%------------------------------------------------------------------------------
% 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    : Knights_Tour/0000_KnightsTour/prob_00919_035857__5867492_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 1575 ( 622 unt; 297 typ;   0 def)
%            Number of atoms       : 3350 (1444 equ;   0 cnn)
%            Maximal formula atoms :    9 (   2 avg)
%            Number of connectives : 11341 ( 269   ~;  70   |; 254   &;9525   @)
%                                         (   0 <=>;1223  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   17 (   6 avg)
%            Number of types       :   42 (  41 usr)
%            Number of type conns  :  856 ( 856   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  259 ( 256 usr;  13 con; 0-8 aty)
%            Number of variables   : 3500 ( 112   ^;3287   !; 101   ?;3500   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 09:00:51.560
%------------------------------------------------------------------------------
% Could-be-implicit typings (41)
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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thf(ty_n_t__List__Olist_I_062_It__Int__Oint_Mt__Nat__Onat_J_J,type,
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thf(ty_n_t__List__Olist_It__List__Olist_It__Int__Oint_J_J,type,
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thf(ty_n_t__Set__Oset_It__List__Olist_It__Int__Oint_J_J,type,
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thf(ty_n_t__Option__Ooption_It__Int__Oint_J,type,
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thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
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thf(ty_n_t__List__Olist_It__Int__Oint_J,type,
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thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
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thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
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thf(ty_n_t__Typerep__Otyperep,type,
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thf(ty_n_t__String__Ochar,type,
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thf(ty_n_t__Nat__Onat,type,
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thf(ty_n_t__Int__Oint,type,
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% Explicit typings (256)
thf(sy_c_BNF__Def_Ocsquare_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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thf(sy_c_BNF__Def_OfstOp_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint,type,
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thf(sy_c_BNF__Def_OsndOp_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint,type,
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thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint,type,
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thf(sy_c_Fun_Ofun__upd_001t__Int__Oint_001t__Option__Ooption_It__Int__Oint_J,type,
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thf(sy_c_Fun_Ofun__upd_001t__Int__Oint_001t__Option__Ooption_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
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thf(sy_c_Fun_Ofun__upd_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Option__Ooption_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
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thf(sy_c_Groups_Oabs__class_Oabs_001t__Int__Oint,type,
    abs_abs_int: int > int ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
    minus_minus_int: int > int > int ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
    minus_minus_nat: nat > nat > nat ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Int__Oint_J,type,
    minus_minus_set_int: set_int > set_int > set_int ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
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thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
    one_one_int: int ).

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_Otimes__class_Otimes_001t__Int__Oint,type,
    times_times_int: int > int > int ).

thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
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thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
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thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
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thf(sy_c_If_001t__List__Olist_It__Int__Oint_J,type,
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thf(sy_c_If_001t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
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thf(sy_c_If_001t__Nat__Onat,type,
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thf(sy_c_KnightsTour_Omirror1,type,
    mirror1: list_P5707943133018811711nt_int > list_P5707943133018811711nt_int ).

thf(sy_c_KnightsTour_Omirror1__aux,type,
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thf(sy_c_KnightsTour_Omirror2,type,
    mirror2: list_P5707943133018811711nt_int > list_P5707943133018811711nt_int ).

thf(sy_c_KnightsTour_Omirror2__aux,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Int__Oint_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
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thf(sy_c_Lattices__Big_Olinorder__class_OMax_001t__Int__Oint,type,
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thf(sy_c_Lattices__Big_Olinorder__class_OMin_001t__Int__Oint,type,
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thf(sy_c_List_Obutlast_001t__Int__Oint,type,
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thf(sy_c_List_Obutlast_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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thf(sy_c_List_Ocan__select_001t__Int__Oint,type,
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thf(sy_c_List_Ocan__select_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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thf(sy_c_List_Ocount__list_001t__Int__Oint,type,
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thf(sy_c_List_Ocount__list_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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thf(sy_c_List_Odistinct_001t__Int__Oint,type,
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thf(sy_c_List_Odistinct_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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thf(sy_c_List_Odistinct_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
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thf(sy_c_List_Oenumerate_001t__Int__Oint,type,
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thf(sy_c_List_Oenumerate_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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thf(sy_c_List_Ofind_001t__Int__Oint,type,
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thf(sy_c_List_Ofind_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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thf(sy_c_List_Ogen__length_001t__Int__Oint,type,
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thf(sy_c_List_Ogen__length_001t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
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thf(sy_c_List_Ogen__length_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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thf(sy_c_List_Ogen__length_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
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thf(sy_c_List_Oinsert_001t__Int__Oint,type,
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thf(sy_c_List_Oinsert_001t__List__Olist_It__Int__Oint_J,type,
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thf(sy_c_List_Oinsert_001t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
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thf(sy_c_List_Oinsert_001t__Nat__Onat,type,
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thf(sy_c_List_Oinsert_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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thf(sy_c_List_Oinsert_001t__Product____Type__Oprod_It__Int__Oint_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
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thf(sy_c_List_Oinsert_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
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thf(sy_c_List_Olenlex_001t__Int__Oint,type,
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thf(sy_c_List_Olenlex_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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thf(sy_c_List_Olex_001t__Int__Oint,type,
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thf(sy_c_List_Olex_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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thf(sy_c_List_Olexord_001t__Int__Oint,type,
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    restri2103409392168779760nt_int: ( product_prod_int_int > option4624381673175914239nt_int ) > set_Pr958786334691620121nt_int > product_prod_int_int > option4624381673175914239nt_int ).

thf(sy_c_Nat_OSuc,type,
    suc: nat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Int__Oint_J,type,
    size_size_list_int: list_int > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
    size_s2969076144586574001nt_int: list_l1670014477004246597nt_int > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    size_s5157815400016825771nt_int: list_P5707943133018811711nt_int > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
    size_s2001693051472072450nt_int: list_P314425023053850222nt_int > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
    size_s6770063216428074713nt_int: list_P2336717926344734829nt_int > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__String__Ochar,type,
    size_size_char: char > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__Typerep__Otyperep,type,
    size_size_typerep: typerep > nat ).

thf(sy_c_Option_Ooption_ONone_001t__Int__Oint,type,
    none_int: option_int ).

thf(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    none_P2377608414092835994nt_int: option4624381673175914239nt_int ).

thf(sy_c_Option_Ooption_OSome_001t__Int__Oint,type,
    some_int: int > option_int ).

thf(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    some_P4184893108420464158nt_int: product_prod_int_int > option4624381673175914239nt_int ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
    ord_less_int: int > int > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
    ord_less_nat: nat > nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Int__Oint_J,type,
    ord_less_set_int: set_int > set_int > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    ord_le7563427860532173253nt_int: set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
    ord_less_eq_int: int > int > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
    ord_less_eq_nat: nat > nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Int__Oint_J,type,
    ord_less_eq_set_int: set_int > set_int > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
    ord_less_eq_set_nat: set_nat > set_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    ord_le2843351958646193337nt_int: set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int > $o ).

thf(sy_c_Orderings_Oord__class_Omin_001t__Int__Oint,type,
    ord_min_int: int > int > int ).

thf(sy_c_Orderings_Oord__class_Omin_001t__Nat__Onat,type,
    ord_min_nat: nat > nat > nat ).

thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Int__Oint_M_Eo_J,type,
    top_top_int_o: int > $o ).

thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J,type,
    top_to1578927101902068148_int_o: product_prod_int_int > $o ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Int__Oint_J,type,
    top_top_set_int: set_int ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
    top_top_set_nat: set_nat ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    top_to4366644338036079209nt_int: set_Pr958786334691620121nt_int ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
    top_to2321149100101803671nt_int: set_Pr2560585780119916871nt_int ).

thf(sy_c_Parity_Ounique__euclidean__semiring__with__nat__division__class_Odivides__aux_001t__Int__Oint,type,
    unique5329631941980267465ux_int: product_prod_int_int > $o ).

thf(sy_c_Parity_Ounique__euclidean__semiring__with__nat__division__class_Odivides__aux_001t__Nat__Onat,type,
    unique5332122412489317741ux_nat: product_prod_nat_nat > $o ).

thf(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Int__Oint,type,
    product_Pair_int_int: int > int > product_prod_int_int ).

thf(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    produc346731203614136500nt_int: int > product_prod_int_int > produc759720530913461378nt_int ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Int__Oint_J_001t__List__Olist_It__Int__Oint_J,type,
    produc364263696895485585st_int: list_int > list_int > produc1186641810826059865st_int ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
    produc2694037385005941721st_nat: list_nat > list_nat > produc1828647624359046049st_nat ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    produc1932183703851549015nt_int: list_P5707943133018811711nt_int > list_P5707943133018811711nt_int > produc1089560213143673063nt_int ).

thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Int__Oint,type,
    product_Pair_nat_int: nat > int > product_prod_nat_int ).

thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Nat__Onat,type,
    product_Pair_nat_nat: nat > nat > product_prod_nat_nat ).

thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    produc6532261156004778512nt_int: nat > product_prod_int_int > produc6945250483304103390nt_int ).

thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Int__Oint,type,
    produc8906795734991021710nt_int: product_prod_int_int > int > produc6754623152930779156nt_int ).

thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    produc3646306378393792727nt_int: product_prod_int_int > product_prod_int_int > produc1219242969750017639nt_int ).

thf(sy_c_Product__Type_Omap__prod_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint,type,
    produc6036585564866528938nt_int: ( int > int ) > ( int > int ) > product_prod_int_int > product_prod_int_int ).

thf(sy_c_Product__Type_Omap__prod_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Int__Oint,type,
    produc8321826182988070966nt_int: ( product_prod_int_int > int ) > ( product_prod_int_int > int ) > produc1219242969750017639nt_int > product_prod_int_int ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001_Eo,type,
    produc4947309494688390418_int_o: ( int > int > $o ) > product_prod_int_int > $o ).

thf(sy_c_Product__Type_Oprod_Ofst_001t__Int__Oint_001t__Int__Oint,type,
    product_fst_int_int: product_prod_int_int > int ).

thf(sy_c_Product__Type_Oprod_Ofst_001t__Nat__Onat_001t__Nat__Onat,type,
    product_fst_nat_nat: product_prod_nat_nat > nat ).

thf(sy_c_Product__Type_Oprod_Ofst_001t__Nat__Onat_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    produc4803745454372187940nt_int: produc6945250483304103390nt_int > nat ).

thf(sy_c_Product__Type_Oprod_Ofst_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Nat__Onat,type,
    produc7180770503867481414nt_nat: produc1709102135585200056nt_nat > product_prod_int_int ).

thf(sy_c_Product__Type_Oprod_Ofst_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    produc698254169746827971nt_int: produc1219242969750017639nt_int > product_prod_int_int ).

thf(sy_c_Product__Type_Oprod_Osnd_001t__Int__Oint_001t__Int__Oint,type,
    product_snd_int_int: product_prod_int_int > int ).

thf(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001t__Nat__Onat,type,
    product_snd_nat_nat: product_prod_nat_nat > nat ).

thf(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    produc3248548876605496418nt_int: produc6945250483304103390nt_int > product_prod_int_int ).

thf(sy_c_Product__Type_Oprod_Osnd_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Nat__Onat,type,
    produc5625573926100789892nt_nat: produc1709102135585200056nt_nat > nat ).

thf(sy_c_Product__Type_Oprod_Osnd_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    produc3892743399831173125nt_int: produc1219242969750017639nt_int > product_prod_int_int ).

thf(sy_c_Product__Type_Oprod_Oswap_001t__Int__Oint_001t__Int__Oint,type,
    product_swap_int_int: product_prod_int_int > product_prod_int_int ).

thf(sy_c_Product__Type_Oprod_Oswap_001t__Nat__Onat_001t__Nat__Onat,type,
    product_swap_nat_nat: product_prod_nat_nat > product_prod_nat_nat ).

thf(sy_c_Product__Type_Oprod_Oswap_001t__Nat__Onat_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    produc854102916164664432nt_int: produc6945250483304103390nt_int > produc1709102135585200056nt_nat ).

thf(sy_c_Product__Type_Oprod_Oswap_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Nat__Onat,type,
    produc3231127965659957906nt_nat: produc1709102135585200056nt_nat > produc6945250483304103390nt_int ).

thf(sy_c_Product__Type_Oprod_Oswap_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    produc4709573950824599159nt_int: produc1219242969750017639nt_int > produc1219242969750017639nt_int ).

thf(sy_c_Relation_Oasym__on_001t__Int__Oint,type,
    asym_on_int: set_int > set_Pr958786334691620121nt_int > $o ).

thf(sy_c_Relation_Oasym__on_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    asym_o5368087371152427499nt_int: set_Pr958786334691620121nt_int > set_Pr2560585780119916871nt_int > $o ).

thf(sy_c_Relation_Oinv__image_001t__Int__Oint_001t__Int__Oint,type,
    inv_image_int_int: set_Pr958786334691620121nt_int > ( int > int ) > set_Pr958786334691620121nt_int ).

thf(sy_c_Relation_Orelcompp_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint,type,
    relcompp_int_int_int: ( int > int > $o ) > ( int > int > $o ) > int > int > $o ).

thf(sy_c_Set_OCollect_001t__Int__Oint,type,
    collect_int: ( int > $o ) > set_int ).

thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    collec213857154873943460nt_int: ( product_prod_int_int > $o ) > set_Pr958786334691620121nt_int ).

thf(sy_c_Set_Oimage_001t__Int__Oint_001t__Int__Oint,type,
    image_int_int: ( int > int ) > set_int > set_int ).

thf(sy_c_Set_Oimage_001t__Int__Oint_001t__List__Olist_It__Int__Oint_J,type,
    image_int_list_int: ( int > list_int ) > set_int > set_list_int ).

thf(sy_c_Set_Oimage_001t__Int__Oint_001t__Nat__Onat,type,
    image_int_nat: ( int > nat ) > set_int > set_nat ).

thf(sy_c_Set_Oimage_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    image_5705468584675977158nt_int: ( int > product_prod_int_int ) > set_int > set_Pr958786334691620121nt_int ).

thf(sy_c_Set_Oimage_001t__List__Olist_It__Int__Oint_J_001t__Int__Oint,type,
    image_list_int_int: ( list_int > int ) > set_list_int > set_int ).

thf(sy_c_Set_Oimage_001t__List__Olist_It__Int__Oint_J_001t__Nat__Onat,type,
    image_list_int_nat: ( list_int > nat ) > set_list_int > set_nat ).

thf(sy_c_Set_Oimage_001t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001t__Int__Oint,type,
    image_589057387511608870nt_int: ( list_P5707943133018811711nt_int > int ) > set_li3855193629254082847nt_int > set_int ).

thf(sy_c_Set_Oimage_001t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001t__Nat__Onat,type,
    image_591547858020659146nt_nat: ( list_P5707943133018811711nt_int > nat ) > set_li3855193629254082847nt_int > set_nat ).

thf(sy_c_Set_Oimage_001t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    image_689400715899363487nt_int: ( list_P5707943133018811711nt_int > set_Pr958786334691620121nt_int ) > set_li3855193629254082847nt_int > set_se6260736226359567993nt_int ).

thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Int__Oint,type,
    image_nat_int: ( nat > int ) > set_nat > set_int ).

thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
    image_nat_nat: ( nat > nat ) > set_nat > set_nat ).

thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    image_2667626500211843362nt_int: ( nat > product_prod_int_int ) > set_nat > set_Pr958786334691620121nt_int ).

thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Int__Oint,type,
    image_5042161079198086560nt_int: ( product_prod_int_int > int ) > set_Pr958786334691620121nt_int > set_int ).

thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Nat__Onat,type,
    image_5044651549707136836nt_nat: ( product_prod_int_int > nat ) > set_Pr958786334691620121nt_int > set_nat ).

thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    image_2653370878348428101nt_int: ( product_prod_int_int > product_prod_int_int ) > set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int ).

thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    image_2053528049745388055nt_int: ( produc1219242969750017639nt_int > product_prod_int_int ) > set_Pr2560585780119916871nt_int > set_Pr958786334691620121nt_int ).

thf(sy_c_Set_Oinsert_001t__Int__Oint,type,
    insert_int2: int > set_int > set_int ).

thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
    insert_nat2: nat > set_nat > set_nat ).

thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    insert5033312907999012233nt_int: product_prod_int_int > set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int ).

