) ) @ ( lex_term_a_b @ R3 ) )
      = ( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R3 )
          & ( ( size_s8906293707977694520rm_a_b @ Xs )
            = ( size_s8906293707977694520rm_a_b @ Ys2 ) ) )
        | ( ( X = Y )
          & ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs @ Ys2 ) @ ( lex_term_a_b @ R3 ) ) ) ) ) ).

% Cons_in_lex
thf(fact_936_Cons__in__lex,axiom,
    ! [X: fset_P3577670608877728751rm_a_b,Xs: list_f7931864845379430901rm_a_b,Y: fset_P3577670608877728751rm_a_b,Ys2: list_f7931864845379430901rm_a_b,R3: set_Pr3641583360614959815rm_a_b] :
      ( ( member6557215607983664784rm_a_b @ ( produc8130127474510664151rm_a_b @ ( cons_f831858257744602021rm_a_b @ X @ Xs ) @ ( cons_f831858257744602021rm_a_b @ Y @ Ys2 ) ) @ ( lex_fs5102511641744928345rm_a_b @ R3 ) )
      = ( ( ( member2354150401426528528rm_a_b @ ( produc6898453836806167895rm_a_b @ X @ Y ) @ R3 )
          & ( ( size_s1967422219310127201rm_a_b @ Xs )
            = ( size_s1967422219310127201rm_a_b @ Ys2 ) ) )
        | ( ( X = Y )
          & ( member6557215607983664784rm_a_b @ ( produc8130127474510664151rm_a_b @ Xs @ Ys2 ) @ ( lex_fs5102511641744928345rm_a_b @ R3 ) ) ) ) ) ).

% Cons_in_lex
thf(fact_937_Nil__lenlex__iff1,axiom,
    ! [Ns: list_s5629610930313288943rm_a_b,R3: set_Pr647378613824986951rm_a_b] :
      ( ( member4739679974255130896rm_a_b @ ( produc510695698380873047rm_a_b @ nil_se7334864872594423151rm_a_b @ Ns ) @ ( lenlex3929883553291215604rm_a_b @ R3 ) )
      = ( Ns != nil_se7334864872594423151rm_a_b ) ) ).

% Nil_lenlex_iff1
thf(fact_938_Nil__lenlex__iff1,axiom,
    ! [Ns: list_f7931864845379430901rm_a_b,R3: set_Pr3641583360614959815rm_a_b] :
      ( ( member6557215607983664784rm_a_b @ ( produc8130127474510664151rm_a_b @ nil_fs4020363321024022517rm_a_b @ Ns ) @ ( lenlex227106943944653562rm_a_b @ R3 ) )
      = ( Ns != nil_fs4020363321024022517rm_a_b ) ) ).

% Nil_lenlex_iff1
thf(fact_939_Nil2__notin__lex,axiom,
    ! [Xs: list_s5629610930313288943rm_a_b,R3: set_Pr647378613824986951rm_a_b] :
      ~ ( member4739679974255130896rm_a_b @ ( produc510695698380873047rm_a_b @ Xs @ nil_se7334864872594423151rm_a_b ) @ ( lex_se2012386382219318739rm_a_b @ R3 ) ) ).

% Nil2_notin_lex
thf(fact_940_Nil2__notin__lex,axiom,
    ! [Xs: list_f7931864845379430901rm_a_b,R3: set_Pr3641583360614959815rm_a_b] :
      ~ ( member6557215607983664784rm_a_b @ ( produc8130127474510664151rm_a_b @ Xs @ nil_fs4020363321024022517rm_a_b ) @ ( lex_fs5102511641744928345rm_a_b @ R3 ) ) ).

% Nil2_notin_lex
thf(fact_941_Nil__notin__lex,axiom,
    ! [Ys2: list_s5629610930313288943rm_a_b,R3: set_Pr647378613824986951rm_a_b] :
      ~ ( member4739679974255130896rm_a_b @ ( produc510695698380873047rm_a_b @ nil_se7334864872594423151rm_a_b @ Ys2 ) @ ( lex_se2012386382219318739rm_a_b @ R3 ) ) ).

% Nil_notin_lex
thf(fact_942_Nil__notin__lex,axiom,
    ! [Ys2: list_f7931864845379430901rm_a_b,R3: set_Pr3641583360614959815rm_a_b] :
      ~ ( member6557215607983664784rm_a_b @ ( produc8130127474510664151rm_a_b @ nil_fs4020363321024022517rm_a_b @ Ys2 ) @ ( lex_fs5102511641744928345rm_a_b @ R3 ) ) ).

% Nil_notin_lex
thf(fact_943_lenlex__irreflexive,axiom,
    ! [R3: set_Pr4386577575007340137rm_a_b,Xs: list_term_a_b] :
      ( ! [X4: term_a_b] :
          ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ X4 ) @ R3 )
     => ~ ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs @ Xs ) @ ( lenlex_term_a_b @ R3 ) ) ) ).

% lenlex_irreflexive
thf(fact_944_Nil__lenlex__iff2,axiom,
    ! [Ns: list_s5629610930313288943rm_a_b,R3: set_Pr647378613824986951rm_a_b] :
      ~ ( member4739679974255130896rm_a_b @ ( produc510695698380873047rm_a_b @ Ns @ nil_se7334864872594423151rm_a_b ) @ ( lenlex3929883553291215604rm_a_b @ R3 ) ) ).

% Nil_lenlex_iff2
thf(fact_945_Nil__lenlex__iff2,axiom,
    ! [Ns: list_f7931864845379430901rm_a_b,R3: set_Pr3641583360614959815rm_a_b] :
      ~ ( member6557215607983664784rm_a_b @ ( produc8130127474510664151rm_a_b @ Ns @ nil_fs4020363321024022517rm_a_b ) @ ( lenlex227106943944653562rm_a_b @ R3 ) ) ).

% Nil_lenlex_iff2
thf(fact_946_lenlex__length,axiom,
    ! [Ms: list_f7931864845379430901rm_a_b,Ns: list_f7931864845379430901rm_a_b,R3: set_Pr3641583360614959815rm_a_b] :
      ( ( member6557215607983664784rm_a_b @ ( produc8130127474510664151rm_a_b @ Ms @ Ns ) @ ( lenlex227106943944653562rm_a_b @ R3 ) )
     => ( ord_less_eq_nat @ ( size_s1967422219310127201rm_a_b @ Ms ) @ ( size_s1967422219310127201rm_a_b @ Ns ) ) ) ).

% lenlex_length
thf(fact_947_concat__index__split__mono__first__arg,axiom,
    ! [I4: nat,Xs: list_l3960025731926578043rm_a_b,O_idx: nat] :
      ( ( ord_less_nat @ I4 @ ( size_s1967422219310127201rm_a_b @ ( concat8439565143799850800rm_a_b @ Xs ) ) )
     => ( ord_less_eq_nat @ O_idx @ ( product_fst_nat_nat @ ( concat3863192905831170157rm_a_b @ ( product_Pair_nat_nat @ O_idx @ I4 ) @ Xs ) ) ) ) ).

% concat_index_split_mono_first_arg
thf(fact_948_concat__index__split__sound__snd__arg,axiom,
    ! [I4: nat,Xs: list_l3960025731926578043rm_a_b] :
      ( ( ord_less_nat @ I4 @ ( size_s1967422219310127201rm_a_b @ ( concat8439565143799850800rm_a_b @ Xs ) ) )
     => ( ord_less_nat @ ( product_snd_nat_nat @ ( concat3863192905831170157rm_a_b @ ( product_Pair_nat_nat @ zero_zero_nat @ I4 ) @ Xs ) ) @ ( size_s1967422219310127201rm_a_b @ ( nth_li947802356562890492rm_a_b @ Xs @ ( product_fst_nat_nat @ ( concat3863192905831170157rm_a_b @ ( product_Pair_nat_nat @ zero_zero_nat @ I4 ) @ Xs ) ) ) ) ) ) ).

% concat_index_split_sound_snd_arg
thf(fact_949_concat__index__split__sound__fst__arg__aux,axiom,
    ! [I4: nat,Xs: list_l3960025731926578043rm_a_b,O_idx: nat] :
      ( ( ord_less_nat @ I4 @ ( size_s1967422219310127201rm_a_b @ ( concat8439565143799850800rm_a_b @ Xs ) ) )
     => ( ord_less_nat @ ( product_fst_nat_nat @ ( concat3863192905831170157rm_a_b @ ( product_Pair_nat_nat @ O_idx @ I4 ) @ Xs ) ) @ ( plus_plus_nat @ ( size_s7600236048634125287rm_a_b @ Xs ) @ O_idx ) ) ) ).

% concat_index_split_sound_fst_arg_aux
thf(fact_950_listrel1__iff__update,axiom,
    ! [Xs: list_term_a_b,Ys2: list_term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
      ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs @ Ys2 ) @ ( listrel1_term_a_b @ R3 ) )
      = ( ? [Y2: term_a_b,N3: nat] :
            ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( nth_term_a_b @ Xs @ N3 ) @ Y2 ) @ R3 )
            & ( ord_less_nat @ N3 @ ( size_s8906293707977694520rm_a_b @ Xs ) )
            & ( Ys2
              = ( list_update_term_a_b @ Xs @ N3 @ Y2 ) ) ) ) ) ).

% listrel1_iff_update
thf(fact_951_listrel1__iff__update,axiom,
    ! [Xs: list_f7931864845379430901rm_a_b,Ys2: list_f7931864845379430901rm_a_b,R3: set_Pr3641583360614959815rm_a_b] :
      ( ( member6557215607983664784rm_a_b @ ( produc8130127474510664151rm_a_b @ Xs @ Ys2 ) @ ( listre1515560112423418318rm_a_b @ R3 ) )
      = ( ? [Y2: fset_P3577670608877728751rm_a_b,N3: nat] :
            ( ( member2354150401426528528rm_a_b @ ( produc6898453836806167895rm_a_b @ ( nth_fs3949069584078227574rm_a_b @ Xs @ N3 ) @ Y2 ) @ R3 )
            & ( ord_less_nat @ N3 @ ( size_s1967422219310127201rm_a_b @ Xs ) )
            & ( Ys2
              = ( list_u7212055091542668893rm_a_b @ Xs @ N3 @ Y2 ) ) ) ) ) ).

% listrel1_iff_update
thf(fact_952_add__right__cancel,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( ( plus_plus_nat @ B @ A )
        = ( plus_plus_nat @ C2 @ A ) )
      = ( B = C2 ) ) ).

% add_right_cancel
thf(fact_953_add__right__cancel,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( ( plus_plus_int @ B @ A )
        = ( plus_plus_int @ C2 @ A ) )
      = ( B = C2 ) ) ).

% add_right_cancel
thf(fact_954_add__left__cancel,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ( plus_plus_nat @ A @ B )
        = ( plus_plus_nat @ A @ C2 ) )
      = ( B = C2 ) ) ).

% add_left_cancel
thf(fact_955_add__left__cancel,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ( plus_plus_int @ A @ B )
        = ( plus_plus_int @ A @ C2 ) )
      = ( B = C2 ) ) ).

% add_left_cancel
thf(fact_956_add__le__cancel__right,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
      = ( ord_less_eq_nat @ A @ B ) ) ).

% add_le_cancel_right
thf(fact_957_add__le__cancel__right,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) )
      = ( ord_less_eq_int @ A @ B ) ) ).

% add_le_cancel_right
thf(fact_958_add__le__cancel__left,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
      = ( ord_less_eq_nat @ A @ B ) ) ).

% add_le_cancel_left
thf(fact_959_add__le__cancel__left,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) )
      = ( ord_less_eq_int @ A @ B ) ) ).

% add_le_cancel_left
thf(fact_960_double__eq__0__iff,axiom,
    ! [A: int] :
      ( ( ( plus_plus_int @ A @ A )
        = zero_zero_int )
      = ( A = zero_zero_int ) ) ).

% double_eq_0_iff
thf(fact_961_add__0,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ A )
      = A ) ).

% add_0
thf(fact_962_add__0,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ zero_zero_int @ A )
      = A ) ).

% add_0
thf(fact_963_zero__eq__add__iff__both__eq__0,axiom,
    ! [X: nat,Y: nat] :
      ( ( zero_zero_nat
        = ( plus_plus_nat @ X @ Y ) )
      = ( ( X = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% zero_eq_add_iff_both_eq_0
thf(fact_964_add__eq__0__iff__both__eq__0,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( plus_plus_nat @ X @ Y )
        = zero_zero_nat )
      = ( ( X = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% add_eq_0_iff_both_eq_0
thf(fact_965_add__cancel__right__right,axiom,
    ! [A: nat,B: nat] :
      ( ( A
        = ( plus_plus_nat @ A @ B ) )
      = ( B = zero_zero_nat ) ) ).

% add_cancel_right_right
thf(fact_966_add__cancel__right__right,axiom,
    ! [A: int,B: int] :
      ( ( A
        = ( plus_plus_int @ A @ B ) )
      = ( B = zero_zero_int ) ) ).

% add_cancel_right_right
thf(fact_967_add__cancel__right__left,axiom,
    ! [A: nat,B: nat] :
      ( ( A
        = ( plus_plus_nat @ B @ A ) )
      = ( B = zero_zero_nat ) ) ).

% add_cancel_right_left
thf(fact_968_add__cancel__right__left,axiom,
    ! [A: int,B: int] :
      ( ( A
        = ( plus_plus_int @ B @ A ) )
      = ( B = zero_zero_int ) ) ).

% add_cancel_right_left
thf(fact_969_add__cancel__left__right,axiom,
    ! [A: nat,B: nat] :
      ( ( ( plus_plus_nat @ A @ B )
        = A )
      = ( B = zero_zero_nat ) ) ).

% add_cancel_left_right
thf(fact_970_add__cancel__left__right,axiom,
    ! [A: int,B: int] :
      ( ( ( plus_plus_int @ A @ B )
        = A )
      = ( B = zero_zero_int ) ) ).

% add_cancel_left_right
thf(fact_971_add__cancel__left__left,axiom,
    ! [B: nat,A: nat] :
      ( ( ( plus_plus_nat @ B @ A )
        = A )
      = ( B = zero_zero_nat ) ) ).

% add_cancel_left_left
thf(fact_972_add__cancel__left__left,axiom,
    ! [B: int,A: int] :
      ( ( ( plus_plus_int @ B @ A )
        = A )
      = ( B = zero_zero_int ) ) ).

% add_cancel_left_left
thf(fact_973_double__zero__sym,axiom,
    ! [A: int] :
      ( ( zero_zero_int
        = ( plus_plus_int @ A @ A ) )
      = ( A = zero_zero_int ) ) ).

% double_zero_sym
thf(fact_974_add_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% add.right_neutral
thf(fact_975_add_Oright__neutral,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ zero_zero_int )
      = A ) ).

% add.right_neutral
thf(fact_976_add__less__cancel__left,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
      = ( ord_less_nat @ A @ B ) ) ).

% add_less_cancel_left
thf(fact_977_add__less__cancel__left,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( ord_less_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) )
      = ( ord_less_int @ A @ B ) ) ).

% add_less_cancel_left
thf(fact_978_add__less__cancel__right,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
      = ( ord_less_nat @ A @ B ) ) ).

% add_less_cancel_right
thf(fact_979_add__less__cancel__right,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( ord_less_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) )
      = ( ord_less_int @ A @ B ) ) ).

% add_less_cancel_right
thf(fact_980_minus__add__distrib,axiom,
    ! [A: int,B: int] :
      ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
      = ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) ) ) ).

% minus_add_distrib
thf(fact_981_minus__add__cancel,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( plus_plus_int @ A @ B ) )
      = B ) ).

% minus_add_cancel
thf(fact_982_add__minus__cancel,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ A @ ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B ) )
      = B ) ).

% add_minus_cancel
thf(fact_983_Nat_Oadd__0__right,axiom,
    ! [M: nat] :
      ( ( plus_plus_nat @ M @ zero_zero_nat )
      = M ) ).

% Nat.add_0_right
thf(fact_984_add__is__0,axiom,
    ! [M: nat,N: nat] :
      ( ( ( plus_plus_nat @ M @ N )
        = zero_zero_nat )
      = ( ( M = zero_zero_nat )
        & ( N = zero_zero_nat ) ) ) ).

% add_is_0
thf(fact_985_add__Suc__right,axiom,
    ! [M: nat,N: nat] :
      ( ( plus_plus_nat @ M @ ( suc @ N ) )
      = ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).

% add_Suc_right
thf(fact_986_nat__add__left__cancel__less,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% nat_add_left_cancel_less
thf(fact_987_nat__add__left__cancel__le,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% nat_add_left_cancel_le
thf(fact_988_add__le__same__cancel1,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
      = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).

% add_le_same_cancel1
thf(fact_989_add__le__same__cancel1,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ B @ A ) @ B )
      = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).

% add_le_same_cancel1
thf(fact_990_add__le__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
      = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).