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_OChar,type,
    char2: $o > $o > $o > $o > $o > $o > $o > $o > char ).

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

thf(sy_c_Wellfounded_Olex__prod_001t__Int__Oint_001t__Int__Oint,type,
    lex_prod_int_int: set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int > set_Pr2560585780119916871nt_int ).

thf(sy_c_Wellfounded_Omeasure_001t__Int__Oint,type,
    measure_int: ( int > nat ) > set_Pr958786334691620121nt_int ).

thf(sy_c_member_001t__Int__Oint,type,
    member_int2: int > set_int > $o ).

thf(sy_c_member_001t__List__Olist_It__Int__Oint_J,type,
    member_list_int2: list_int > set_list_int > $o ).

thf(sy_c_member_001t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    member2764346250752101224nt_int: list_P5707943133018811711nt_int > set_li3855193629254082847nt_int > $o ).

thf(sy_c_member_001t__Nat__Onat,type,
    member_nat2: nat > set_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    member5262025264175285858nt_int: product_prod_int_int > set_Pr958786334691620121nt_int > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Int__Oint_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    member4957952664650131097nt_int: produc759720530913461378nt_int > set_Pr5025984505184017080nt_int > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J,type,
    member6698963635872716290st_int: produc1186641810826059865st_int > set_Pr765067013931698361st_int > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
    member1390679175989562640nt_int: produc1089560213143673063nt_int > set_Pr6304946757569631943nt_int > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    member1920110580185997301nt_int: produc6945250483304103390nt_int > set_Pr6123067485828015636nt_int > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Int__Oint_J,type,
    member1729483249812673067nt_int: produc6754623152930779156nt_int > set_Pr53225590989397322nt_int > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    member8566619992076573584nt_int: produc1219242969750017639nt_int > set_Pr2560585780119916871nt_int > $o ).

thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    member2340774599025711042nt_int: set_Pr958786334691620121nt_int > set_se6260736226359567993nt_int > $o ).

thf(sy_v_ps,type,
    ps: list_P5707943133018811711nt_int ).

% Relevant facts (1270)
thf(fact_0_length__mirror1__aux,axiom,
    ! [N: int] :
      ( size_s5157815400016825771nt_int
      = ( ^ [Ps: list_P5707943133018811711nt_int] : ( size_s5157815400016825771nt_int @ ( mirror1_aux @ N @ Ps ) ) ) ) ).

% length_mirror1_aux
thf(fact_1_mirror1__def,axiom,
    ( mirror1
    = ( ^ [Ps: list_P5707943133018811711nt_int] : ( mirror1_aux @ ( plus_plus_int @ ( lattic8263393255366662781ax_int @ ( image_5042161079198086560nt_int @ product_fst_int_int @ ( set_Pr2470121279949933262nt_int @ Ps ) ) ) @ ( lattic8718645017227715691in_int @ ( image_5042161079198086560nt_int @ product_fst_int_int @ ( set_Pr2470121279949933262nt_int @ Ps ) ) ) ) @ Ps ) ) ) ).

% mirror1_def
thf(fact_2_image__eqI,axiom,
    ! [B: int,F: product_prod_int_int > int,X: product_prod_int_int,A: set_Pr958786334691620121nt_int] :
      ( ( B
        = ( F @ X ) )
     => ( ( member5262025264175285858nt_int @ X @ A )
       => ( member_int2 @ B @ ( image_5042161079198086560nt_int @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_3_image__eqI,axiom,
    ! [B: int,F: int > int,X: int,A: set_int] :
      ( ( B
        = ( F @ X ) )
     => ( ( member_int2 @ X @ A )
       => ( member_int2 @ B @ ( image_int_int @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_4_image__eqI,axiom,
    ! [B: product_prod_int_int,F: int > product_prod_int_int,X: int,A: set_int] :
      ( ( B
        = ( F @ X ) )
     => ( ( member_int2 @ X @ A )
       => ( member5262025264175285858nt_int @ B @ ( image_5705468584675977158nt_int @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_5_image__eqI,axiom,
    ! [B: product_prod_int_int,F: product_prod_int_int > product_prod_int_int,X: product_prod_int_int,A: set_Pr958786334691620121nt_int] :
      ( ( B
        = ( F @ X ) )
     => ( ( member5262025264175285858nt_int @ X @ A )
       => ( member5262025264175285858nt_int @ B @ ( image_2653370878348428101nt_int @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_6_image__eqI,axiom,
    ! [B: nat,F: int > nat,X: int,A: set_int] :
      ( ( B
        = ( F @ X ) )
     => ( ( member_int2 @ X @ A )
       => ( member_nat2 @ B @ ( image_int_nat @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_7_image__eqI,axiom,
    ! [B: int,F: nat > int,X: nat,A: set_nat] :
      ( ( B
        = ( F @ X ) )
     => ( ( member_nat2 @ X @ A )
       => ( member_int2 @ B @ ( image_nat_int @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_8_image__eqI,axiom,
    ! [B: nat,F: nat > nat,X: nat,A: set_nat] :
      ( ( B
        = ( F @ X ) )
     => ( ( member_nat2 @ X @ A )
       => ( member_nat2 @ B @ ( image_nat_nat @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_9_image__eqI,axiom,
    ! [B: list_int,F: int > list_int,X: int,A: set_int] :
      ( ( B
        = ( F @ X ) )
     => ( ( member_int2 @ X @ A )
       => ( member_list_int2 @ B @ ( image_int_list_int @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_10_image__eqI,axiom,
    ! [B: int,F: list_int > int,X: list_int,A: set_list_int] :
      ( ( B
        = ( F @ X ) )
     => ( ( member_list_int2 @ X @ A )
       => ( member_int2 @ B @ ( image_list_int_int @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_11_image__eqI,axiom,
    ! [B: nat,F: list_int > nat,X: list_int,A: set_list_int] :
      ( ( B
        = ( F @ X ) )
     => ( ( member_list_int2 @ X @ A )
       => ( member_nat2 @ B @ ( image_list_int_nat @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_12_add__left__cancel,axiom,
    ! [A2: int,B: int,C: int] :
      ( ( ( plus_plus_int @ A2 @ B )
        = ( plus_plus_int @ A2 @ C ) )
      = ( B = C ) ) ).

% add_left_cancel
thf(fact_13_add__left__cancel,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( ( ( plus_plus_nat @ A2 @ B )
        = ( plus_plus_nat @ A2 @ C ) )
      = ( B = C ) ) ).

% add_left_cancel
thf(fact_14_add__right__cancel,axiom,
    ! [B: int,A2: int,C: int] :
      ( ( ( plus_plus_int @ B @ A2 )
        = ( plus_plus_int @ C @ A2 ) )
      = ( B = C ) ) ).

% add_right_cancel
thf(fact_15_add__right__cancel,axiom,
    ! [B: nat,A2: nat,C: nat] :
      ( ( ( plus_plus_nat @ B @ A2 )
        = ( plus_plus_nat @ C @ A2 ) )
      = ( B = C ) ) ).

% add_right_cancel
thf(fact_16_mirror2__def,axiom,
    ( mirror2
    = ( ^ [Ps: list_P5707943133018811711nt_int] : ( mirror2_aux @ ( plus_plus_int @ ( lattic8263393255366662781ax_int @ ( image_5042161079198086560nt_int @ product_snd_int_int @ ( set_Pr2470121279949933262nt_int @ Ps ) ) ) @ ( lattic8718645017227715691in_int @ ( image_5042161079198086560nt_int @ product_snd_int_int @ ( set_Pr2470121279949933262nt_int @ Ps ) ) ) ) @ Ps ) ) ) ).

% mirror2_def
thf(fact_17_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs: list_int] :
      ( ( size_size_list_int @ Xs )
      = N ) ).

% Ex_list_of_length
thf(fact_18_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs: list_P2336717926344734829nt_int] :
      ( ( size_s6770063216428074713nt_int @ Xs )
      = N ) ).

% Ex_list_of_length
thf(fact_19_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs: list_l1670014477004246597nt_int] :
      ( ( size_s2969076144586574001nt_int @ Xs )
      = N ) ).

% Ex_list_of_length
thf(fact_20_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs: list_P314425023053850222nt_int] :
      ( ( size_s2001693051472072450nt_int @ Xs )
      = N ) ).

% Ex_list_of_length
thf(fact_21_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs: list_P5707943133018811711nt_int] :
      ( ( size_s5157815400016825771nt_int @ Xs )
      = N ) ).

% Ex_list_of_length
thf(fact_22_neq__if__length__neq,axiom,
    ! [Xs2: list_int,Ys: list_int] :
      ( ( ( size_size_list_int @ Xs2 )
       != ( size_size_list_int @ Ys ) )
     => ( Xs2 != Ys ) ) ).

% neq_if_length_neq
thf(fact_23_neq__if__length__neq,axiom,
    ! [Xs2: list_P2336717926344734829nt_int,Ys: list_P2336717926344734829nt_int] :
      ( ( ( size_s6770063216428074713nt_int @ Xs2 )
       != ( size_s6770063216428074713nt_int @ Ys ) )
     => ( Xs2 != Ys ) ) ).

% neq_if_length_neq
thf(fact_24_neq__if__length__neq,axiom,
    ! [Xs2: list_l1670014477004246597nt_int,Ys: list_l1670014477004246597nt_int] :
      ( ( ( size_s2969076144586574001nt_int @ Xs2 )
       != ( size_s2969076144586574001nt_int @ Ys ) )
     => ( Xs2 != Ys ) ) ).

% neq_if_length_neq
thf(fact_25_neq__if__length__neq,axiom,
    ! [Xs2: list_P314425023053850222nt_int,Ys: list_P314425023053850222nt_int] :
      ( ( ( size_s2001693051472072450nt_int @ Xs2 )
       != ( size_s2001693051472072450nt_int @ Ys ) )
     => ( Xs2 != Ys ) ) ).

% neq_if_length_neq
thf(fact_26_neq__if__length__neq,axiom,
    ! [Xs2: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int] :
      ( ( ( size_s5157815400016825771nt_int @ Xs2 )
       != ( size_s5157815400016825771nt_int @ Ys ) )
     => ( Xs2 != Ys ) ) ).

% neq_if_length_neq
thf(fact_27_size__neq__size__imp__neq,axiom,
    ! [X: list_int,Y: list_int] :
      ( ( ( size_size_list_int @ X )
       != ( size_size_list_int @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_28_size__neq__size__imp__neq,axiom,
    ! [X: list_P2336717926344734829nt_int,Y: list_P2336717926344734829nt_int] :
      ( ( ( size_s6770063216428074713nt_int @ X )
       != ( size_s6770063216428074713nt_int @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_29_size__neq__size__imp__neq,axiom,
    ! [X: list_l1670014477004246597nt_int,Y: list_l1670014477004246597nt_int] :
      ( ( ( size_s2969076144586574001nt_int @ X )
       != ( size_s2969076144586574001nt_int @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_30_size__neq__size__imp__neq,axiom,
    ! [X: list_P314425023053850222nt_int,Y: list_P314425023053850222nt_int] :
      ( ( ( size_s2001693051472072450nt_int @ X )
       != ( size_s2001693051472072450nt_int @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_31_size__neq__size__imp__neq,axiom,
    ! [X: list_P5707943133018811711nt_int,Y: list_P5707943133018811711nt_int] :
      ( ( ( size_s5157815400016825771nt_int @ X )
       != ( size_s5157815400016825771nt_int @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_32_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_33_size__neq__size__imp__neq,axiom,
    ! [X: typerep,Y: typerep] :
      ( ( ( size_size_typerep @ X )
       != ( size_size_typerep @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_34_Inf_OINF__cong,axiom,
    ! [A: set_int,B2: set_int,C2: int > product_prod_int_int,D: int > product_prod_int_int,Inf: set_Pr958786334691620121nt_int > product_prod_int_int] :
      ( ( A = B2 )
     => ( ! [X2: int] :
            ( ( member_int2 @ X2 @ B2 )
           => ( ( C2 @ X2 )
              = ( D @ X2 ) ) )
       => ( ( Inf @ ( image_5705468584675977158nt_int @ C2 @ A ) )
          = ( Inf @ ( image_5705468584675977158nt_int @ D @ B2 ) ) ) ) ) ).

% Inf.INF_cong
thf(fact_35_Inf_OINF__cong,axiom,
    ! [A: set_int,B2: set_int,C2: int > int,D: int > int,Inf: set_int > int] :
      ( ( A = B2 )
     => ( ! [X2: int] :
            ( ( member_int2 @ X2 @ B2 )
           => ( ( C2 @ X2 )
              = ( D @ X2 ) ) )
       => ( ( Inf @ ( image_int_int @ C2 @ A ) )
          = ( Inf @ ( image_int_int @ D @ B2 ) ) ) ) ) ).