% add_le_same_cancel2
thf(fact_991_add__le__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ B )
      = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).

% add_le_same_cancel2
thf(fact_992_le__add__same__cancel1,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
      = ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).

% le_add_same_cancel1
thf(fact_993_le__add__same__cancel1,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ ( plus_plus_int @ A @ B ) )
      = ( ord_less_eq_int @ zero_zero_int @ B ) ) ).

% le_add_same_cancel1
thf(fact_994_le__add__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
      = ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).

% le_add_same_cancel2
thf(fact_995_le__add__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ ( plus_plus_int @ B @ A ) )
      = ( ord_less_eq_int @ zero_zero_int @ B ) ) ).

% le_add_same_cancel2
thf(fact_996_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
      = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).

% double_add_le_zero_iff_single_add_le_zero
thf(fact_997_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
      = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).

% zero_le_double_add_iff_zero_le_single_add
thf(fact_998_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: int] :
      ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
      = ( ord_less_int @ zero_zero_int @ A ) ) ).

% zero_less_double_add_iff_zero_less_single_add
thf(fact_999_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: int] :
      ( ( ord_less_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
      = ( ord_less_int @ A @ zero_zero_int ) ) ).

% double_add_less_zero_iff_single_add_less_zero
thf(fact_1000_less__add__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
      = ( ord_less_nat @ zero_zero_nat @ B ) ) ).

% less_add_same_cancel2
thf(fact_1001_less__add__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ ( plus_plus_int @ B @ A ) )
      = ( ord_less_int @ zero_zero_int @ B ) ) ).

% less_add_same_cancel2
thf(fact_1002_less__add__same__cancel1,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
      = ( ord_less_nat @ zero_zero_nat @ B ) ) ).

% less_add_same_cancel1
thf(fact_1003_less__add__same__cancel1,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ ( plus_plus_int @ A @ B ) )
      = ( ord_less_int @ zero_zero_int @ B ) ) ).

% less_add_same_cancel1
thf(fact_1004_add__less__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
      = ( ord_less_nat @ A @ zero_zero_nat ) ) ).

% add_less_same_cancel2
thf(fact_1005_add__less__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ ( plus_plus_int @ A @ B ) @ B )
      = ( ord_less_int @ A @ zero_zero_int ) ) ).

% add_less_same_cancel2
thf(fact_1006_add__less__same__cancel1,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
      = ( ord_less_nat @ A @ zero_zero_nat ) ) ).

% add_less_same_cancel1
thf(fact_1007_add__less__same__cancel1,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ ( plus_plus_int @ B @ A ) @ B )
      = ( ord_less_int @ A @ zero_zero_int ) ) ).

% add_less_same_cancel1
thf(fact_1008_ab__left__minus,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
      = zero_zero_int ) ).

% ab_left_minus
thf(fact_1009_add_Oright__inverse,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ ( uminus_uminus_int @ A ) )
      = zero_zero_int ) ).

% add.right_inverse
thf(fact_1010_add__gr__0,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ M )
        | ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% add_gr_0
thf(fact_1011_of__nat__add,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M @ N ) )
      = ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).

% of_nat_add
thf(fact_1012_of__nat__add,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).

% of_nat_add
thf(fact_1013_Cons__listrel1__Cons,axiom,
    ! [X: term_a_b,Xs: list_term_a_b,Y: term_a_b,Ys2: list_term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
      ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ ( cons_term_a_b @ X @ Xs ) @ ( cons_term_a_b @ Y @ Ys2 ) ) @ ( listrel1_term_a_b @ R3 ) )
      = ( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R3 )
          & ( Xs = Ys2 ) )
        | ( ( X = Y )
          & ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs @ Ys2 ) @ ( listrel1_term_a_b @ R3 ) ) ) ) ) ).

% Cons_listrel1_Cons
thf(fact_1014_nth__drop,axiom,
    ! [N: nat,Xs: list_f7931864845379430901rm_a_b,I4: nat] :
      ( ( ord_less_eq_nat @ N @ ( size_s1967422219310127201rm_a_b @ Xs ) )
     => ( ( nth_fs3949069584078227574rm_a_b @ ( drop_f1037359496611548469rm_a_b @ N @ Xs ) @ I4 )
        = ( nth_fs3949069584078227574rm_a_b @ Xs @ ( plus_plus_nat @ N @ I4 ) ) ) ) ).

% nth_drop
thf(fact_1015_add__nonpos__eq__0__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
       => ( ( ( plus_plus_nat @ X @ Y )
            = zero_zero_nat )
          = ( ( X = zero_zero_nat )
            & ( Y = zero_zero_nat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_1016_add__nonpos__eq__0__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ zero_zero_int )
     => ( ( ord_less_eq_int @ Y @ zero_zero_int )
       => ( ( ( plus_plus_int @ X @ Y )
            = zero_zero_int )
          = ( ( X = zero_zero_int )
            & ( Y = zero_zero_int ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_1017_add__nonneg__eq__0__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ X )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
       => ( ( ( plus_plus_nat @ X @ Y )
            = zero_zero_nat )
          = ( ( X = zero_zero_nat )
            & ( Y = zero_zero_nat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_1018_add__nonneg__eq__0__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ( ( plus_plus_int @ X @ Y )
            = zero_zero_int )
          = ( ( X = zero_zero_int )
            & ( Y = zero_zero_int ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_1019_add__nonpos__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% add_nonpos_nonpos
thf(fact_1020_add__nonpos__nonpos,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_eq_int @ B @ zero_zero_int )
       => ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).

% add_nonpos_nonpos
thf(fact_1021_add__nonneg__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_1022_add__nonneg__nonneg,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_1023_add__increasing2,axiom,
    ! [C2: nat,B: nat,A: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
     => ( ( ord_less_eq_nat @ B @ A )
       => ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).

% add_increasing2
thf(fact_1024_add__increasing2,axiom,
    ! [C2: int,B: int,A: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C2 )
     => ( ( ord_less_eq_int @ B @ A )
       => ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C2 ) ) ) ) ).

% add_increasing2
thf(fact_1025_add__decreasing2,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ C2 @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ A @ B )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ B ) ) ) ).

% add_decreasing2
thf(fact_1026_add__decreasing2,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( ord_less_eq_int @ C2 @ zero_zero_int )
     => ( ( ord_less_eq_int @ A @ B )
       => ( ord_less_eq_int @ ( plus_plus_int @ A @ C2 ) @ B ) ) ) ).

% add_decreasing2
thf(fact_1027_add__increasing,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).

% add_increasing
thf(fact_1028_add__increasing,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ B @ C2 )
       => ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C2 ) ) ) ) ).

% add_increasing
thf(fact_1029_add__decreasing,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ C2 @ B )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ B ) ) ) ).

% add_decreasing
thf(fact_1030_add__decreasing,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_eq_int @ C2 @ B )
       => ( ord_less_eq_int @ ( plus_plus_int @ A @ C2 ) @ B ) ) ) ).

% add_decreasing
thf(fact_1031_add__less__le__mono,axiom,
    ! [A: nat,B: nat,C2: nat,D: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C2 @ D )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).

% add_less_le_mono
thf(fact_1032_add__less__le__mono,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_eq_int @ C2 @ D )
       => ( ord_less_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ D ) ) ) ) ).

% add_less_le_mono
thf(fact_1033_add__le__less__mono,axiom,
    ! [A: nat,B: nat,C2: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ C2 @ D )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).

% add_le_less_mono
thf(fact_1034_add__le__less__mono,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_int @ C2 @ D )
       => ( ord_less_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ D ) ) ) ) ).

% add_le_less_mono
thf(fact_1035_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I4: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_nat @ I4 @ J )
        & ( ord_less_eq_nat @ K @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_1036_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I4: int,J: int,K: int,L: int] :
      ( ( ( ord_less_int @ I4 @ J )
        & ( ord_less_eq_int @ K @ L ) )
     => ( ord_less_int @ ( plus_plus_int @ I4 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_1037_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I4: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_eq_nat @ I4 @ J )
        & ( ord_less_nat @ K @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_1038_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I4: int,J: int,K: int,L: int] :
      ( ( ( ord_less_eq_int @ I4 @ J )
        & ( ord_less_int @ K @ L ) )
     => ( ord_less_int @ ( plus_plus_int @ I4 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_1039_add__neg__neg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ zero_zero_nat )
     => ( ( ord_less_nat @ B @ zero_zero_nat )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% add_neg_neg
thf(fact_1040_add__neg__neg,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ B @ zero_zero_int )
       => ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).