% Inf.INF_cong
thf(fact_36_Inf_OINF__cong,axiom,
    ! [A: set_li3855193629254082847nt_int,B2: set_li3855193629254082847nt_int,C2: list_P5707943133018811711nt_int > set_Pr958786334691620121nt_int,D: list_P5707943133018811711nt_int > set_Pr958786334691620121nt_int,Inf: set_se6260736226359567993nt_int > set_Pr958786334691620121nt_int] :
      ( ( A = B2 )
     => ( ! [X2: list_P5707943133018811711nt_int] :
            ( ( member2764346250752101224nt_int @ X2 @ B2 )
           => ( ( C2 @ X2 )
              = ( D @ X2 ) ) )
       => ( ( Inf @ ( image_689400715899363487nt_int @ C2 @ A ) )
          = ( Inf @ ( image_689400715899363487nt_int @ D @ B2 ) ) ) ) ) ).

% Inf.INF_cong
thf(fact_37_Inf_OINF__cong,axiom,
    ! [A: set_nat,B2: set_nat,C2: nat > product_prod_int_int,D: nat > product_prod_int_int,Inf: set_Pr958786334691620121nt_int > product_prod_int_int] :
      ( ( A = B2 )
     => ( ! [X2: nat] :
            ( ( member_nat2 @ X2 @ B2 )
           => ( ( C2 @ X2 )
              = ( D @ X2 ) ) )
       => ( ( Inf @ ( image_2667626500211843362nt_int @ C2 @ A ) )
          = ( Inf @ ( image_2667626500211843362nt_int @ D @ B2 ) ) ) ) ) ).

% Inf.INF_cong
thf(fact_38_Inf_OINF__cong,axiom,
    ! [A: set_nat,B2: set_nat,C2: nat > nat,D: nat > nat,Inf: set_nat > nat] :
      ( ( A = B2 )
     => ( ! [X2: nat] :
            ( ( member_nat2 @ X2 @ B2 )
           => ( ( C2 @ X2 )
              = ( D @ X2 ) ) )
       => ( ( Inf @ ( image_nat_nat @ C2 @ A ) )
          = ( Inf @ ( image_nat_nat @ D @ B2 ) ) ) ) ) ).

% Inf.INF_cong
thf(fact_39_Inf_OINF__cong,axiom,
    ! [A: set_Pr958786334691620121nt_int,B2: set_Pr958786334691620121nt_int,C2: product_prod_int_int > int,D: product_prod_int_int > int,Inf: set_int > int] :
      ( ( A = B2 )
     => ( ! [X2: product_prod_int_int] :
            ( ( member5262025264175285858nt_int @ X2 @ B2 )
           => ( ( C2 @ X2 )
              = ( D @ X2 ) ) )
       => ( ( Inf @ ( image_5042161079198086560nt_int @ C2 @ A ) )
          = ( Inf @ ( image_5042161079198086560nt_int @ D @ B2 ) ) ) ) ) ).

% Inf.INF_cong
thf(fact_40_Sup_OSUP__cong,axiom,
    ! [A: set_int,B2: set_int,C2: int > product_prod_int_int,D: int > product_prod_int_int,Sup: set_Pr958786334691620121nt_int > product_prod_int_int] :
      ( ( A = B2 )
     => ( ! [X2: int] :
            ( ( member_int2 @ X2 @ B2 )
           => ( ( C2 @ X2 )
              = ( D @ X2 ) ) )
       => ( ( Sup @ ( image_5705468584675977158nt_int @ C2 @ A ) )
          = ( Sup @ ( image_5705468584675977158nt_int @ D @ B2 ) ) ) ) ) ).

% Sup.SUP_cong
thf(fact_41_Sup_OSUP__cong,axiom,
    ! [A: set_int,B2: set_int,C2: int > int,D: int > int,Sup: set_int > int] :
      ( ( A = B2 )
     => ( ! [X2: int] :
            ( ( member_int2 @ X2 @ B2 )
           => ( ( C2 @ X2 )
              = ( D @ X2 ) ) )
       => ( ( Sup @ ( image_int_int @ C2 @ A ) )
          = ( Sup @ ( image_int_int @ D @ B2 ) ) ) ) ) ).

% Sup.SUP_cong
thf(fact_42_Sup_OSUP__cong,axiom,
    ! [A: set_li3855193629254082847nt_int,B2: set_li3855193629254082847nt_int,C2: list_P5707943133018811711nt_int > set_Pr958786334691620121nt_int,D: list_P5707943133018811711nt_int > set_Pr958786334691620121nt_int,Sup: set_se6260736226359567993nt_int > set_Pr958786334691620121nt_int] :
      ( ( A = B2 )
     => ( ! [X2: list_P5707943133018811711nt_int] :
            ( ( member2764346250752101224nt_int @ X2 @ B2 )
           => ( ( C2 @ X2 )
              = ( D @ X2 ) ) )
       => ( ( Sup @ ( image_689400715899363487nt_int @ C2 @ A ) )
          = ( Sup @ ( image_689400715899363487nt_int @ D @ B2 ) ) ) ) ) ).

% Sup.SUP_cong
thf(fact_43_Sup_OSUP__cong,axiom,
    ! [A: set_nat,B2: set_nat,C2: nat > product_prod_int_int,D: nat > product_prod_int_int,Sup: set_Pr958786334691620121nt_int > product_prod_int_int] :
      ( ( A = B2 )
     => ( ! [X2: nat] :
            ( ( member_nat2 @ X2 @ B2 )
           => ( ( C2 @ X2 )
              = ( D @ X2 ) ) )
       => ( ( Sup @ ( image_2667626500211843362nt_int @ C2 @ A ) )
          = ( Sup @ ( image_2667626500211843362nt_int @ D @ B2 ) ) ) ) ) ).

% Sup.SUP_cong
thf(fact_44_Sup_OSUP__cong,axiom,
    ! [A: set_nat,B2: set_nat,C2: nat > nat,D: nat > nat,Sup: set_nat > nat] :
      ( ( A = B2 )
     => ( ! [X2: nat] :
            ( ( member_nat2 @ X2 @ B2 )
           => ( ( C2 @ X2 )
              = ( D @ X2 ) ) )
       => ( ( Sup @ ( image_nat_nat @ C2 @ A ) )
          = ( Sup @ ( image_nat_nat @ D @ B2 ) ) ) ) ) ).

% Sup.SUP_cong
thf(fact_45_Sup_OSUP__cong,axiom,
    ! [A: set_Pr958786334691620121nt_int,B2: set_Pr958786334691620121nt_int,C2: product_prod_int_int > int,D: product_prod_int_int > int,Sup: set_int > int] :
      ( ( A = B2 )
     => ( ! [X2: product_prod_int_int] :
            ( ( member5262025264175285858nt_int @ X2 @ B2 )
           => ( ( C2 @ X2 )
              = ( D @ X2 ) ) )
       => ( ( Sup @ ( image_5042161079198086560nt_int @ C2 @ A ) )
          = ( Sup @ ( image_5042161079198086560nt_int @ D @ B2 ) ) ) ) ) ).

% Sup.SUP_cong
thf(fact_46_imageI,axiom,
    ! [X: int,A: set_int,F: int > int] :
      ( ( member_int2 @ X @ A )
     => ( member_int2 @ ( F @ X ) @ ( image_int_int @ F @ A ) ) ) ).

% imageI
thf(fact_47_imageI,axiom,
    ! [X: product_prod_int_int,A: set_Pr958786334691620121nt_int,F: product_prod_int_int > int] :
      ( ( member5262025264175285858nt_int @ X @ A )
     => ( member_int2 @ ( F @ X ) @ ( image_5042161079198086560nt_int @ F @ A ) ) ) ).

% imageI
thf(fact_48_imageI,axiom,
    ! [X: int,A: set_int,F: int > product_prod_int_int] :
      ( ( member_int2 @ X @ A )
     => ( member5262025264175285858nt_int @ ( F @ X ) @ ( image_5705468584675977158nt_int @ F @ A ) ) ) ).

% imageI
thf(fact_49_imageI,axiom,
    ! [X: product_prod_int_int,A: set_Pr958786334691620121nt_int,F: product_prod_int_int > product_prod_int_int] :
      ( ( member5262025264175285858nt_int @ X @ A )
     => ( member5262025264175285858nt_int @ ( F @ X ) @ ( image_2653370878348428101nt_int @ F @ A ) ) ) ).

% imageI
thf(fact_50_imageI,axiom,
    ! [X: int,A: set_int,F: int > nat] :
      ( ( member_int2 @ X @ A )
     => ( member_nat2 @ ( F @ X ) @ ( image_int_nat @ F @ A ) ) ) ).

% imageI
thf(fact_51_imageI,axiom,
    ! [X: nat,A: set_nat,F: nat > int] :
      ( ( member_nat2 @ X @ A )
     => ( member_int2 @ ( F @ X ) @ ( image_nat_int @ F @ A ) ) ) ).

% imageI
thf(fact_52_imageI,axiom,
    ! [X: nat,A: set_nat,F: nat > nat] :
      ( ( member_nat2 @ X @ A )
     => ( member_nat2 @ ( F @ X ) @ ( image_nat_nat @ F @ A ) ) ) ).

% imageI
thf(fact_53_imageI,axiom,
    ! [X: int,A: set_int,F: int > list_int] :
      ( ( member_int2 @ X @ A )
     => ( member_list_int2 @ ( F @ X ) @ ( image_int_list_int @ F @ A ) ) ) ).

% imageI
thf(fact_54_imageI,axiom,
    ! [X: list_int,A: set_list_int,F: list_int > int] :
      ( ( member_list_int2 @ X @ A )
     => ( member_int2 @ ( F @ X ) @ ( image_list_int_int @ F @ A ) ) ) ).

% imageI
thf(fact_55_imageI,axiom,
    ! [X: list_int,A: set_list_int,F: list_int > nat] :
      ( ( member_list_int2 @ X @ A )
     => ( member_nat2 @ ( F @ X ) @ ( image_list_int_nat @ F @ A ) ) ) ).

% imageI
thf(fact_56_image__iff,axiom,
    ! [Z: product_prod_int_int,F: nat > product_prod_int_int,A: set_nat] :
      ( ( member5262025264175285858nt_int @ Z @ ( image_2667626500211843362nt_int @ F @ A ) )
      = ( ? [X3: nat] :
            ( ( member_nat2 @ X3 @ A )
            & ( Z
              = ( F @ X3 ) ) ) ) ) ).

% image_iff
thf(fact_57_image__iff,axiom,
    ! [Z: product_prod_int_int,F: int > product_prod_int_int,A: set_int] :
      ( ( member5262025264175285858nt_int @ Z @ ( image_5705468584675977158nt_int @ F @ A ) )
      = ( ? [X3: int] :
            ( ( member_int2 @ X3 @ A )
            & ( Z
              = ( F @ X3 ) ) ) ) ) ).

% image_iff
thf(fact_58_image__iff,axiom,
    ! [Z: int,F: int > int,A: set_int] :
      ( ( member_int2 @ Z @ ( image_int_int @ F @ A ) )
      = ( ? [X3: int] :
            ( ( member_int2 @ X3 @ A )
            & ( Z
              = ( F @ X3 ) ) ) ) ) ).

% image_iff
thf(fact_59_image__iff,axiom,
    ! [Z: set_Pr958786334691620121nt_int,F: list_P5707943133018811711nt_int > set_Pr958786334691620121nt_int,A: set_li3855193629254082847nt_int] :
      ( ( member2340774599025711042nt_int @ Z @ ( image_689400715899363487nt_int @ F @ A ) )
      = ( ? [X3: list_P5707943133018811711nt_int] :
            ( ( member2764346250752101224nt_int @ X3 @ A )
            & ( Z
              = ( F @ X3 ) ) ) ) ) ).

% image_iff
thf(fact_60_image__iff,axiom,
    ! [Z: nat,F: nat > nat,A: set_nat] :
      ( ( member_nat2 @ Z @ ( image_nat_nat @ F @ A ) )
      = ( ? [X3: nat] :
            ( ( member_nat2 @ X3 @ A )
            & ( Z
              = ( F @ X3 ) ) ) ) ) ).

% image_iff
thf(fact_61_image__iff,axiom,
    ! [Z: int,F: product_prod_int_int > int,A: set_Pr958786334691620121nt_int] :
      ( ( member_int2 @ Z @ ( image_5042161079198086560nt_int @ F @ A ) )
      = ( ? [X3: product_prod_int_int] :
            ( ( member5262025264175285858nt_int @ X3 @ A )
            & ( Z
              = ( F @ X3 ) ) ) ) ) ).

% image_iff
thf(fact_62_add__right__imp__eq,axiom,
    ! [B: int,A2: int,C: int] :
      ( ( ( plus_plus_int @ B @ A2 )
        = ( plus_plus_int @ C @ A2 ) )
     => ( B = C ) ) ).

% add_right_imp_eq
thf(fact_63_add__right__imp__eq,axiom,
    ! [B: nat,A2: nat,C: nat] :
      ( ( ( plus_plus_nat @ B @ A2 )
        = ( plus_plus_nat @ C @ A2 ) )
     => ( B = C ) ) ).

% add_right_imp_eq
thf(fact_64_add__left__imp__eq,axiom,
    ! [A2: int,B: int,C: int] :
      ( ( ( plus_plus_int @ A2 @ B )
        = ( plus_plus_int @ A2 @ C ) )
     => ( B = C ) ) ).

% add_left_imp_eq
thf(fact_65_add__left__imp__eq,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( ( ( plus_plus_nat @ A2 @ B )
        = ( plus_plus_nat @ A2 @ C ) )
     => ( B = C ) ) ).

% add_left_imp_eq
thf(fact_66_add_Oleft__commute,axiom,
    ! [B: int,A2: int,C: int] :
      ( ( plus_plus_int @ B @ ( plus_plus_int @ A2 @ C ) )
      = ( plus_plus_int @ A2 @ ( plus_plus_int @ B @ C ) ) ) ).

% add.left_commute
thf(fact_67_add_Oleft__commute,axiom,
    ! [B: nat,A2: nat,C: nat] :
      ( ( plus_plus_nat @ B @ ( plus_plus_nat @ A2 @ C ) )
      = ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B @ C ) ) ) ).