% add_neg_neg
thf(fact_1041_add__pos__pos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ B )
       => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% add_pos_pos
thf(fact_1042_add__pos__pos,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).

% add_pos_pos
thf(fact_1043_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ! [C3: nat] :
            ( ( B
              = ( plus_plus_nat @ A @ C3 ) )
           => ( C3 = zero_zero_nat ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
thf(fact_1044_pos__add__strict,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ B @ C2 )
       => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).

% pos_add_strict
thf(fact_1045_pos__add__strict,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ B @ C2 )
       => ( ord_less_int @ B @ ( plus_plus_int @ A @ C2 ) ) ) ) ).

% pos_add_strict
thf(fact_1046_add__less__zeroD,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ ( plus_plus_int @ X @ Y ) @ zero_zero_int )
     => ( ( ord_less_int @ X @ zero_zero_int )
        | ( ord_less_int @ Y @ zero_zero_int ) ) ) ).

% add_less_zeroD
thf(fact_1047_group__cancel_Oneg1,axiom,
    ! [A2: int,K: int,A: int] :
      ( ( A2
        = ( plus_plus_int @ K @ A ) )
     => ( ( uminus_uminus_int @ A2 )
        = ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( uminus_uminus_int @ A ) ) ) ) ).

% group_cancel.neg1
thf(fact_1048_add_Oinverse__distrib__swap,axiom,
    ! [A: int,B: int] :
      ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
      = ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).

% add.inverse_distrib_swap
thf(fact_1049_is__num__normalize_I8_J,axiom,
    ! [A: int,B: int] :
      ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
      = ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).

% is_num_normalize(8)
thf(fact_1050_is__num__normalize_I1_J,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C2 )
      = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C2 ) ) ) ).

% is_num_normalize(1)
thf(fact_1051_add__right__imp__eq,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( ( plus_plus_nat @ B @ A )
        = ( plus_plus_nat @ C2 @ A ) )
     => ( B = C2 ) ) ).

% add_right_imp_eq
thf(fact_1052_add__right__imp__eq,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( ( plus_plus_int @ B @ A )
        = ( plus_plus_int @ C2 @ A ) )
     => ( B = C2 ) ) ).

% add_right_imp_eq
thf(fact_1053_add__left__imp__eq,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ( plus_plus_nat @ A @ B )
        = ( plus_plus_nat @ A @ C2 ) )
     => ( B = C2 ) ) ).

% add_left_imp_eq
thf(fact_1054_add__left__imp__eq,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ( plus_plus_int @ A @ B )
        = ( plus_plus_int @ A @ C2 ) )
     => ( B = C2 ) ) ).

% add_left_imp_eq
thf(fact_1055_add_Oleft__commute,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C2 ) )
      = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ).

% add.left_commute
thf(fact_1056_add_Oleft__commute,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( plus_plus_int @ B @ ( plus_plus_int @ A @ C2 ) )
      = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C2 ) ) ) ).

% add.left_commute
thf(fact_1057_add_Ocommute,axiom,
    ( plus_plus_nat
    = ( ^ [A4: nat,B3: nat] : ( plus_plus_nat @ B3 @ A4 ) ) ) ).

% add.commute
thf(fact_1058_add_Ocommute,axiom,
    ( plus_plus_int
    = ( ^ [A4: int,B3: int] : ( plus_plus_int @ B3 @ A4 ) ) ) ).

% add.commute
thf(fact_1059_add_Oright__cancel,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( ( plus_plus_int @ B @ A )
        = ( plus_plus_int @ C2 @ A ) )
      = ( B = C2 ) ) ).

% add.right_cancel
thf(fact_1060_add_Oleft__cancel,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ( plus_plus_int @ A @ B )
        = ( plus_plus_int @ A @ C2 ) )
      = ( B = C2 ) ) ).

% add.left_cancel
thf(fact_1061_add_Oassoc,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C2 )
      = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ).

% add.assoc
thf(fact_1062_add_Oassoc,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C2 )
      = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C2 ) ) ) ).

% add.assoc
thf(fact_1063_group__cancel_Oadd2,axiom,
    ! [B2: nat,K: nat,B: nat,A: nat] :
      ( ( B2
        = ( plus_plus_nat @ K @ B ) )
     => ( ( plus_plus_nat @ A @ B2 )
        = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_1064_group__cancel_Oadd2,axiom,
    ! [B2: int,K: int,B: int,A: int] :
      ( ( B2
        = ( plus_plus_int @ K @ B ) )
     => ( ( plus_plus_int @ A @ B2 )
        = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_1065_group__cancel_Oadd1,axiom,
    ! [A2: nat,K: nat,A: nat,B: nat] :
      ( ( A2
        = ( plus_plus_nat @ K @ A ) )
     => ( ( plus_plus_nat @ A2 @ B )
        = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% group_cancel.add1
thf(fact_1066_group__cancel_Oadd1,axiom,
    ! [A2: int,K: int,A: int,B: int] :
      ( ( A2
        = ( plus_plus_int @ K @ A ) )
     => ( ( plus_plus_int @ A2 @ B )
        = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).

% group_cancel.add1
thf(fact_1067_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I4: nat,J: nat,K: nat,L: nat] :
      ( ( ( I4 = J )
        & ( K = L ) )
     => ( ( plus_plus_nat @ I4 @ K )
        = ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_1068_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I4: int,J: int,K: int,L: int] :
      ( ( ( I4 = J )
        & ( K = L ) )
     => ( ( plus_plus_int @ I4 @ K )
        = ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_1069_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C2 )
      = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ).

% ab_semigroup_add_class.add_ac(1)
thf(fact_1070_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C2 )
      = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C2 ) ) ) ).

% ab_semigroup_add_class.add_ac(1)
thf(fact_1071_nat__arith_Osuc1,axiom,
    ! [A2: nat,K: nat,A: nat] :
      ( ( A2
        = ( plus_plus_nat @ K @ A ) )
     => ( ( suc @ A2 )
        = ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).

% nat_arith.suc1
thf(fact_1072_add__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( plus_plus_nat @ ( suc @ M ) @ N )
      = ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).

% add_Suc
thf(fact_1073_add__Suc__shift,axiom,
    ! [M: nat,N: nat] :
      ( ( plus_plus_nat @ ( suc @ M ) @ N )
      = ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).

% add_Suc_shift
thf(fact_1074_add__eq__self__zero,axiom,
    ! [M: nat,N: nat] :
      ( ( ( plus_plus_nat @ M @ N )
        = M )
     => ( N = zero_zero_nat ) ) ).

% add_eq_self_zero
thf(fact_1075_plus__nat_Oadd__0,axiom,
    ! [N: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ N )
      = N ) ).

% plus_nat.add_0
thf(fact_1076_add_Ogroup__left__neutral,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ zero_zero_int @ A )
      = A ) ).

% add.group_left_neutral
thf(fact_1077_add_Ocomm__neutral,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% add.comm_neutral
thf(fact_1078_add_Ocomm__neutral,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ zero_zero_int )
      = A ) ).

% add.comm_neutral
thf(fact_1079_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ A )
      = A ) ).

% comm_monoid_add_class.add_0
thf(fact_1080_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ zero_zero_int @ A )
      = A ) ).

% comm_monoid_add_class.add_0
thf(fact_1081_verit__sum__simplify,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% verit_sum_simplify
thf(fact_1082_verit__sum__simplify,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ zero_zero_int )
      = A ) ).

% verit_sum_simplify
thf(fact_1083_Euclid__induct,axiom,
    ! [P: nat > nat > $o,A: nat,B: nat] :
      ( ! [A7: nat,B5: nat] :
          ( ( P @ A7 @ B5 )
          = ( P @ B5 @ A7 ) )
     => ( ! [A7: nat] : ( P @ A7 @ zero_zero_nat )
       => ( ! [A7: nat,B5: nat] :
              ( ( P @ A7 @ B5 )
             => ( P @ A7 @ ( plus_plus_nat @ A7 @ B5 ) ) )
         => ( P @ A @ B ) ) ) ) ).

% Euclid_induct
thf(fact_1084_add__le__imp__le__right,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
     => ( ord_less_eq_nat @ A @ B ) ) ).

% add_le_imp_le_right
thf(fact_1085_add__le__imp__le__right,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) )
     => ( ord_less_eq_int @ A @ B ) ) ).

% add_le_imp_le_right
thf(fact_1086_add__le__imp__le__left,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
     => ( ord_less_eq_nat @ A @ B ) ) ).

% add_le_imp_le_left
thf(fact_1087_add__le__imp__le__left,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) )
     => ( ord_less_eq_int @ A @ B ) ) ).

% add_le_imp_le_left
thf(fact_1088_le__iff__add,axiom,
    ( ord_less_eq_nat
    = ( ^ [A4: nat,B3: nat] :
        ? [C: nat] :
          ( B3
          = ( plus_plus_nat @ A4 @ C ) ) ) ) ).

% le_iff_add
thf(fact_1089_add__right__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) ) ) ).

% add_right_mono
thf(fact_1090_add__right__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ord_less_eq_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) ) ) ).

% add_right_mono
thf(fact_1091_less__eqE,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ~ ! [C3: nat] :
            ( B
           != ( plus_plus_nat @ A @ C3 ) ) ) ).

% less_eqE
thf(fact_1092_add__left__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) ) ) ).

% add_left_mono
thf(fact_1093_add__left__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ord_less_eq_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) ) ) ).

% add_left_mono
thf(fact_1094_add__mono,axiom,
    ! [A: nat,B: nat,C2: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C2 @ D )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).

% add_mono
thf(fact_1095_add__mono,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ C2 @ D )
       => ( ord_less_eq_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ D ) ) ) ) ).

% add_mono
thf(fact_1096_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I4: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_eq_nat @ I4 @ J )
        & ( ord_less_eq_nat @ K @ L ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_1097_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I4: int,J: int,K: int,L: int] :
      ( ( ( ord_less_eq_int @ I4 @ J )
        & ( ord_less_eq_int @ K @ L ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ I4 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_1098_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I4: nat,J: nat,K: nat,L: nat] :
      ( ( ( I4 = J )
        & ( ord_less_eq_nat @ K @ L ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_1099_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I4: int,J: int,K: int,L: int] :
      ( ( ( I4 = J )
        & ( ord_less_eq_int @ K @ L ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ I4 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_1100_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I4: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_eq_nat @ I4 @ J )
        & ( K = L ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_1101_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I4: int,J: int,K: int,L: int] :
      ( ( ( ord_less_eq_int @ I4 @ J )
        & ( K = L ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ I4 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_1102_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I4: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_nat @ I4 @ J )
        & ( ord_less_nat @ K @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_1103_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I4: int,J: int,K: int,L: int] :
      ( ( ( ord_less_int @ I4 @ J )
        & ( ord_less_int @ K @ L ) )
     => ( ord_less_int @ ( plus_plus_int @ I4 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_1104_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I4: nat,J: nat,K: nat,L: nat] :
      ( ( ( I4 = J )
        & ( ord_less_nat @ K @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_1105_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I4: int,J: int,K: int,L: int] :
      ( ( ( I4 = J )
        & ( ord_less_int @ K @ L ) )
     => ( ord_less_int @ ( plus_plus_int @ I4 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_1106_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I4: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_nat @ I4 @ J )
        & ( K = L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_1107_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I4: int,J: int,K: int,L: int] :
      ( ( ( ord_less_int @ I4 @ J )
        & ( K = L ) )
     => ( ord_less_int @ ( plus_plus_int @ I4 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_1108_add__strict__mono,axiom,
    ! [A: nat,B: nat,C2: nat,D: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ C2 @ D )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).

% add_strict_mono
thf(fact_1109_add__strict__mono,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ C2 @ D )
       => ( ord_less_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ D ) ) ) ) ).

% add_strict_mono
thf(fact_1110_add__strict__left__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ord_less_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) ) ) ).

% add_strict_left_mono
thf(fact_1111_add__strict__left__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_int @ A @ B )
     => ( ord_less_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) ) ) ).

% add_strict_left_mono
thf(fact_1112_add__strict__right__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) ) ) ).

% add_strict_right_mono
thf(fact_1113_add__strict__right__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_int @ A @ B )
     => ( ord_less_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) ) ) ).

% add_strict_right_mono
thf(fact_1114_add__less__imp__less__left,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
     => ( ord_less_nat @ A @ B ) ) ).

% add_less_imp_less_left
thf(fact_1115_add__less__imp__less__left,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( ord_less_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) )
     => ( ord_less_int @ A @ B ) ) ).

% add_less_imp_less_left
thf(fact_1116_add__less__imp__less__right,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
     => ( ord_less_nat @ A @ B ) ) ).

% add_less_imp_less_right
thf(fact_1117_add__less__imp__less__right,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( ord_less_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) )
     => ( ord_less_int @ A @ B ) ) ).

% add_less_imp_less_right
thf(fact_1118_add__lessD1,axiom,
    ! [I4: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ I4 @ J ) @ K )
     => ( ord_less_nat @ I4 @ K ) ) ).

% add_lessD1
thf(fact_1119_add__less__mono,axiom,
    ! [I4: nat,J: nat,K: nat,L: nat] :
      ( ( ord_less_nat @ I4 @ J )
     => ( ( ord_less_nat @ K @ L )
       => ( ord_less_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).

% add_less_mono
thf(fact_1120_not__add__less1,axiom,
    ! [I4: nat,J: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ I4 @ J ) @ I4 ) ).

% not_add_less1
thf(fact_1121_not__add__less2,axiom,
    ! [J: nat,I4: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ J @ I4 ) @ I4 ) ).

% not_add_less2
thf(fact_1122_add__less__mono1,axiom,
    ! [I4: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I4 @ J )
     => ( ord_less_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).

% add_less_mono1
thf(fact_1123_trans__less__add1,axiom,
    ! [I4: nat,J: nat,M: nat] :
      ( ( ord_less_nat @ I4 @ J )
     => ( ord_less_nat @ I4 @ ( plus_plus_nat @ J @ M ) ) ) ).

% trans_less_add1
thf(fact_1124_trans__less__add2,axiom,
    ! [I4: nat,J: nat,M: nat] :
      ( ( ord_less_nat @ I4 @ J )
     => ( ord_less_nat @ I4 @ ( plus_plus_nat @ M @ J ) ) ) ).

% trans_less_add2
thf(fact_1125_less__add__eq__less,axiom,
    ! [K: nat,L: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ K @ L )
     => ( ( ( plus_plus_nat @ M @ L )
          = ( plus_plus_nat @ K @ N ) )
       => ( ord_less_nat @ M @ N ) ) ) ).

% less_add_eq_less
thf(fact_1126_add__leE,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
     => ~ ( ( ord_less_eq_nat @ M @ N )
         => ~ ( ord_less_eq_nat @ K @ N ) ) ) ).

% add_leE
thf(fact_1127_le__add1,axiom,
    ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).

% le_add1
thf(fact_1128_le__add2,axiom,
    ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).

% le_add2
thf(fact_1129_add__leD1,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% add_leD1
thf(fact_1130_add__leD2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
     => ( ord_less_eq_nat @ K @ N ) ) ).

% add_leD2
thf(fact_1131_le__Suc__ex,axiom,
    ! [K: nat,L: nat] :
      ( ( ord_less_eq_nat @ K @ L )
     => ? [N2: nat] :
          ( L
          = ( plus_plus_nat @ K @ N2 ) ) ) ).

% le_Suc_ex
thf(fact_1132_add__le__mono,axiom,
    ! [I4: nat,J: nat,K: nat,L: nat] :
      ( ( ord_less_eq_nat @ I4 @ J )
     => ( ( ord_less_eq_nat @ K @ L )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).

% add_le_mono
thf(fact_1133_add__le__mono1,axiom,
    ! [I4: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I4 @ J )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).

% add_le_mono1
thf(fact_1134_trans__le__add1,axiom,
    ! [I4: nat,J: nat,M: nat] :
      ( ( ord_less_eq_nat @ I4 @ J )
     => ( ord_less_eq_nat @ I4 @ ( plus_plus_nat @ J @ M ) ) ) ).

% trans_le_add1
thf(fact_1135_trans__le__add2,axiom,
    ! [I4: nat,J: nat,M: nat] :
      ( ( ord_less_eq_nat @ I4 @ J )
     => ( ord_less_eq_nat @ I4 @ ( plus_plus_nat @ M @ J ) ) ) ).

% trans_le_add2
thf(fact_1136_nat__le__iff__add,axiom,
    ( ord_less_eq_nat
    = ( ^ [M3: nat,N3: nat] :
        ? [K5: nat] :
          ( N3
          = ( plus_plus_nat @ M3 @ K5 ) ) ) ) ).

% nat_le_iff_add
thf(fact_1137_add__eq__0__iff,axiom,
    ! [A: int,B: int] :
      ( ( ( plus_plus_int @ A @ B )
        = zero_zero_int )
      = ( B
        = ( uminus_uminus_int @ A ) ) ) ).