% add.left_commute
thf(fact_68_add_Ocommute,axiom,
    ( plus_plus_int
    = ( ^ [A3: int,B3: int] : ( plus_plus_int @ B3 @ A3 ) ) ) ).

% add.commute
thf(fact_69_add_Ocommute,axiom,
    ( plus_plus_nat
    = ( ^ [A3: nat,B3: nat] : ( plus_plus_nat @ B3 @ A3 ) ) ) ).

% add.commute
thf(fact_70_add_Oright__cancel,axiom,
    ! [B: int,A2: int,C: int] :
      ( ( ( plus_plus_int @ B @ A2 )
        = ( plus_plus_int @ C @ A2 ) )
      = ( B = C ) ) ).

% add.right_cancel
thf(fact_71_add_Oleft__cancel,axiom,
    ! [A2: int,B: int,C: int] :
      ( ( ( plus_plus_int @ A2 @ B )
        = ( plus_plus_int @ A2 @ C ) )
      = ( B = C ) ) ).

% add.left_cancel
thf(fact_72_add_Oassoc,axiom,
    ! [A2: int,B: int,C: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ A2 @ B ) @ C )
      = ( plus_plus_int @ A2 @ ( plus_plus_int @ B @ C ) ) ) ).

% add.assoc
thf(fact_73_add_Oassoc,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ A2 @ B ) @ C )
      = ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B @ C ) ) ) ).

% add.assoc
thf(fact_74_group__cancel_Oadd2,axiom,
    ! [B2: int,K: int,B: int,A2: int] :
      ( ( B2
        = ( plus_plus_int @ K @ B ) )
     => ( ( plus_plus_int @ A2 @ B2 )
        = ( plus_plus_int @ K @ ( plus_plus_int @ A2 @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_75_group__cancel_Oadd2,axiom,
    ! [B2: nat,K: nat,B: nat,A2: nat] :
      ( ( B2
        = ( plus_plus_nat @ K @ B ) )
     => ( ( plus_plus_nat @ A2 @ B2 )
        = ( plus_plus_nat @ K @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_76_group__cancel_Oadd1,axiom,
    ! [A: int,K: int,A2: int,B: int] :
      ( ( A
        = ( plus_plus_int @ K @ A2 ) )
     => ( ( plus_plus_int @ A @ B )
        = ( plus_plus_int @ K @ ( plus_plus_int @ A2 @ B ) ) ) ) ).

% group_cancel.add1
thf(fact_77_group__cancel_Oadd1,axiom,
    ! [A: nat,K: nat,A2: nat,B: nat] :
      ( ( A
        = ( plus_plus_nat @ K @ A2 ) )
     => ( ( plus_plus_nat @ A @ B )
        = ( plus_plus_nat @ K @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).

% group_cancel.add1
thf(fact_78_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( ( plus_plus_int @ I @ K )
        = ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_79_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( ( plus_plus_nat @ I @ K )
        = ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_80_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A2: int,B: int,C: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ A2 @ B ) @ C )
      = ( plus_plus_int @ A2 @ ( plus_plus_int @ B @ C ) ) ) ).

% ab_semigroup_add_class.add_ac(1)
thf(fact_81_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ A2 @ B ) @ C )
      = ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B @ C ) ) ) ).

% ab_semigroup_add_class.add_ac(1)
thf(fact_82_rev__image__eqI,axiom,
    ! [X: int,A: set_int,B: int,F: int > int] :
      ( ( member_int2 @ X @ A )
     => ( ( B
          = ( F @ X ) )
       => ( member_int2 @ B @ ( image_int_int @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_83_rev__image__eqI,axiom,
    ! [X: product_prod_int_int,A: set_Pr958786334691620121nt_int,B: int,F: product_prod_int_int > int] :
      ( ( member5262025264175285858nt_int @ X @ A )
     => ( ( B
          = ( F @ X ) )
       => ( member_int2 @ B @ ( image_5042161079198086560nt_int @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_84_rev__image__eqI,axiom,
    ! [X: int,A: set_int,B: product_prod_int_int,F: int > product_prod_int_int] :
      ( ( member_int2 @ X @ A )
     => ( ( B
          = ( F @ X ) )
       => ( member5262025264175285858nt_int @ B @ ( image_5705468584675977158nt_int @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_85_rev__image__eqI,axiom,
    ! [X: product_prod_int_int,A: set_Pr958786334691620121nt_int,B: product_prod_int_int,F: product_prod_int_int > product_prod_int_int] :
      ( ( member5262025264175285858nt_int @ X @ A )
     => ( ( B
          = ( F @ X ) )
       => ( member5262025264175285858nt_int @ B @ ( image_2653370878348428101nt_int @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_86_rev__image__eqI,axiom,
    ! [X: int,A: set_int,B: nat,F: int > nat] :
      ( ( member_int2 @ X @ A )
     => ( ( B
          = ( F @ X ) )
       => ( member_nat2 @ B @ ( image_int_nat @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_87_rev__image__eqI,axiom,
    ! [X: nat,A: set_nat,B: int,F: nat > int] :
      ( ( member_nat2 @ X @ A )
     => ( ( B
          = ( F @ X ) )
       => ( member_int2 @ B @ ( image_nat_int @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_88_rev__image__eqI,axiom,
    ! [X: nat,A: set_nat,B: nat,F: nat > nat] :
      ( ( member_nat2 @ X @ A )
     => ( ( B
          = ( F @ X ) )
       => ( member_nat2 @ B @ ( image_nat_nat @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_89_rev__image__eqI,axiom,
    ! [X: int,A: set_int,B: list_int,F: int > list_int] :
      ( ( member_int2 @ X @ A )
     => ( ( B
          = ( F @ X ) )
       => ( member_list_int2 @ B @ ( image_int_list_int @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_90_rev__image__eqI,axiom,
    ! [X: list_int,A: set_list_int,B: int,F: list_int > int] :
      ( ( member_list_int2 @ X @ A )
     => ( ( B
          = ( F @ X ) )
       => ( member_int2 @ B @ ( image_list_int_int @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_91_rev__image__eqI,axiom,
    ! [X: list_int,A: set_list_int,B: nat,F: list_int > nat] :
      ( ( member_list_int2 @ X @ A )
     => ( ( B
          = ( F @ X ) )
       => ( member_nat2 @ B @ ( image_list_int_nat @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_92_ball__imageD,axiom,
    ! [F: list_P5707943133018811711nt_int > set_Pr958786334691620121nt_int,A: set_li3855193629254082847nt_int,P: set_Pr958786334691620121nt_int > $o] :
      ( ! [X2: set_Pr958786334691620121nt_int] :
          ( ( member2340774599025711042nt_int @ X2 @ ( image_689400715899363487nt_int @ F @ A ) )
         => ( P @ X2 ) )
     => ! [X4: list_P5707943133018811711nt_int] :
          ( ( member2764346250752101224nt_int @ X4 @ A )
         => ( P @ ( F @ X4 ) ) ) ) ).

% ball_imageD
thf(fact_93_ball__imageD,axiom,
    ! [F: nat > product_prod_int_int,A: set_nat,P: product_prod_int_int > $o] :
      ( ! [X2: product_prod_int_int] :
          ( ( member5262025264175285858nt_int @ X2 @ ( image_2667626500211843362nt_int @ F @ A ) )
         => ( P @ X2 ) )
     => ! [X4: nat] :
          ( ( member_nat2 @ X4 @ A )
         => ( P @ ( F @ X4 ) ) ) ) ).

% ball_imageD
thf(fact_94_ball__imageD,axiom,
    ! [F: nat > nat,A: set_nat,P: nat > $o] :
      ( ! [X2: nat] :
          ( ( member_nat2 @ X2 @ ( image_nat_nat @ F @ A ) )
         => ( P @ X2 ) )
     => ! [X4: nat] :
          ( ( member_nat2 @ X4 @ A )
         => ( P @ ( F @ X4 ) ) ) ) ).

% ball_imageD
thf(fact_95_ball__imageD,axiom,
    ! [F: int > product_prod_int_int,A: set_int,P: product_prod_int_int > $o] :
      ( ! [X2: product_prod_int_int] :
          ( ( member5262025264175285858nt_int @ X2 @ ( image_5705468584675977158nt_int @ F @ A ) )
         => ( P @ X2 ) )
     => ! [X4: int] :
          ( ( member_int2 @ X4 @ A )
         => ( P @ ( F @ X4 ) ) ) ) ).

% ball_imageD
thf(fact_96_ball__imageD,axiom,
    ! [F: int > int,A: set_int,P: int > $o] :
      ( ! [X2: int] :
          ( ( member_int2 @ X2 @ ( image_int_int @ F @ A ) )
         => ( P @ X2 ) )
     => ! [X4: int] :
          ( ( member_int2 @ X4 @ A )
         => ( P @ ( F @ X4 ) ) ) ) ).

% ball_imageD
thf(fact_97_ball__imageD,axiom,
    ! [F: product_prod_int_int > int,A: set_Pr958786334691620121nt_int,P: int > $o] :
      ( ! [X2: int] :
          ( ( member_int2 @ X2 @ ( image_5042161079198086560nt_int @ F @ A ) )
         => ( P @ X2 ) )
     => ! [X4: product_prod_int_int] :
          ( ( member5262025264175285858nt_int @ X4 @ A )
         => ( P @ ( F @ X4 ) ) ) ) ).

% ball_imageD
thf(fact_98_image__cong,axiom,
    ! [M: set_int,N2: set_int,F: int > product_prod_int_int,G: int > product_prod_int_int] :
      ( ( M = N2 )
     => ( ! [X2: int] :
            ( ( member_int2 @ X2 @ N2 )
           => ( ( F @ X2 )
              = ( G @ X2 ) ) )
       => ( ( image_5705468584675977158nt_int @ F @ M )
          = ( image_5705468584675977158nt_int @ G @ N2 ) ) ) ) ).

% image_cong
thf(fact_99_image__cong,axiom,
    ! [M: set_int,N2: set_int,F: int > int,G: int > int] :
      ( ( M = N2 )
     => ( ! [X2: int] :
            ( ( member_int2 @ X2 @ N2 )
           => ( ( F @ X2 )
              = ( G @ X2 ) ) )
       => ( ( image_int_int @ F @ M )
          = ( image_int_int @ G @ N2 ) ) ) ) ).

% image_cong
thf(fact_100_image__cong,axiom,
    ! [M: set_li3855193629254082847nt_int,N2: set_li3855193629254082847nt_int,F: list_P5707943133018811711nt_int > set_Pr958786334691620121nt_int,G: list_P5707943133018811711nt_int > set_Pr958786334691620121nt_int] :
      ( ( M = N2 )
     => ( ! [X2: list_P5707943133018811711nt_int] :
            ( ( member2764346250752101224nt_int @ X2 @ N2 )
           => ( ( F @ X2 )
              = ( G @ X2 ) ) )
       => ( ( image_689400715899363487nt_int @ F @ M )
          = ( image_689400715899363487nt_int @ G @ N2 ) ) ) ) ).

% image_cong
thf(fact_101_image__cong,axiom,
    ! [M: set_nat,N2: set_nat,F: nat > product_prod_int_int,G: nat > product_prod_int_int] :
      ( ( M = N2 )
     => ( ! [X2: nat] :
            ( ( member_nat2 @ X2 @ N2 )
           => ( ( F @ X2 )
              = ( G @ X2 ) ) )
       => ( ( image_2667626500211843362nt_int @ F @ M )
          = ( image_2667626500211843362nt_int @ G @ N2 ) ) ) ) ).

% image_cong
thf(fact_102_image__cong,axiom,
    ! [M: set_nat,N2: set_nat,F: nat > nat,G: nat > nat] :
      ( ( M = N2 )
     => ( ! [X2: nat] :
            ( ( member_nat2 @ X2 @ N2 )
           => ( ( F @ X2 )
              = ( G @ X2 ) ) )
       => ( ( image_nat_nat @ F @ M )
          = ( image_nat_nat @ G @ N2 ) ) ) ) ).

% image_cong
thf(fact_103_image__cong,axiom,
    ! [M: set_Pr958786334691620121nt_int,N2: set_Pr958786334691620121nt_int,F: product_prod_int_int > int,G: product_prod_int_int > int] :
      ( ( M = N2 )
     => ( ! [X2: product_prod_int_int] :
            ( ( member5262025264175285858nt_int @ X2 @ N2 )
           => ( ( F @ X2 )
              = ( G @ X2 ) ) )
       => ( ( image_5042161079198086560nt_int @ F @ M )
          = ( image_5042161079198086560nt_int @ G @ N2 ) ) ) ) ).

% image_cong
thf(fact_104_bex__imageD,axiom,
    ! [F: list_P5707943133018811711nt_int > set_Pr958786334691620121nt_int,A: set_li3855193629254082847nt_int,P: set_Pr958786334691620121nt_int > $o] :
      ( ? [X4: set_Pr958786334691620121nt_int] :
          ( ( member2340774599025711042nt_int @ X4 @ ( image_689400715899363487nt_int @ F @ A ) )
          & ( P @ X4 ) )
     => ? [X2: list_P5707943133018811711nt_int] :
          ( ( member2764346250752101224nt_int @ X2 @ A )
          & ( P @ ( F @ X2 ) ) ) ) ).

% bex_imageD
thf(fact_105_bex__imageD,axiom,
    ! [F: nat > product_prod_int_int,A: set_nat,P: product_prod_int_int > $o] :
      ( ? [X4: product_prod_int_int] :
          ( ( member5262025264175285858nt_int @ X4 @ ( image_2667626500211843362nt_int @ F @ A ) )
          & ( P @ X4 ) )
     => ? [X2: nat] :
          ( ( member_nat2 @ X2 @ A )
          & ( P @ ( F @ X2 ) ) ) ) ).

% bex_imageD
thf(fact_106_bex__imageD,axiom,
    ! [F: nat > nat,A: set_nat,P: nat > $o] :
      ( ? [X4: nat] :
          ( ( member_nat2 @ X4 @ ( image_nat_nat @ F @ A ) )
          & ( P @ X4 ) )
     => ? [X2: nat] :
          ( ( member_nat2 @ X2 @ A )
          & ( P @ ( F @ X2 ) ) ) ) ).

% bex_imageD
thf(fact_107_bex__imageD,axiom,
    ! [F: int > product_prod_int_int,A: set_int,P: product_prod_int_int > $o] :
      ( ? [X4: product_prod_int_int] :
          ( ( member5262025264175285858nt_int @ X4 @ ( image_5705468584675977158nt_int @ F @ A ) )
          & ( P @ X4 ) )
     => ? [X2: int] :
          ( ( member_int2 @ X2 @ A )
          & ( P @ ( F @ X2 ) ) ) ) ).