% add_eq_0_iff
thf(fact_1138_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
      = zero_zero_int ) ).

% ab_group_add_class.ab_left_minus
thf(fact_1139_add_Oinverse__unique,axiom,
    ! [A: int,B: int] :
      ( ( ( plus_plus_int @ A @ B )
        = zero_zero_int )
     => ( ( uminus_uminus_int @ A )
        = B ) ) ).

% add.inverse_unique
thf(fact_1140_eq__neg__iff__add__eq__0,axiom,
    ! [A: int,B: int] :
      ( ( A
        = ( uminus_uminus_int @ B ) )
      = ( ( plus_plus_int @ A @ B )
        = zero_zero_int ) ) ).

% eq_neg_iff_add_eq_0
thf(fact_1141_neg__eq__iff__add__eq__0,axiom,
    ! [A: int,B: int] :
      ( ( ( uminus_uminus_int @ A )
        = B )
      = ( ( plus_plus_int @ A @ B )
        = zero_zero_int ) ) ).

% neg_eq_iff_add_eq_0
thf(fact_1142_one__is__add,axiom,
    ! [M: nat,N: nat] :
      ( ( ( suc @ zero_zero_nat )
        = ( plus_plus_nat @ M @ N ) )
      = ( ( ( M
            = ( suc @ zero_zero_nat ) )
          & ( N = zero_zero_nat ) )
        | ( ( M = zero_zero_nat )
          & ( N
            = ( suc @ zero_zero_nat ) ) ) ) ) ).

% one_is_add
thf(fact_1143_add__is__1,axiom,
    ! [M: nat,N: nat] :
      ( ( ( plus_plus_nat @ M @ N )
        = ( suc @ zero_zero_nat ) )
      = ( ( ( M
            = ( suc @ zero_zero_nat ) )
          & ( N = zero_zero_nat ) )
        | ( ( M = zero_zero_nat )
          & ( N
            = ( suc @ zero_zero_nat ) ) ) ) ) ).

% add_is_1
thf(fact_1144_less__imp__add__positive,axiom,
    ! [I4: nat,J: nat] :
      ( ( ord_less_nat @ I4 @ J )
     => ? [K2: nat] :
          ( ( ord_less_nat @ zero_zero_nat @ K2 )
          & ( ( plus_plus_nat @ I4 @ K2 )
            = J ) ) ) ).

% less_imp_add_positive
thf(fact_1145_less__natE,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ~ ! [Q4: nat] :
            ( N
           != ( suc @ ( plus_plus_nat @ M @ Q4 ) ) ) ) ).

% less_natE
thf(fact_1146_less__add__Suc1,axiom,
    ! [I4: nat,M: nat] : ( ord_less_nat @ I4 @ ( suc @ ( plus_plus_nat @ I4 @ M ) ) ) ).

% less_add_Suc1
thf(fact_1147_less__add__Suc2,axiom,
    ! [I4: nat,M: nat] : ( ord_less_nat @ I4 @ ( suc @ ( plus_plus_nat @ M @ I4 ) ) ) ).

% less_add_Suc2
thf(fact_1148_less__iff__Suc__add,axiom,
    ( ord_less_nat
    = ( ^ [M3: nat,N3: nat] :
        ? [K5: nat] :
          ( N3
          = ( suc @ ( plus_plus_nat @ M3 @ K5 ) ) ) ) ) ).

% less_iff_Suc_add
thf(fact_1149_less__imp__Suc__add,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ? [K2: nat] :
          ( N
          = ( suc @ ( plus_plus_nat @ M @ K2 ) ) ) ) ).

% less_imp_Suc_add
thf(fact_1150_mono__nat__linear__lb,axiom,
    ! [F: nat > nat,M: nat,K: nat] :
      ( ! [M4: nat,N2: nat] :
          ( ( ord_less_nat @ M4 @ N2 )
         => ( ord_less_nat @ ( F @ M4 ) @ ( F @ N2 ) ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).

% mono_nat_linear_lb
thf(fact_1151_not__Nil__listrel1,axiom,
    ! [Xs: list_s5629610930313288943rm_a_b,R3: set_Pr647378613824986951rm_a_b] :
      ~ ( member4739679974255130896rm_a_b @ ( produc510695698380873047rm_a_b @ nil_se7334864872594423151rm_a_b @ Xs ) @ ( listre214978426196003784rm_a_b @ R3 ) ) ).

% not_Nil_listrel1
thf(fact_1152_not__Nil__listrel1,axiom,
    ! [Xs: list_f7931864845379430901rm_a_b,R3: set_Pr3641583360614959815rm_a_b] :
      ~ ( member6557215607983664784rm_a_b @ ( produc8130127474510664151rm_a_b @ nil_fs4020363321024022517rm_a_b @ Xs ) @ ( listre1515560112423418318rm_a_b @ R3 ) ) ).

% not_Nil_listrel1
thf(fact_1153_not__listrel1__Nil,axiom,
    ! [Xs: list_s5629610930313288943rm_a_b,R3: set_Pr647378613824986951rm_a_b] :
      ~ ( member4739679974255130896rm_a_b @ ( produc510695698380873047rm_a_b @ Xs @ nil_se7334864872594423151rm_a_b ) @ ( listre214978426196003784rm_a_b @ R3 ) ) ).

% not_listrel1_Nil
thf(fact_1154_not__listrel1__Nil,axiom,
    ! [Xs: list_f7931864845379430901rm_a_b,R3: set_Pr3641583360614959815rm_a_b] :
      ~ ( member6557215607983664784rm_a_b @ ( produc8130127474510664151rm_a_b @ Xs @ nil_fs4020363321024022517rm_a_b ) @ ( listre1515560112423418318rm_a_b @ R3 ) ) ).

% not_listrel1_Nil
thf(fact_1155_listrel1__eq__len,axiom,
    ! [Xs: list_f7931864845379430901rm_a_b,Ys2: list_f7931864845379430901rm_a_b,R3: set_Pr3641583360614959815rm_a_b] :
      ( ( member6557215607983664784rm_a_b @ ( produc8130127474510664151rm_a_b @ Xs @ Ys2 ) @ ( listre1515560112423418318rm_a_b @ R3 ) )
     => ( ( size_s1967422219310127201rm_a_b @ Xs )
        = ( size_s1967422219310127201rm_a_b @ Ys2 ) ) ) ).

% listrel1_eq_len
thf(fact_1156_gen__length__def,axiom,
    ( gen_le1078766959997619202rm_a_b
    = ( ^ [N3: nat,Xs3: list_f7931864845379430901rm_a_b] : ( plus_plus_nat @ N3 @ ( size_s1967422219310127201rm_a_b @ Xs3 ) ) ) ) ).

% gen_length_def
thf(fact_1157_divides__aux__def,axiom,
    ( unique5332122412489317741ux_nat
    = ( ^ [Qr: product_prod_nat_nat] :
          ( ( product_snd_nat_nat @ Qr )
          = zero_zero_nat ) ) ) ).

% divides_aux_def
thf(fact_1158_divides__aux__def,axiom,
    ( unique5329631941980267465ux_int
    = ( ^ [Qr: product_prod_int_int] :
          ( ( product_snd_int_int @ Qr )
          = zero_zero_int ) ) ) ).

% divides_aux_def
thf(fact_1159_add__strict__increasing2,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ B @ C2 )
       => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).

% add_strict_increasing2
thf(fact_1160_add__strict__increasing2,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ B @ C2 )
       => ( ord_less_int @ B @ ( plus_plus_int @ A @ C2 ) ) ) ) ).

% add_strict_increasing2
thf(fact_1161_add__strict__increasing,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).

% add_strict_increasing
thf(fact_1162_add__strict__increasing,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ B @ C2 )
       => ( ord_less_int @ B @ ( plus_plus_int @ A @ C2 ) ) ) ) ).

% add_strict_increasing
thf(fact_1163_add__pos__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% add_pos_nonneg
thf(fact_1164_add__pos__nonneg,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).

% add_pos_nonneg
thf(fact_1165_add__nonpos__neg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( ( ord_less_nat @ B @ zero_zero_nat )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% add_nonpos_neg
thf(fact_1166_add__nonpos__neg,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ B @ zero_zero_int )
       => ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).

% add_nonpos_neg
thf(fact_1167_add__nonneg__pos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ B )
       => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% add_nonneg_pos
thf(fact_1168_add__nonneg__pos,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).