% bex_imageD
thf(fact_108_bex__imageD,axiom,
    ! [F: int > int,A: set_int,P: int > $o] :
      ( ? [X4: int] :
          ( ( member_int2 @ X4 @ ( image_int_int @ F @ A ) )
          & ( P @ X4 ) )
     => ? [X2: int] :
          ( ( member_int2 @ X2 @ A )
          & ( P @ ( F @ X2 ) ) ) ) ).

% bex_imageD
thf(fact_109_bex__imageD,axiom,
    ! [F: product_prod_int_int > int,A: set_Pr958786334691620121nt_int,P: int > $o] :
      ( ? [X4: int] :
          ( ( member_int2 @ X4 @ ( image_5042161079198086560nt_int @ F @ A ) )
          & ( P @ X4 ) )
     => ? [X2: product_prod_int_int] :
          ( ( member5262025264175285858nt_int @ X2 @ A )
          & ( P @ ( F @ X2 ) ) ) ) ).

% bex_imageD
thf(fact_110_prod__eq__iff,axiom,
    ( ( ^ [Y2: product_prod_nat_nat,Z2: product_prod_nat_nat] : ( Y2 = Z2 ) )
    = ( ^ [S: product_prod_nat_nat,T: product_prod_nat_nat] :
          ( ( ( product_fst_nat_nat @ S )
            = ( product_fst_nat_nat @ T ) )
          & ( ( product_snd_nat_nat @ S )
            = ( product_snd_nat_nat @ T ) ) ) ) ) ).

% prod_eq_iff
thf(fact_111_prod__eq__iff,axiom,
    ( ( ^ [Y2: produc1219242969750017639nt_int,Z2: produc1219242969750017639nt_int] : ( Y2 = Z2 ) )
    = ( ^ [S: produc1219242969750017639nt_int,T: produc1219242969750017639nt_int] :
          ( ( ( produc698254169746827971nt_int @ S )
            = ( produc698254169746827971nt_int @ T ) )
          & ( ( produc3892743399831173125nt_int @ S )
            = ( produc3892743399831173125nt_int @ T ) ) ) ) ) ).

% prod_eq_iff
thf(fact_112_prod__eq__iff,axiom,
    ( ( ^ [Y2: produc6945250483304103390nt_int,Z2: produc6945250483304103390nt_int] : ( Y2 = Z2 ) )
    = ( ^ [S: produc6945250483304103390nt_int,T: produc6945250483304103390nt_int] :
          ( ( ( produc4803745454372187940nt_int @ S )
            = ( produc4803745454372187940nt_int @ T ) )
          & ( ( produc3248548876605496418nt_int @ S )
            = ( produc3248548876605496418nt_int @ T ) ) ) ) ) ).

% prod_eq_iff
thf(fact_113_prod__eq__iff,axiom,
    ( ( ^ [Y2: product_prod_int_int,Z2: product_prod_int_int] : ( Y2 = Z2 ) )
    = ( ^ [S: product_prod_int_int,T: product_prod_int_int] :
          ( ( ( product_fst_int_int @ S )
            = ( product_fst_int_int @ T ) )
          & ( ( product_snd_int_int @ S )
            = ( product_snd_int_int @ T ) ) ) ) ) ).

% prod_eq_iff
thf(fact_114_exE__realizer_H,axiom,
    ! [P: nat > nat > $o,P2: product_prod_nat_nat] :
      ( ( P @ ( product_snd_nat_nat @ P2 ) @ ( product_fst_nat_nat @ P2 ) )
     => ~ ! [X2: nat,Y3: nat] :
            ~ ( P @ Y3 @ X2 ) ) ).

% exE_realizer'
thf(fact_115_exE__realizer_H,axiom,
    ! [P: product_prod_int_int > product_prod_int_int > $o,P2: produc1219242969750017639nt_int] :
      ( ( P @ ( produc3892743399831173125nt_int @ P2 ) @ ( produc698254169746827971nt_int @ P2 ) )
     => ~ ! [X2: product_prod_int_int,Y3: product_prod_int_int] :
            ~ ( P @ Y3 @ X2 ) ) ).

% exE_realizer'
thf(fact_116_exE__realizer_H,axiom,
    ! [P: product_prod_int_int > nat > $o,P2: produc6945250483304103390nt_int] :
      ( ( P @ ( produc3248548876605496418nt_int @ P2 ) @ ( produc4803745454372187940nt_int @ P2 ) )
     => ~ ! [X2: nat,Y3: product_prod_int_int] :
            ~ ( P @ Y3 @ X2 ) ) ).

% exE_realizer'
thf(fact_117_exE__realizer_H,axiom,
    ! [P: int > int > $o,P2: product_prod_int_int] :
      ( ( P @ ( product_snd_int_int @ P2 ) @ ( product_fst_int_int @ P2 ) )
     => ~ ! [X2: int,Y3: int] :
            ~ ( P @ Y3 @ X2 ) ) ).

% exE_realizer'
thf(fact_118_prod__eqI,axiom,
    ! [P2: product_prod_nat_nat,Q: product_prod_nat_nat] :
      ( ( ( product_fst_nat_nat @ P2 )
        = ( product_fst_nat_nat @ Q ) )
     => ( ( ( product_snd_nat_nat @ P2 )
          = ( product_snd_nat_nat @ Q ) )
       => ( P2 = Q ) ) ) ).

% prod_eqI
thf(fact_119_prod__eqI,axiom,
    ! [P2: produc1219242969750017639nt_int,Q: produc1219242969750017639nt_int] :
      ( ( ( produc698254169746827971nt_int @ P2 )
        = ( produc698254169746827971nt_int @ Q ) )
     => ( ( ( produc3892743399831173125nt_int @ P2 )
          = ( produc3892743399831173125nt_int @ Q ) )
       => ( P2 = Q ) ) ) ).

% prod_eqI
thf(fact_120_prod__eqI,axiom,
    ! [P2: produc6945250483304103390nt_int,Q: produc6945250483304103390nt_int] :
      ( ( ( produc4803745454372187940nt_int @ P2 )
        = ( produc4803745454372187940nt_int @ Q ) )
     => ( ( ( produc3248548876605496418nt_int @ P2 )
          = ( produc3248548876605496418nt_int @ Q ) )
       => ( P2 = Q ) ) ) ).

% prod_eqI
thf(fact_121_prod__eqI,axiom,
    ! [P2: product_prod_int_int,Q: product_prod_int_int] :
      ( ( ( product_fst_int_int @ P2 )
        = ( product_fst_int_int @ Q ) )
     => ( ( ( product_snd_int_int @ P2 )
          = ( product_snd_int_int @ Q ) )
       => ( P2 = Q ) ) ) ).

% prod_eqI
thf(fact_122_prod_Oexpand,axiom,
    ! [Prod: product_prod_nat_nat,Prod2: product_prod_nat_nat] :
      ( ( ( ( product_fst_nat_nat @ Prod )
          = ( product_fst_nat_nat @ Prod2 ) )
        & ( ( product_snd_nat_nat @ Prod )
          = ( product_snd_nat_nat @ Prod2 ) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
thf(fact_123_prod_Oexpand,axiom,
    ! [Prod: produc1219242969750017639nt_int,Prod2: produc1219242969750017639nt_int] :
      ( ( ( ( produc698254169746827971nt_int @ Prod )
          = ( produc698254169746827971nt_int @ Prod2 ) )
        & ( ( produc3892743399831173125nt_int @ Prod )
          = ( produc3892743399831173125nt_int @ Prod2 ) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
thf(fact_124_prod_Oexpand,axiom,
    ! [Prod: produc6945250483304103390nt_int,Prod2: produc6945250483304103390nt_int] :
      ( ( ( ( produc4803745454372187940nt_int @ Prod )
          = ( produc4803745454372187940nt_int @ Prod2 ) )
        & ( ( produc3248548876605496418nt_int @ Prod )
          = ( produc3248548876605496418nt_int @ Prod2 ) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
thf(fact_125_prod_Oexpand,axiom,
    ! [Prod: product_prod_int_int,Prod2: product_prod_int_int] :
      ( ( ( ( product_fst_int_int @ Prod )
          = ( product_fst_int_int @ Prod2 ) )
        & ( ( product_snd_int_int @ Prod )
          = ( product_snd_int_int @ Prod2 ) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
thf(fact_126_insort__insert__key__triv,axiom,
    ! [F: product_prod_int_int > int,X: product_prod_int_int,Xs2: list_P5707943133018811711nt_int] :
      ( ( member_int2 @ ( F @ X ) @ ( image_5042161079198086560nt_int @ F @ ( set_Pr2470121279949933262nt_int @ Xs2 ) ) )
     => ( ( linord5950828073485037667nt_int @ F @ X @ Xs2 )
        = Xs2 ) ) ).

% insort_insert_key_triv
thf(fact_127_insort__insert__key__triv,axiom,
    ! [F: nat > int,X: nat,Xs2: list_nat] :
      ( ( member_int2 @ ( F @ X ) @ ( image_nat_int @ F @ ( set_nat2 @ Xs2 ) ) )
     => ( ( linord1919045884167398656at_int @ F @ X @ Xs2 )
        = Xs2 ) ) ).

% insort_insert_key_triv
thf(fact_128_insort__insert__key__triv,axiom,
    ! [F: nat > nat,X: nat,Xs2: list_nat] :
      ( ( member_nat2 @ ( F @ X ) @ ( image_nat_nat @ F @ ( set_nat2 @ Xs2 ) ) )
     => ( ( linord1921536354676448932at_nat @ F @ X @ Xs2 )
        = Xs2 ) ) ).

% insort_insert_key_triv
thf(fact_129_insort__insert__key__triv,axiom,
    ! [F: int > int,X: int,Xs2: list_int] :
      ( ( member_int2 @ ( F @ X ) @ ( image_int_int @ F @ ( set_int2 @ Xs2 ) ) )
     => ( ( linord2918399596068453212nt_int @ F @ X @ Xs2 )
        = Xs2 ) ) ).

% insort_insert_key_triv
thf(fact_130_insort__insert__key__triv,axiom,
    ! [F: int > nat,X: int,Xs2: list_int] :
      ( ( member_nat2 @ ( F @ X ) @ ( image_int_nat @ F @ ( set_int2 @ Xs2 ) ) )
     => ( ( linord2920890066577503488nt_nat @ F @ X @ Xs2 )
        = Xs2 ) ) ).

% insort_insert_key_triv
thf(fact_131_insort__insert__key__triv,axiom,
    ! [F: list_int > int,X: list_int,Xs2: list_list_int] :
      ( ( member_int2 @ ( F @ X ) @ ( image_list_int_int @ F @ ( set_list_int2 @ Xs2 ) ) )
     => ( ( linord5437656553537133292nt_int @ F @ X @ Xs2 )
        = Xs2 ) ) ).

% insort_insert_key_triv
thf(fact_132_insort__insert__key__triv,axiom,
    ! [F: list_int > nat,X: list_int,Xs2: list_list_int] :
      ( ( member_nat2 @ ( F @ X ) @ ( image_list_int_nat @ F @ ( set_list_int2 @ Xs2 ) ) )
     => ( ( linord5440147024046183568nt_nat @ F @ X @ Xs2 )
        = Xs2 ) ) ).

% insort_insert_key_triv
thf(fact_133_insort__insert__key__triv,axiom,
    ! [F: product_prod_int_int > nat,X: product_prod_int_int,Xs2: list_P5707943133018811711nt_int] :
      ( ( member_nat2 @ ( F @ X ) @ ( image_5044651549707136836nt_nat @ F @ ( set_Pr2470121279949933262nt_int @ Xs2 ) ) )
     => ( ( linord5953318543994087943nt_nat @ F @ X @ Xs2 )
        = Xs2 ) ) ).

% insort_insert_key_triv
thf(fact_134_insort__insert__key__triv,axiom,
    ! [F: list_P5707943133018811711nt_int > int,X: list_P5707943133018811711nt_int,Xs2: list_l1670014477004246597nt_int] :
      ( ( member_int2 @ ( F @ X ) @ ( image_589057387511608870nt_int @ F @ ( set_li2659200638379878868nt_int @ Xs2 ) ) )
     => ( ( linord4396723656979755881nt_int @ F @ X @ Xs2 )
        = Xs2 ) ) ).

% insort_insert_key_triv
thf(fact_135_insort__insert__key__triv,axiom,
    ! [F: list_P5707943133018811711nt_int > nat,X: list_P5707943133018811711nt_int,Xs2: list_l1670014477004246597nt_int] :
      ( ( member_nat2 @ ( F @ X ) @ ( image_591547858020659146nt_nat @ F @ ( set_li2659200638379878868nt_int @ Xs2 ) ) )
     => ( ( linord4399214127488806157nt_nat @ F @ X @ Xs2 )
        = Xs2 ) ) ).

% insort_insert_key_triv
thf(fact_136_in__set__member,axiom,
    ! [X: set_Pr958786334691620121nt_int,Xs2: list_s8839374986383574687nt_int] :
      ( ( member2340774599025711042nt_int @ X @ ( set_se4746441305218329646nt_int @ Xs2 ) )
      = ( member4349927117382914868nt_int @ Xs2 @ X ) ) ).

% in_set_member
thf(fact_137_in__set__member,axiom,
    ! [X: produc759720530913461378nt_int,Xs2: list_P8440714079264627474nt_int] :
      ( ( member4957952664650131097nt_int @ X @ ( set_Pr222267339979123501nt_int @ Xs2 ) )
      = ( member7896400568998120615nt_int @ Xs2 @ X ) ) ).

% in_set_member
thf(fact_138_in__set__member,axiom,
    ! [X: list_P5707943133018811711nt_int,Xs2: list_l1670014477004246597nt_int] :
      ( ( member2764346250752101224nt_int @ X @ ( set_li2659200638379878868nt_int @ Xs2 ) )
      = ( member7098935381885431258nt_int @ Xs2 @ X ) ) ).

% in_set_member
thf(fact_139_in__set__member,axiom,
    ! [X: list_int,Xs2: list_list_int] :
      ( ( member_list_int2 @ X @ ( set_list_int2 @ Xs2 ) )
      = ( member_list_int @ Xs2 @ X ) ) ).

% in_set_member
thf(fact_140_in__set__member,axiom,
    ! [X: nat,Xs2: list_nat] :
      ( ( member_nat2 @ X @ ( set_nat2 @ Xs2 ) )
      = ( member_nat @ Xs2 @ X ) ) ).

% in_set_member
thf(fact_141_in__set__member,axiom,
    ! [X: int,Xs2: list_int] :
      ( ( member_int2 @ X @ ( set_int2 @ Xs2 ) )
      = ( member_int @ Xs2 @ X ) ) ).

% in_set_member
thf(fact_142_in__set__member,axiom,
    ! [X: product_prod_int_int,Xs2: list_P5707943133018811711nt_int] :
      ( ( member5262025264175285858nt_int @ X @ ( set_Pr2470121279949933262nt_int @ Xs2 ) )
      = ( member2925713097805433300nt_int @ Xs2 @ X ) ) ).

% in_set_member
thf(fact_143_is__num__normalize_I1_J,axiom,
    ! [A2: int,B: int,C: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ A2 @ B ) @ C )
      = ( plus_plus_int @ A2 @ ( plus_plus_int @ B @ C ) ) ) ).

% is_num_normalize(1)
thf(fact_144_gen__length__def,axiom,
    ( gen_length_int
    = ( ^ [N3: nat,Xs3: list_int] : ( plus_plus_nat @ N3 @ ( size_size_list_int @ Xs3 ) ) ) ) ).

% gen_length_def
thf(fact_145_gen__length__def,axiom,
    ( gen_le4899709912081684090nt_int
    = ( ^ [N3: nat,Xs3: list_P2336717926344734829nt_int] : ( plus_plus_nat @ N3 @ ( size_s6770063216428074713nt_int @ Xs3 ) ) ) ) ).

% gen_length_def
thf(fact_146_gen__length__def,axiom,
    ( gen_le5940319154913792082nt_int
    = ( ^ [N3: nat,Xs3: list_l1670014477004246597nt_int] : ( plus_plus_nat @ N3 @ ( size_s2969076144586574001nt_int @ Xs3 ) ) ) ) ).

% gen_length_def
thf(fact_147_gen__length__def,axiom,
    ( gen_le2044925581308924299nt_int
    = ( ^ [N3: nat,Xs3: list_P314425023053850222nt_int] : ( plus_plus_nat @ N3 @ ( size_s2001693051472072450nt_int @ Xs3 ) ) ) ) ).

% gen_length_def
thf(fact_148_gen__length__def,axiom,
    ( gen_le8428774395332151372nt_int
    = ( ^ [N3: nat,Xs3: list_P5707943133018811711nt_int] : ( plus_plus_nat @ N3 @ ( size_s5157815400016825771nt_int @ Xs3 ) ) ) ) ).

% gen_length_def
thf(fact_149_in__set__insert,axiom,
    ! [X: set_Pr958786334691620121nt_int,Xs2: list_s8839374986383574687nt_int] :
      ( ( member2340774599025711042nt_int @ X @ ( set_se4746441305218329646nt_int @ Xs2 ) )
     => ( ( insert6853257344505722933nt_int @ X @ Xs2 )
        = Xs2 ) ) ).

% in_set_insert
thf(fact_150_in__set__insert,axiom,
    ! [X: produc759720530913461378nt_int,Xs2: list_P8440714079264627474nt_int] :
      ( ( member4957952664650131097nt_int @ X @ ( set_Pr222267339979123501nt_int @ Xs2 ) )
     => ( ( insert110774618870972006nt_int @ X @ Xs2 )
        = Xs2 ) ) ).

% in_set_insert
thf(fact_151_in__set__insert,axiom,
    ! [X: list_P5707943133018811711nt_int,Xs2: list_l1670014477004246597nt_int] :
      ( ( member2764346250752101224nt_int @ X @ ( set_li2659200638379878868nt_int @ Xs2 ) )
     => ( ( insert2185137395918070363nt_int @ X @ Xs2 )
        = Xs2 ) ) ).

% in_set_insert
thf(fact_152_in__set__insert,axiom,
    ! [X: list_int,Xs2: list_list_int] :
      ( ( member_list_int2 @ X @ ( set_list_int2 @ Xs2 ) )
     => ( ( insert_list_int @ X @ Xs2 )
        = Xs2 ) ) ).

% in_set_insert
thf(fact_153_in__set__insert,axiom,
    ! [X: nat,Xs2: list_nat] :
      ( ( member_nat2 @ X @ ( set_nat2 @ Xs2 ) )
     => ( ( insert_nat @ X @ Xs2 )
        = Xs2 ) ) ).

% in_set_insert
thf(fact_154_in__set__insert,axiom,
    ! [X: int,Xs2: list_int] :
      ( ( member_int2 @ X @ ( set_int2 @ Xs2 ) )
     => ( ( insert_int @ X @ Xs2 )
        = Xs2 ) ) ).

% in_set_insert
thf(fact_155_in__set__insert,axiom,
    ! [X: product_prod_int_int,Xs2: list_P5707943133018811711nt_int] :
      ( ( member5262025264175285858nt_int @ X @ ( set_Pr2470121279949933262nt_int @ Xs2 ) )
     => ( ( insert5765537519290168021nt_int @ X @ Xs2 )
        = Xs2 ) ) ).

% in_set_insert
thf(fact_156_list__ex1__iff,axiom,
    ( list_e2144717942298772962nt_int
    = ( ^ [P3: set_Pr958786334691620121nt_int > $o,Xs3: list_s8839374986383574687nt_int] :
        ? [X3: set_Pr958786334691620121nt_int] :
          ( ( member2340774599025711042nt_int @ X3 @ ( set_se4746441305218329646nt_int @ Xs3 ) )
          & ( P3 @ X3 )
          & ! [Y4: set_Pr958786334691620121nt_int] :
              ( ( ( member2340774599025711042nt_int @ Y4 @ ( set_se4746441305218329646nt_int @ Xs3 ) )
                & ( P3 @ Y4 ) )
             => ( Y4 = X3 ) ) ) ) ) ).

% list_ex1_iff
thf(fact_157_list__ex1__iff,axiom,
    ( list_e2884510438846156921nt_int
    = ( ^ [P3: produc759720530913461378nt_int > $o,Xs3: list_P8440714079264627474nt_int] :
        ? [X3: produc759720530913461378nt_int] :
          ( ( member4957952664650131097nt_int @ X3 @ ( set_Pr222267339979123501nt_int @ Xs3 ) )
          & ( P3 @ X3 )
          & ! [Y4: produc759720530913461378nt_int] :
              ( ( ( member4957952664650131097nt_int @ Y4 @ ( set_Pr222267339979123501nt_int @ Xs3 ) )
                & ( P3 @ Y4 ) )
             => ( Y4 = X3 ) ) ) ) ) ).

% list_ex1_iff
thf(fact_158_list__ex1__iff,axiom,
    ( list_e3855813444280166792nt_int
    = ( ^ [P3: list_P5707943133018811711nt_int > $o,Xs3: list_l1670014477004246597nt_int] :
        ? [X3: list_P5707943133018811711nt_int] :
          ( ( member2764346250752101224nt_int @ X3 @ ( set_li2659200638379878868nt_int @ Xs3 ) )
          & ( P3 @ X3 )
          & ! [Y4: list_P5707943133018811711nt_int] :
              ( ( ( member2764346250752101224nt_int @ Y4 @ ( set_li2659200638379878868nt_int @ Xs3 ) )
                & ( P3 @ Y4 ) )
             => ( Y4 = X3 ) ) ) ) ) ).

% list_ex1_iff
thf(fact_159_list__ex1__iff,axiom,
    ( list_ex1_list_int
    = ( ^ [P3: list_int > $o,Xs3: list_list_int] :
        ? [X3: list_int] :
          ( ( member_list_int2 @ X3 @ ( set_list_int2 @ Xs3 ) )
          & ( P3 @ X3 )
          & ! [Y4: list_int] :
              ( ( ( member_list_int2 @ Y4 @ ( set_list_int2 @ Xs3 ) )
                & ( P3 @ Y4 ) )
             => ( Y4 = X3 ) ) ) ) ) ).

% list_ex1_iff
thf(fact_160_list__ex1__iff,axiom,
    ( list_ex1_nat
    = ( ^ [P3: nat > $o,Xs3: list_nat] :
        ? [X3: nat] :
          ( ( member_nat2 @ X3 @ ( set_nat2 @ Xs3 ) )
          & ( P3 @ X3 )
          & ! [Y4: nat] :
              ( ( ( member_nat2 @ Y4 @ ( set_nat2 @ Xs3 ) )
                & ( P3 @ Y4 ) )
             => ( Y4 = X3 ) ) ) ) ) ).

% list_ex1_iff
thf(fact_161_list__ex1__iff,axiom,
    ( list_ex1_int
    = ( ^ [P3: int > $o,Xs3: list_int] :
        ? [X3: int] :
          ( ( member_int2 @ X3 @ ( set_int2 @ Xs3 ) )
          & ( P3 @ X3 )
          & ! [Y4: int] :
              ( ( ( member_int2 @ Y4 @ ( set_int2 @ Xs3 ) )
                & ( P3 @ Y4 ) )
             => ( Y4 = X3 ) ) ) ) ) ).

% list_ex1_iff
thf(fact_162_list__ex1__iff,axiom,
    ( list_e5465588451548443778nt_int
    = ( ^ [P3: product_prod_int_int > $o,Xs3: list_P5707943133018811711nt_int] :
        ? [X3: product_prod_int_int] :
          ( ( member5262025264175285858nt_int @ X3 @ ( set_Pr2470121279949933262nt_int @ Xs3 ) )
          & ( P3 @ X3 )
          & ! [Y4: product_prod_int_int] :
              ( ( ( member5262025264175285858nt_int @ Y4 @ ( set_Pr2470121279949933262nt_int @ Xs3 ) )
                & ( P3 @ Y4 ) )
             => ( Y4 = X3 ) ) ) ) ) ).

% list_ex1_iff
thf(fact_163_image__add__0,axiom,
    ! [S2: set_nat] :
      ( ( image_nat_nat @ ( plus_plus_nat @ zero_zero_nat ) @ S2 )
      = S2 ) ).

% image_add_0
thf(fact_164_image__add__0,axiom,
    ! [S2: set_int] :
      ( ( image_int_int @ ( plus_plus_int @ zero_zero_int ) @ S2 )
      = S2 ) ).

% image_add_0
thf(fact_165_snd__swap,axiom,
    ! [X: produc6945250483304103390nt_int] :
      ( ( produc5625573926100789892nt_nat @ ( produc854102916164664432nt_int @ X ) )
      = ( produc4803745454372187940nt_int @ X ) ) ).

% snd_swap
thf(fact_166_snd__swap,axiom,
    ! [X: product_prod_nat_nat] :
      ( ( product_snd_nat_nat @ ( product_swap_nat_nat @ X ) )
      = ( product_fst_nat_nat @ X ) ) ).

% snd_swap
thf(fact_167_snd__swap,axiom,
    ! [X: produc1219242969750017639nt_int] :
      ( ( produc3892743399831173125nt_int @ ( produc4709573950824599159nt_int @ X ) )
      = ( produc698254169746827971nt_int @ X ) ) ).

% snd_swap
thf(fact_168_snd__swap,axiom,
    ! [X: produc1709102135585200056nt_nat] :
      ( ( produc3248548876605496418nt_int @ ( produc3231127965659957906nt_nat @ X ) )
      = ( produc7180770503867481414nt_nat @ X ) ) ).

% snd_swap
thf(fact_169_snd__swap,axiom,
    ! [X: product_prod_int_int] :
      ( ( product_snd_int_int @ ( product_swap_int_int @ X ) )
      = ( product_fst_int_int @ X ) ) ).

% snd_swap
thf(fact_170_add__is__0,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( plus_plus_nat @ M2 @ N )
        = zero_zero_nat )
      = ( ( M2 = zero_zero_nat )
        & ( N = zero_zero_nat ) ) ) ).

% add_is_0
thf(fact_171_Nat_Oadd__0__right,axiom,
    ! [M2: nat] :
      ( ( plus_plus_nat @ M2 @ zero_zero_nat )
      = M2 ) ).

% Nat.add_0_right
thf(fact_172_add_Oright__neutral,axiom,
    ! [A2: nat] :
      ( ( plus_plus_nat @ A2 @ zero_zero_nat )
      = A2 ) ).

% add.right_neutral
thf(fact_173_add_Oright__neutral,axiom,
    ! [A2: int] :
      ( ( plus_plus_int @ A2 @ zero_zero_int )
      = A2 ) ).

% add.right_neutral
thf(fact_174_mem__Collect__eq,axiom,
    ! [A2: product_prod_int_int,P: product_prod_int_int > $o] :
      ( ( member5262025264175285858nt_int @ A2 @ ( collec213857154873943460nt_int @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_175_mem__Collect__eq,axiom,
    ! [A2: int,P: int > $o] :
      ( ( member_int2 @ A2 @ ( collect_int @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_176_Collect__mem__eq,axiom,
    ! [A: set_Pr958786334691620121nt_int] :
      ( ( collec213857154873943460nt_int
        @ ^ [X3: product_prod_int_int] : ( member5262025264175285858nt_int @ X3 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_177_Collect__mem__eq,axiom,
    ! [A: set_int] :
      ( ( collect_int
        @ ^ [X3: int] : ( member_int2 @ X3 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_178_double__zero__sym,axiom,
    ! [A2: int] :
      ( ( zero_zero_int
        = ( plus_plus_int @ A2 @ A2 ) )
      = ( A2 = zero_zero_int ) ) ).

% double_zero_sym
thf(fact_179_add__cancel__left__left,axiom,
    ! [B: nat,A2: nat] :
      ( ( ( plus_plus_nat @ B @ A2 )
        = A2 )
      = ( B = zero_zero_nat ) ) ).

% add_cancel_left_left
thf(fact_180_add__cancel__left__left,axiom,
    ! [B: int,A2: int] :
      ( ( ( plus_plus_int @ B @ A2 )
        = A2 )
      = ( B = zero_zero_int ) ) ).

% add_cancel_left_left
thf(fact_181_add__cancel__left__right,axiom,
    ! [A2: nat,B: nat] :
      ( ( ( plus_plus_nat @ A2 @ B )
        = A2 )
      = ( B = zero_zero_nat ) ) ).

% add_cancel_left_right
thf(fact_182_add__cancel__left__right,axiom,
    ! [A2: int,B: int] :
      ( ( ( plus_plus_int @ A2 @ B )
        = A2 )
      = ( B = zero_zero_int ) ) ).