% add_nonneg_pos
thf(fact_1169_add__neg__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% add_neg_nonpos
thf(fact_1170_add__neg__nonpos,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( ord_less_eq_int @ B @ zero_zero_int )
       => ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).

% add_neg_nonpos
thf(fact_1171_Cons__listrel1E2,axiom,
    ! [Xs: list_term_a_b,Y: term_a_b,Ys2: list_term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
      ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs @ ( cons_term_a_b @ Y @ Ys2 ) ) @ ( listrel1_term_a_b @ R3 ) )
     => ( ! [X4: term_a_b] :
            ( ( Xs
              = ( cons_term_a_b @ X4 @ Ys2 ) )
           => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ Y ) @ R3 ) )
       => ~ ! [Zs: list_term_a_b] :
              ( ( Xs
                = ( cons_term_a_b @ Y @ Zs ) )
             => ~ ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Zs @ Ys2 ) @ ( listrel1_term_a_b @ R3 ) ) ) ) ) ).

% Cons_listrel1E2
thf(fact_1172_Cons__listrel1E1,axiom,
    ! [X: term_a_b,Xs: list_term_a_b,Ys2: list_term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
      ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ ( cons_term_a_b @ X @ Xs ) @ Ys2 ) @ ( listrel1_term_a_b @ R3 ) )
     => ( ! [Y4: term_a_b] :
            ( ( Ys2
              = ( cons_term_a_b @ Y4 @ Xs ) )
           => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y4 ) @ R3 ) )
       => ~ ! [Zs: list_term_a_b] :
              ( ( Ys2
                = ( cons_term_a_b @ X @ Zs ) )
             => ~ ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs @ Zs ) @ ( listrel1_term_a_b @ R3 ) ) ) ) ) ).

% Cons_listrel1E1
thf(fact_1173_listrel1I1,axiom,
    ! [X: term_a_b,Y: term_a_b,R3: set_Pr4386577575007340137rm_a_b,Xs: list_term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R3 )
     => ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ ( cons_term_a_b @ X @ Xs ) @ ( cons_term_a_b @ Y @ Xs ) ) @ ( listrel1_term_a_b @ R3 ) ) ) ).

% listrel1I1
thf(fact_1174_less__prod__def_H,axiom,
    ( ord_le2202778214669499401nt_nat
    = ( ^ [X3: product_prod_int_nat,Y2: product_prod_int_nat] :
          ( ( ord_less_int @ ( product_fst_int_nat @ X3 ) @ ( product_fst_int_nat @ Y2 ) )
          | ( ( ( product_fst_int_nat @ X3 )
              = ( product_fst_int_nat @ Y2 ) )
            & ( ord_less_nat @ ( product_snd_int_nat @ X3 ) @ ( product_snd_int_nat @ Y2 ) ) ) ) ) ) ).

% less_prod_def'
thf(fact_1175_less__prod__def_H,axiom,
    ( ord_le7248299232015078501nt_int
    = ( ^ [X3: product_prod_int_int,Y2: product_prod_int_int] :
          ( ( ord_less_int @ ( product_fst_int_int @ X3 ) @ ( product_fst_int_int @ Y2 ) )
          | ( ( ( product_fst_int_int @ X3 )
              = ( product_fst_int_int @ Y2 ) )
            & ( ord_less_int @ ( product_snd_int_int @ X3 ) @ ( product_snd_int_int @ Y2 ) ) ) ) ) ) ).

% less_prod_def'
thf(fact_1176_diff__0__eq__0,axiom,
    ! [N: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ N )
      = zero_zero_nat ) ).

% diff_0_eq_0
thf(fact_1177_diff__self__eq__0,axiom,
    ! [M: nat] :
      ( ( minus_minus_nat @ M @ M )
      = zero_zero_nat ) ).

% diff_self_eq_0
thf(fact_1178_Suc__diff__diff,axiom,
    ! [M: nat,N: nat,K: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
      = ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).

% Suc_diff_diff
thf(fact_1179_diff__Suc__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
      = ( minus_minus_nat @ M @ N ) ) ).

% diff_Suc_Suc
thf(fact_1180_diff__diff__left,axiom,
    ! [I4: nat,J: nat,K: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ I4 @ J ) @ K )
      = ( minus_minus_nat @ I4 @ ( plus_plus_nat @ J @ K ) ) ) ).

% diff_diff_left
thf(fact_1181_diff__diff__cancel,axiom,
    ! [I4: nat,N: nat] :
      ( ( ord_less_eq_nat @ I4 @ N )
     => ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I4 ) )
        = I4 ) ) ).

% diff_diff_cancel
thf(fact_1182_zero__less__diff,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
      = ( ord_less_nat @ M @ N ) ) ).

% zero_less_diff
thf(fact_1183_diff__is__0__eq_H,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( minus_minus_nat @ M @ N )
        = zero_zero_nat ) ) ).

% diff_is_0_eq'
thf(fact_1184_diff__is__0__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ( minus_minus_nat @ M @ N )
        = zero_zero_nat )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% diff_is_0_eq
thf(fact_1185_Nat_Oadd__diff__assoc,axiom,
    ! [K: nat,J: nat,I4: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( plus_plus_nat @ I4 @ ( minus_minus_nat @ J @ K ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I4 @ J ) @ K ) ) ) ).

% Nat.add_diff_assoc
thf(fact_1186_Nat_Oadd__diff__assoc2,axiom,
    ! [K: nat,J: nat,I4: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I4 )
        = ( minus_minus_nat @ ( plus_plus_nat @ J @ I4 ) @ K ) ) ) ).

% Nat.add_diff_assoc2
thf(fact_1187_Nat_Odiff__diff__right,axiom,
    ! [K: nat,J: nat,I4: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ I4 @ ( minus_minus_nat @ J @ K ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I4 @ K ) @ J ) ) ) ).

% Nat.diff_diff_right
thf(fact_1188_Suc__pred,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
        = N ) ) ).

% Suc_pred
thf(fact_1189_diff__Suc__diff__eq2,axiom,
    ! [K: nat,J: nat,I4: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I4 )
        = ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I4 ) ) ) ) ).

% diff_Suc_diff_eq2
thf(fact_1190_diff__Suc__diff__eq1,axiom,
    ! [K: nat,J: nat,I4: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ I4 @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I4 @ K ) @ ( suc @ J ) ) ) ) ).

% diff_Suc_diff_eq1
thf(fact_1191_diff__add__0,axiom,
    ! [N: nat,M: nat] :
      ( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
      = zero_zero_nat ) ).

% diff_add_0
thf(fact_1192_diff__add__inverse2,axiom,
    ! [M: nat,N: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
      = M ) ).

% diff_add_inverse2
thf(fact_1193_diff__add__inverse,axiom,
    ! [N: nat,M: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
      = M ) ).

% diff_add_inverse
thf(fact_1194_diff__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
      = ( minus_minus_nat @ M @ N ) ) ).

% diff_cancel2
thf(fact_1195_Nat_Odiff__cancel,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
      = ( minus_minus_nat @ M @ N ) ) ).

% Nat.diff_cancel
thf(fact_1196_add__diff__inverse__nat,axiom,
    ! [M: nat,N: nat] :
      ( ~ ( ord_less_nat @ M @ N )
     => ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
        = M ) ) ).

% add_diff_inverse_nat
thf(fact_1197_less__diff__conv,axiom,
    ! [I4: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I4 @ ( minus_minus_nat @ J @ K ) )
      = ( ord_less_nat @ ( plus_plus_nat @ I4 @ K ) @ J ) ) ).

% less_diff_conv
thf(fact_1198_diff__less,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_nat @ zero_zero_nat @ M )
       => ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).

% diff_less
thf(fact_1199_Suc__diff__Suc,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ N @ M )
     => ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N ) ) )
        = ( minus_minus_nat @ M @ N ) ) ) ).

% Suc_diff_Suc
thf(fact_1200_diff__less__Suc,axiom,
    ! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).

% diff_less_Suc
thf(fact_1201_Suc__diff__le,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( minus_minus_nat @ ( suc @ M ) @ N )
        = ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).

% Suc_diff_le
thf(fact_1202_diff__less__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C2 @ A )
       => ( ord_less_nat @ ( minus_minus_nat @ A @ C2 ) @ ( minus_minus_nat @ B @ C2 ) ) ) ) ).

% diff_less_mono
thf(fact_1203_less__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
          = ( ord_less_nat @ M @ N ) ) ) ) ).

% less_diff_iff
thf(fact_1204_diff__le__mono2,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).