% add_cancel_left_right
thf(fact_183_add__cancel__right__left,axiom,
    ! [A2: nat,B: nat] :
      ( ( A2
        = ( plus_plus_nat @ B @ A2 ) )
      = ( B = zero_zero_nat ) ) ).

% add_cancel_right_left
thf(fact_184_add__cancel__right__left,axiom,
    ! [A2: int,B: int] :
      ( ( A2
        = ( plus_plus_int @ B @ A2 ) )
      = ( B = zero_zero_int ) ) ).

% add_cancel_right_left
thf(fact_185_add__cancel__right__right,axiom,
    ! [A2: nat,B: nat] :
      ( ( A2
        = ( plus_plus_nat @ A2 @ B ) )
      = ( B = zero_zero_nat ) ) ).

% add_cancel_right_right
thf(fact_186_add__cancel__right__right,axiom,
    ! [A2: int,B: int] :
      ( ( A2
        = ( plus_plus_int @ A2 @ B ) )
      = ( B = zero_zero_int ) ) ).

% add_cancel_right_right
thf(fact_187_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_188_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_189_add__0,axiom,
    ! [A2: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ A2 )
      = A2 ) ).

% add_0
thf(fact_190_add__0,axiom,
    ! [A2: int] :
      ( ( plus_plus_int @ zero_zero_int @ A2 )
      = A2 ) ).

% add_0
thf(fact_191_fst__swap,axiom,
    ! [X: product_prod_int_int] :
      ( ( product_fst_int_int @ ( product_swap_int_int @ X ) )
      = ( product_snd_int_int @ X ) ) ).

% fst_swap
thf(fact_192_plus__nat_Oadd__0,axiom,
    ! [N: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ N )
      = N ) ).

% plus_nat.add_0
thf(fact_193_zero__reorient,axiom,
    ! [X: nat] :
      ( ( zero_zero_nat = X )
      = ( X = zero_zero_nat ) ) ).

% zero_reorient
thf(fact_194_zero__reorient,axiom,
    ! [X: int] :
      ( ( zero_zero_int = X )
      = ( X = zero_zero_int ) ) ).

% zero_reorient
thf(fact_195_add__eq__self__zero,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( plus_plus_nat @ M2 @ N )
        = M2 )
     => ( N = zero_zero_nat ) ) ).

% add_eq_self_zero
thf(fact_196_length__code,axiom,
    ( size_s5157815400016825771nt_int
    = ( gen_le8428774395332151372nt_int @ zero_zero_nat ) ) ).

% length_code
thf(fact_197_comm__monoid__add__class_Oadd__0,axiom,
    ! [A2: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ A2 )
      = A2 ) ).

% comm_monoid_add_class.add_0
thf(fact_198_comm__monoid__add__class_Oadd__0,axiom,
    ! [A2: int] :
      ( ( plus_plus_int @ zero_zero_int @ A2 )
      = A2 ) ).

% comm_monoid_add_class.add_0
thf(fact_199_add_Ocomm__neutral,axiom,
    ! [A2: nat] :
      ( ( plus_plus_nat @ A2 @ zero_zero_nat )
      = A2 ) ).

% add.comm_neutral
thf(fact_200_add_Ocomm__neutral,axiom,
    ! [A2: int] :
      ( ( plus_plus_int @ A2 @ zero_zero_int )
      = A2 ) ).

% add.comm_neutral
thf(fact_201_add_Ogroup__left__neutral,axiom,
    ! [A2: int] :
      ( ( plus_plus_int @ zero_zero_int @ A2 )
      = A2 ) ).

% add.group_left_neutral
thf(fact_202_double__eq__0__iff,axiom,
    ! [A2: int] :
      ( ( ( plus_plus_int @ A2 @ A2 )
        = zero_zero_int )
      = ( A2 = zero_zero_int ) ) ).

% double_eq_0_iff
thf(fact_203_can__select__set__list__ex1,axiom,
    ! [P: product_prod_int_int > $o,A: list_P5707943133018811711nt_int] :
      ( ( can_se1576335439491303350nt_int @ P @ ( set_Pr2470121279949933262nt_int @ A ) )
      = ( list_e5465588451548443778nt_int @ P @ A ) ) ).

% can_select_set_list_ex1
thf(fact_204_add__0__iff,axiom,
    ! [B: nat,A2: nat] :
      ( ( B
        = ( plus_plus_nat @ B @ A2 ) )
      = ( A2 = zero_zero_nat ) ) ).

% add_0_iff
thf(fact_205_add__0__iff,axiom,
    ! [B: int,A2: int] :
      ( ( B
        = ( plus_plus_int @ B @ A2 ) )
      = ( A2 = zero_zero_int ) ) ).

% add_0_iff
thf(fact_206_verit__sum__simplify,axiom,
    ! [A2: nat] :
      ( ( plus_plus_nat @ A2 @ zero_zero_nat )
      = A2 ) ).

% verit_sum_simplify
thf(fact_207_verit__sum__simplify,axiom,
    ! [A2: int] :
      ( ( plus_plus_int @ A2 @ zero_zero_int )
      = A2 ) ).

% verit_sum_simplify
thf(fact_208_length__splice,axiom,
    ! [Xs2: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int] :
      ( ( size_s5157815400016825771nt_int @ ( splice6983101402924261266nt_int @ Xs2 @ Ys ) )
      = ( plus_plus_nat @ ( size_s5157815400016825771nt_int @ Xs2 ) @ ( size_s5157815400016825771nt_int @ Ys ) ) ) ).

% length_splice
thf(fact_209_insort__insert__key__def,axiom,
    ( linord5950828073485037667nt_int
    = ( ^ [F2: product_prod_int_int > int,X3: product_prod_int_int,Xs3: list_P5707943133018811711nt_int] : ( if_lis8883190402267401221nt_int @ ( member_int2 @ ( F2 @ X3 ) @ ( image_5042161079198086560nt_int @ F2 @ ( set_Pr2470121279949933262nt_int @ Xs3 ) ) ) @ Xs3 @ ( linord5209605980427960106nt_int @ F2 @ X3 @ Xs3 ) ) ) ) ).

% insort_insert_key_def
thf(fact_210_insort__insert__insort__key,axiom,
    ! [F: product_prod_int_int > int,X: product_prod_int_int,Xs2: list_P5707943133018811711nt_int] :
      ( ~ ( member_int2 @ ( F @ X ) @ ( image_5042161079198086560nt_int @ F @ ( set_Pr2470121279949933262nt_int @ Xs2 ) ) )
     => ( ( linord5950828073485037667nt_int @ F @ X @ Xs2 )
        = ( linord5209605980427960106nt_int @ F @ X @ Xs2 ) ) ) ).

% insort_insert_insort_key
thf(fact_211_prod_Oswap__def,axiom,
    ( product_swap_int_int
    = ( ^ [P4: product_prod_int_int] : ( product_Pair_int_int @ ( product_snd_int_int @ P4 ) @ ( product_fst_int_int @ P4 ) ) ) ) ).

% prod.swap_def
thf(fact_212_plus__int__code_I2_J,axiom,
    ! [L: int] :
      ( ( plus_plus_int @ zero_zero_int @ L )
      = L ) ).

% plus_int_code(2)
thf(fact_213_plus__int__code_I1_J,axiom,
    ! [K: int] :
      ( ( plus_plus_int @ K @ zero_zero_int )
      = K ) ).

% plus_int_code(1)
thf(fact_214_Euclid__induct,axiom,
    ! [P: nat > nat > $o,A2: nat,B: nat] :
      ( ! [A4: nat,B4: nat] :
          ( ( P @ A4 @ B4 )
          = ( P @ B4 @ A4 ) )
     => ( ! [A4: nat] : ( P @ A4 @ zero_zero_nat )
       => ( ! [A4: nat,B4: nat] :
              ( ( P @ A4 @ B4 )
             => ( P @ A4 @ ( plus_plus_nat @ A4 @ B4 ) ) )
         => ( P @ A2 @ B ) ) ) ) ).

% Euclid_induct
thf(fact_215_pair__in__swap__image,axiom,
    ! [Y: int,X: int,A: set_Pr958786334691620121nt_int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y @ X ) @ ( image_2653370878348428101nt_int @ product_swap_int_int @ A ) )
      = ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ A ) ) ).

% pair_in_swap_image
thf(fact_216_prod_Ocollapse,axiom,
    ! [Prod: product_prod_int_int] :
      ( ( product_Pair_int_int @ ( product_fst_int_int @ Prod ) @ ( product_snd_int_int @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_217_can__select__def,axiom,
    ( can_se1576335439491303350nt_int
    = ( ^ [P3: product_prod_int_int > $o,A5: set_Pr958786334691620121nt_int] :
        ? [X3: product_prod_int_int] :
          ( ( member5262025264175285858nt_int @ X3 @ A5 )
          & ( P3 @ X3 )
          & ! [Y4: product_prod_int_int] :
              ( ( ( member5262025264175285858nt_int @ Y4 @ A5 )
                & ( P3 @ Y4 ) )
             => ( Y4 = X3 ) ) ) ) ) ).

% can_select_def
thf(fact_218_can__select__def,axiom,
    ( can_select_int
    = ( ^ [P3: int > $o,A5: set_int] :
        ? [X3: int] :
          ( ( member_int2 @ X3 @ A5 )
          & ( P3 @ X3 )
          & ! [Y4: int] :
              ( ( ( member_int2 @ Y4 @ A5 )
                & ( P3 @ Y4 ) )
             => ( Y4 = X3 ) ) ) ) ) ).

% can_select_def
thf(fact_219_fst__conv,axiom,
    ! [X1: int,X22: int] :
      ( ( product_fst_int_int @ ( product_Pair_int_int @ X1 @ X22 ) )
      = X1 ) ).

% fst_conv
thf(fact_220_fst__eqD,axiom,
    ! [X: int,Y: int,A2: int] :
      ( ( ( product_fst_int_int @ ( product_Pair_int_int @ X @ Y ) )
        = A2 )
     => ( X = A2 ) ) ).

% fst_eqD
thf(fact_221_snd__eqD,axiom,
    ! [X: int,Y: int,A2: int] :
      ( ( ( product_snd_int_int @ ( product_Pair_int_int @ X @ Y ) )
        = A2 )
     => ( Y = A2 ) ) ).

% snd_eqD
thf(fact_222_snd__conv,axiom,
    ! [X1: int,X22: int] :
      ( ( product_snd_int_int @ ( product_Pair_int_int @ X1 @ X22 ) )
      = X22 ) ).

% snd_conv
thf(fact_223_prod_Oexhaust__sel,axiom,
    ! [Prod: product_prod_int_int] :
      ( Prod
      = ( product_Pair_int_int @ ( product_fst_int_int @ Prod ) @ ( product_snd_int_int @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_224_exI__realizer,axiom,
    ! [P: int > int > $o,Y: int,X: int] :
      ( ( P @ Y @ X )
     => ( P @ ( product_snd_int_int @ ( product_Pair_int_int @ X @ Y ) ) @ ( product_fst_int_int @ ( product_Pair_int_int @ X @ Y ) ) ) ) ).

% exI_realizer
thf(fact_225_conjI__realizer,axiom,
    ! [P: int > $o,P2: int,Q2: int > $o,Q: int] :
      ( ( P @ P2 )
     => ( ( Q2 @ Q )
       => ( ( P @ ( product_fst_int_int @ ( product_Pair_int_int @ P2 @ Q ) ) )
          & ( Q2 @ ( product_snd_int_int @ ( product_Pair_int_int @ P2 @ Q ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_226_surjective__pairing,axiom,
    ! [T2: product_prod_int_int] :
      ( T2
      = ( product_Pair_int_int @ ( product_fst_int_int @ T2 ) @ ( product_snd_int_int @ T2 ) ) ) ).

% surjective_pairing
thf(fact_227_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P: int > int > $o,X: int,Y: int,A2: product_prod_int_int] :
      ( ( P @ X @ Y )
     => ( ( A2
          = ( product_Pair_int_int @ X @ Y ) )
       => ( P @ ( product_fst_int_int @ A2 ) @ ( product_snd_int_int @ A2 ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_228_size__char__eq__0,axiom,
    ( size_size_char
    = ( ^ [C3: char] : zero_zero_nat ) ) ).

% size_char_eq_0
thf(fact_229_sndI,axiom,
    ! [X: product_prod_int_int,Y: int,Z: int] :
      ( ( X
        = ( product_Pair_int_int @ Y @ Z ) )
     => ( ( product_snd_int_int @ X )
        = Z ) ) ).

% sndI
thf(fact_230_eq__snd__iff,axiom,
    ! [B: int,P2: product_prod_int_int] :
      ( ( B
        = ( product_snd_int_int @ P2 ) )
      = ( ? [A3: int] :
            ( P2
            = ( product_Pair_int_int @ A3 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_231_eq__fst__iff,axiom,
    ! [A2: int,P2: product_prod_int_int] :
      ( ( A2
        = ( product_fst_int_int @ P2 ) )
      = ( ? [B3: int] :
            ( P2
            = ( product_Pair_int_int @ A2 @ B3 ) ) ) ) ).

% eq_fst_iff
thf(fact_232_fstI,axiom,
    ! [X: product_prod_int_int,Y: int,Z: int] :
      ( ( X
        = ( product_Pair_int_int @ Y @ Z ) )
     => ( ( product_fst_int_int @ X )
        = Y ) ) ).

% fstI
thf(fact_233_divides__aux__eq,axiom,
    ! [Q: nat,R: nat] :
      ( ( unique5332122412489317741ux_nat @ ( product_Pair_nat_nat @ Q @ R ) )
      = ( R = zero_zero_nat ) ) ).

% divides_aux_eq
thf(fact_234_divides__aux__eq,axiom,
    ! [Q: int,R: int] :
      ( ( unique5329631941980267465ux_int @ ( product_Pair_int_int @ Q @ R ) )
      = ( R = zero_zero_int ) ) ).

% divides_aux_eq
thf(fact_235_count__notin,axiom,
    ! [X: int,Xs2: list_int] :
      ( ~ ( member_int2 @ X @ ( set_int2 @ Xs2 ) )
     => ( ( count_list_int @ Xs2 @ X )
        = zero_zero_nat ) ) ).

% count_notin
thf(fact_236_count__notin,axiom,
    ! [X: product_prod_int_int,Xs2: list_P5707943133018811711nt_int] :
      ( ~ ( member5262025264175285858nt_int @ X @ ( set_Pr2470121279949933262nt_int @ Xs2 ) )
     => ( ( count_1024995598469094197nt_int @ Xs2 @ X )
        = zero_zero_nat ) ) ).