% diff_le_mono2
thf(fact_1205_le__diff__iff_H,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ C2 )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ C2 @ A ) @ ( minus_minus_nat @ C2 @ B ) )
          = ( ord_less_eq_nat @ B @ A ) ) ) ) ).

% le_diff_iff'
thf(fact_1206_diff__le__self,axiom,
    ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).

% diff_le_self
thf(fact_1207_diff__le__mono,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).

% diff_le_mono
thf(fact_1208_Nat_Odiff__diff__eq,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
          = ( minus_minus_nat @ M @ N ) ) ) ) ).

% Nat.diff_diff_eq
thf(fact_1209_le__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
          = ( ord_less_eq_nat @ M @ N ) ) ) ) ).

% le_diff_iff
thf(fact_1210_eq__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( ( minus_minus_nat @ M @ K )
            = ( minus_minus_nat @ N @ K ) )
          = ( M = N ) ) ) ) ).

% eq_diff_iff
thf(fact_1211_less__imp__diff__less,axiom,
    ! [J: nat,K: nat,N: nat] :
      ( ( ord_less_nat @ J @ K )
     => ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).

% less_imp_diff_less
thf(fact_1212_diff__less__mono2,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ( ord_less_nat @ M @ L )
       => ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).

% diff_less_mono2
thf(fact_1213_minus__nat_Odiff__0,axiom,
    ! [M: nat] :
      ( ( minus_minus_nat @ M @ zero_zero_nat )
      = M ) ).

% minus_nat.diff_0
thf(fact_1214_diffs0__imp__equal,axiom,
    ! [M: nat,N: nat] :
      ( ( ( minus_minus_nat @ M @ N )
        = zero_zero_nat )
     => ( ( ( minus_minus_nat @ N @ M )
          = zero_zero_nat )
       => ( M = N ) ) ) ).

% diffs0_imp_equal
thf(fact_1215_zero__induct__lemma,axiom,
    ! [P: nat > $o,K: nat,I4: nat] :
      ( ( P @ K )
     => ( ! [N2: nat] :
            ( ( P @ ( suc @ N2 ) )
           => ( P @ N2 ) )
       => ( P @ ( minus_minus_nat @ K @ I4 ) ) ) ) ).

% zero_induct_lemma
thf(fact_1216_diff__commute,axiom,
    ! [I4: nat,J: nat,K: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ I4 @ J ) @ K )
      = ( minus_minus_nat @ ( minus_minus_nat @ I4 @ K ) @ J ) ) ).

% diff_commute
thf(fact_1217_le__diff__conv,axiom,
    ! [J: nat,K: nat,I4: nat] :
      ( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I4 )
      = ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I4 @ K ) ) ) ).

% le_diff_conv
thf(fact_1218_Nat_Ole__diff__conv2,axiom,
    ! [K: nat,J: nat,I4: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( ord_less_eq_nat @ I4 @ ( minus_minus_nat @ J @ K ) )
        = ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ K ) @ J ) ) ) ).

% Nat.le_diff_conv2
thf(fact_1219_Nat_Odiff__add__assoc,axiom,
    ! [K: nat,J: nat,I4: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ I4 @ J ) @ K )
        = ( plus_plus_nat @ I4 @ ( minus_minus_nat @ J @ K ) ) ) ) ).

% Nat.diff_add_assoc
thf(fact_1220_Nat_Odiff__add__assoc2,axiom,
    ! [K: nat,J: nat,I4: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I4 ) @ K )
        = ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I4 ) ) ) ).

% Nat.diff_add_assoc2
thf(fact_1221_Nat_Ole__imp__diff__is__add,axiom,
    ! [I4: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I4 @ J )
     => ( ( ( minus_minus_nat @ J @ I4 )
          = K )
        = ( J
          = ( plus_plus_nat @ K @ I4 ) ) ) ) ).

% Nat.le_imp_diff_is_add
thf(fact_1222_int__plus,axiom,
    ! [N: nat,M: nat] :
      ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N @ M ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M ) ) ) ).

% int_plus
thf(fact_1223_int__ops_I5_J,axiom,
    ! [A: nat,B: nat] :
      ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ A @ B ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).

% int_ops(5)
thf(fact_1224_zadd__int__left,axiom,
    ! [M: nat,N: nat,Z4: int] :
      ( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ Z4 ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) ) @ Z4 ) ) ).

% zadd_int_left
thf(fact_1225_mod__if,axiom,
    ( modulo_modulo_nat
    = ( ^ [M3: nat,N3: nat] : ( if_nat @ ( ord_less_nat @ M3 @ N3 ) @ M3 @ ( modulo_modulo_nat @ ( minus_minus_nat @ M3 @ N3 ) @ N3 ) ) ) ) ).

% mod_if
thf(fact_1226_zle__iff__zadd,axiom,
    ( ord_less_eq_int
    = ( ^ [W2: int,Z3: int] :
        ? [N3: nat] :
          ( Z3
          = ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).

% zle_iff_zadd
thf(fact_1227_diff__Suc__less,axiom,
    ! [N: nat,I4: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) @ N ) ) ).

% diff_Suc_less
thf(fact_1228_nat__diff__split,axiom,
    ! [P: nat > $o,A: nat,B: nat] :
      ( ( P @ ( minus_minus_nat @ A @ B ) )
      = ( ( ( ord_less_nat @ A @ B )
         => ( P @ zero_zero_nat ) )
        & ! [D3: nat] :
            ( ( A
              = ( plus_plus_nat @ B @ D3 ) )
           => ( P @ D3 ) ) ) ) ).

% nat_diff_split
thf(fact_1229_nat__diff__split__asm,axiom,
    ! [P: nat > $o,A: nat,B: nat] :
      ( ( P @ ( minus_minus_nat @ A @ B ) )
      = ( ~ ( ( ( ord_less_nat @ A @ B )
              & ~ ( P @ zero_zero_nat ) )
            | ? [D3: nat] :
                ( ( A
                  = ( plus_plus_nat @ B @ D3 ) )
                & ~ ( P @ D3 ) ) ) ) ) ).

% nat_diff_split_asm
thf(fact_1230_less__diff__conv2,axiom,
    ! [K: nat,J: nat,I4: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I4 )
        = ( ord_less_nat @ J @ ( plus_plus_nat @ I4 @ K ) ) ) ) ).

% less_diff_conv2
thf(fact_1231_zless__iff__Suc__zadd,axiom,
    ( ord_less_int
    = ( ^ [W2: int,Z3: int] :
        ? [N3: nat] :
          ( Z3
          = ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ) ).

% zless_iff_Suc_zadd
thf(fact_1232_adjust__idx__rev__def,axiom,
    ( missin3815256168798769645dx_rev
    = ( ^ [I3: nat,J3: nat] : ( if_nat @ ( ord_less_nat @ J3 @ I3 ) @ J3 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) ) ).

% adjust_idx_rev_def
thf(fact_1233_int__diff__cases,axiom,
    ! [Z4: int] :
      ~ ! [M4: nat,N2: nat] :
          ( Z4
         != ( minus_minus_int @ ( semiri1314217659103216013at_int @ M4 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).

% int_diff_cases
thf(fact_1234_int__ops_I6_J,axiom,
    ! [A: nat,B: nat] :
      ( ( ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
       => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
          = zero_zero_int ) )
      & ( ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
       => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
          = ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ) ) ).

% int_ops(6)
thf(fact_1235_zdiff__int__split,axiom,
    ! [P: int > $o,X: nat,Y: nat] :
      ( ( P @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X @ Y ) ) )
      = ( ( ( ord_less_eq_nat @ Y @ X )
         => ( P @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X ) @ ( semiri1314217659103216013at_int @ Y ) ) ) )
        & ( ( ord_less_nat @ X @ Y )
         => ( P @ zero_zero_int ) ) ) ) ).

% zdiff_int_split
thf(fact_1236_less__one,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ N @ one_one_nat )
      = ( N = zero_zero_nat ) ) ).

% less_one
thf(fact_1237_diff__Suc__1,axiom,
    ! [N: nat] :
      ( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
      = N ) ).

% diff_Suc_1

% Helper facts (3)
thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
    ! [P: $o] :
      ( ( P = $true )
      | ( P = $false ) ) ).

thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] :
      ( ( if_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] :
      ( ( if_nat @ $true @ X @ Y )
      = X ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    ( ( fOR_is_to_trs_a_b @ ( map_fs2147491833908496126rm_a_b @ fset_P6170650533370837496rm_a_b @ rs ) @ is )
    = ( fset_P6170650533370837496rm_a_b @ ( fOR_is5184614021136371515rs_a_b @ rs @ is ) ) ) ).

%------------------------------------------------------------------------------