% count_notin
thf(fact_237_count__list__0__iff,axiom,
    ! [Xs2: list_int,X: int] :
      ( ( ( count_list_int @ Xs2 @ X )
        = zero_zero_nat )
      = ( ~ ( member_int2 @ X @ ( set_int2 @ Xs2 ) ) ) ) ).

% count_list_0_iff
thf(fact_238_count__list__0__iff,axiom,
    ! [Xs2: list_P5707943133018811711nt_int,X: product_prod_int_int] :
      ( ( ( count_1024995598469094197nt_int @ Xs2 @ X )
        = zero_zero_nat )
      = ( ~ ( member5262025264175285858nt_int @ X @ ( set_Pr2470121279949933262nt_int @ Xs2 ) ) ) ) ).

% count_list_0_iff
thf(fact_239_divides__aux__def,axiom,
    ( unique5332122412489317741ux_nat
    = ( ^ [Qr: product_prod_nat_nat] :
          ( ( product_snd_nat_nat @ Qr )
          = zero_zero_nat ) ) ) ).

% divides_aux_def
thf(fact_240_divides__aux__def,axiom,
    ( unique5329631941980267465ux_int
    = ( ^ [Qr: product_prod_int_int] :
          ( ( product_snd_int_int @ Qr )
          = zero_zero_int ) ) ) ).

% divides_aux_def
thf(fact_241_size_H__char__eq__0,axiom,
    ( size_char
    = ( ^ [C3: char] : zero_zero_nat ) ) ).

% size'_char_eq_0
thf(fact_242_char_Osize_I2_J,axiom,
    ! [X1: $o,X22: $o,X32: $o,X42: $o,X5: $o,X6: $o,X7: $o,X8: $o] :
      ( ( size_size_char @ ( char2 @ X1 @ X22 @ X32 @ X42 @ X5 @ X6 @ X7 @ X8 ) )
      = zero_zero_nat ) ).

% char.size(2)
thf(fact_243_in__set__impl__in__set__zip1,axiom,
    ! [Xs2: list_int,Ys: list_int,X: int] :
      ( ( ( size_size_list_int @ Xs2 )
        = ( size_size_list_int @ Ys ) )
     => ( ( member_int2 @ X @ ( set_int2 @ Xs2 ) )
       => ~ ! [Y3: int] :
              ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y3 ) @ ( set_Pr2470121279949933262nt_int @ ( zip_int_int @ Xs2 @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_244_in__set__impl__in__set__zip1,axiom,
    ! [Xs2: list_int,Ys: list_P5707943133018811711nt_int,X: int] :
      ( ( ( size_size_list_int @ Xs2 )
        = ( size_s5157815400016825771nt_int @ Ys ) )
     => ( ( member_int2 @ X @ ( set_int2 @ Xs2 ) )
       => ~ ! [Y3: product_prod_int_int] :
              ~ ( member4957952664650131097nt_int @ ( produc346731203614136500nt_int @ X @ Y3 ) @ ( set_Pr222267339979123501nt_int @ ( zip_in865470750896630164nt_int @ Xs2 @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_245_in__set__impl__in__set__zip1,axiom,
    ! [Xs2: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int,X: product_prod_int_int] :
      ( ( ( size_s5157815400016825771nt_int @ Xs2 )
        = ( size_s5157815400016825771nt_int @ Ys ) )
     => ( ( member5262025264175285858nt_int @ X @ ( set_Pr2470121279949933262nt_int @ Xs2 ) )
       => ~ ! [Y3: product_prod_int_int] :
              ~ ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ X @ Y3 ) @ ( set_Pr5878228222108503548nt_int @ ( zip_Pr2148635498163357687nt_int @ Xs2 @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_246_in__set__impl__in__set__zip2,axiom,
    ! [Xs2: list_int,Ys: list_int,Y: int] :
      ( ( ( size_size_list_int @ Xs2 )
        = ( size_size_list_int @ Ys ) )
     => ( ( member_int2 @ Y @ ( set_int2 @ Ys ) )
       => ~ ! [X2: int] :
              ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y ) @ ( set_Pr2470121279949933262nt_int @ ( zip_int_int @ Xs2 @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_247_in__set__impl__in__set__zip2,axiom,
    ! [Xs2: list_P5707943133018811711nt_int,Ys: list_int,Y: int] :
      ( ( ( size_s5157815400016825771nt_int @ Xs2 )
        = ( size_size_list_int @ Ys ) )
     => ( ( member_int2 @ Y @ ( set_int2 @ Ys ) )
       => ~ ! [X2: product_prod_int_int] :
              ~ ( member1729483249812673067nt_int @ ( produc8906795734991021710nt_int @ X2 @ Y ) @ ( set_Pr6217169961996441279nt_int @ ( zip_Pr202163245418739566nt_int @ Xs2 @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_248_in__set__impl__in__set__zip2,axiom,
    ! [Xs2: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int,Y: product_prod_int_int] :
      ( ( ( size_s5157815400016825771nt_int @ Xs2 )
        = ( size_s5157815400016825771nt_int @ Ys ) )
     => ( ( member5262025264175285858nt_int @ Y @ ( set_Pr2470121279949933262nt_int @ Ys ) )
       => ~ ! [X2: product_prod_int_int] :
              ~ ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ X2 @ Y ) @ ( set_Pr5878228222108503548nt_int @ ( zip_Pr2148635498163357687nt_int @ Xs2 @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_249_image2__eqI,axiom,
    ! [B: int,F: product_prod_int_int > int,X: product_prod_int_int,C: int,G: product_prod_int_int > int,A: set_Pr958786334691620121nt_int] :
      ( ( B
        = ( F @ X ) )
     => ( ( C
          = ( G @ X ) )
       => ( ( member5262025264175285858nt_int @ X @ A )
         => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B @ C ) @ ( bNF_Gr4426612819127333049nt_int @ A @ F @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_250_image2__eqI,axiom,
    ! [B: int,F: int > int,X: int,C: int,G: int > int,A: set_int] :
      ( ( B
        = ( F @ X ) )
     => ( ( C
          = ( G @ X ) )
       => ( ( member_int2 @ X @ A )
         => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B @ C ) @ ( bNF_Gr8686673574714534880nt_int @ A @ F @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_251_zero__integer_Orsp,axiom,
    zero_zero_int = zero_zero_int ).

% zero_integer.rsp
thf(fact_252_UNIV__I,axiom,
    ! [X: product_prod_int_int] : ( member5262025264175285858nt_int @ X @ top_to4366644338036079209nt_int ) ).

% UNIV_I
thf(fact_253_UNIV__I,axiom,
    ! [X: int] : ( member_int2 @ X @ top_top_set_int ) ).

% UNIV_I
thf(fact_254_char_Oinject,axiom,
    ! [X1: $o,X22: $o,X32: $o,X42: $o,X5: $o,X6: $o,X7: $o,X8: $o,Y1: $o,Y22: $o,Y32: $o,Y42: $o,Y5: $o,Y6: $o,Y7: $o,Y8: $o] :
      ( ( ( char2 @ X1 @ X22 @ X32 @ X42 @ X5 @ X6 @ X7 @ X8 )
        = ( char2 @ Y1 @ Y22 @ Y32 @ Y42 @ Y5 @ Y6 @ Y7 @ Y8 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y22 )
        & ( X32 = Y32 )
        & ( X42 = Y42 )
        & ( X5 = Y5 )
        & ( X6 = Y6 )
        & ( X7 = Y7 )
        & ( X8 = Y8 ) ) ) ).

% char.inject
thf(fact_255_char_Oexhaust,axiom,
    ! [Y: char] :
      ~ ! [X12: $o,X23: $o,X33: $o,X43: $o,X52: $o,X62: $o,X72: $o,X82: $o] :
          ( Y
         != ( char2 @ X12 @ X23 @ X33 @ X43 @ X52 @ X62 @ X72 @ X82 ) ) ).

% char.exhaust
thf(fact_256_UNIV__witness,axiom,
    ? [X2: product_prod_int_int] : ( member5262025264175285858nt_int @ X2 @ top_to4366644338036079209nt_int ) ).

% UNIV_witness
thf(fact_257_UNIV__witness,axiom,
    ? [X2: int] : ( member_int2 @ X2 @ top_top_set_int ) ).

% UNIV_witness
thf(fact_258_UNIV__eq__I,axiom,
    ! [A: set_Pr958786334691620121nt_int] :
      ( ! [X2: product_prod_int_int] : ( member5262025264175285858nt_int @ X2 @ A )
     => ( top_to4366644338036079209nt_int = A ) ) ).

% UNIV_eq_I
thf(fact_259_UNIV__eq__I,axiom,
    ! [A: set_int] :
      ( ! [X2: int] : ( member_int2 @ X2 @ A )
     => ( top_top_set_int = A ) ) ).

% UNIV_eq_I
thf(fact_260_char_Osize__gen,axiom,
    ! [X1: $o,X22: $o,X32: $o,X42: $o,X5: $o,X6: $o,X7: $o,X8: $o] :
      ( ( size_char @ ( char2 @ X1 @ X22 @ X32 @ X42 @ X5 @ X6 @ X7 @ X8 ) )
      = zero_zero_nat ) ).

% char.size_gen
thf(fact_261_range__eqI,axiom,
    ! [B: int,F: product_prod_int_int > int,X: product_prod_int_int] :
      ( ( B
        = ( F @ X ) )
     => ( member_int2 @ B @ ( image_5042161079198086560nt_int @ F @ top_to4366644338036079209nt_int ) ) ) ).

% range_eqI
thf(fact_262_rangeI,axiom,
    ! [F: product_prod_int_int > int,X: product_prod_int_int] : ( member_int2 @ ( F @ X ) @ ( image_5042161079198086560nt_int @ F @ top_to4366644338036079209nt_int ) ) ).

% rangeI
thf(fact_263_set__zip__rightD,axiom,
    ! [X: int,Y: int,Xs2: list_int,Ys: list_int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( set_Pr2470121279949933262nt_int @ ( zip_int_int @ Xs2 @ Ys ) ) )
     => ( member_int2 @ Y @ ( set_int2 @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_264_set__zip__leftD,axiom,
    ! [X: int,Y: int,Xs2: list_int,Ys: list_int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( set_Pr2470121279949933262nt_int @ ( zip_int_int @ Xs2 @ Ys ) ) )
     => ( member_int2 @ X @ ( set_int2 @ Xs2 ) ) ) ).

% set_zip_leftD
thf(fact_265_in__set__zipE,axiom,
    ! [X: int,Y: product_prod_int_int,Xs2: list_int,Ys: list_P5707943133018811711nt_int] :
      ( ( member4957952664650131097nt_int @ ( produc346731203614136500nt_int @ X @ Y ) @ ( set_Pr222267339979123501nt_int @ ( zip_in865470750896630164nt_int @ Xs2 @ Ys ) ) )
     => ~ ( ( member_int2 @ X @ ( set_int2 @ Xs2 ) )
         => ~ ( member5262025264175285858nt_int @ Y @ ( set_Pr2470121279949933262nt_int @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_266_in__set__zipE,axiom,
    ! [X: product_prod_int_int,Y: int,Xs2: list_P5707943133018811711nt_int,Ys: list_int] :
      ( ( member1729483249812673067nt_int @ ( produc8906795734991021710nt_int @ X @ Y ) @ ( set_Pr6217169961996441279nt_int @ ( zip_Pr202163245418739566nt_int @ Xs2 @ Ys ) ) )
     => ~ ( ( member5262025264175285858nt_int @ X @ ( set_Pr2470121279949933262nt_int @ Xs2 ) )
         => ~ ( member_int2 @ Y @ ( set_int2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_267_in__set__zipE,axiom,
    ! [X: product_prod_int_int,Y: product_prod_int_int,Xs2: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int] :
      ( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ X @ Y ) @ ( set_Pr5878228222108503548nt_int @ ( zip_Pr2148635498163357687nt_int @ Xs2 @ Ys ) ) )
     => ~ ( ( member5262025264175285858nt_int @ X @ ( set_Pr2470121279949933262nt_int @ Xs2 ) )
         => ~ ( member5262025264175285858nt_int @ Y @ ( set_Pr2470121279949933262nt_int @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_268_in__set__zipE,axiom,
    ! [X: int,Y: int,Xs2: list_int,Ys: list_int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( set_Pr2470121279949933262nt_int @ ( zip_int_int @ Xs2 @ Ys ) ) )
     => ~ ( ( member_int2 @ X @ ( set_int2 @ Xs2 ) )
         => ~ ( member_int2 @ Y @ ( set_int2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_269_zip__same,axiom,
    ! [A2: product_prod_int_int,B: product_prod_int_int,Xs2: list_P5707943133018811711nt_int] :
      ( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ A2 @ B ) @ ( set_Pr5878228222108503548nt_int @ ( zip_Pr2148635498163357687nt_int @ Xs2 @ Xs2 ) ) )
      = ( ( member5262025264175285858nt_int @ A2 @ ( set_Pr2470121279949933262nt_int @ Xs2 ) )
        & ( A2 = B ) ) ) ).

% zip_same
thf(fact_270_zip__same,axiom,
    ! [A2: int,B: int,Xs2: list_int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ ( set_Pr2470121279949933262nt_int @ ( zip_int_int @ Xs2 @ Xs2 ) ) )
      = ( ( member_int2 @ A2 @ ( set_int2 @ Xs2 ) )
        & ( A2 = B ) ) ) ).

% zip_same
thf(fact_271_zero__natural_Orsp,axiom,
    zero_zero_nat = zero_zero_nat ).

% zero_natural.rsp
thf(fact_272_range__snd,axiom,
    ( ( image_5042161079198086560nt_int @ product_snd_int_int @ top_to4366644338036079209nt_int )
    = top_top_set_int ) ).

% range_snd
thf(fact_273_range__fst,axiom,
    ( ( image_5042161079198086560nt_int @ product_fst_int_int @ top_to4366644338036079209nt_int )
    = top_top_set_int ) ).

% range_fst
thf(fact_274_surj__plus,axiom,
    ! [A2: int] :
      ( ( image_int_int @ ( plus_plus_int @ A2 ) @ top_top_set_int )
      = top_top_set_int ) ).

% surj_plus
thf(fact_275_iso__tuple__UNIV__I,axiom,
    ! [X: product_prod_int_int] : ( member5262025264175285858nt_int @ X @ top_to4366644338036079209nt_int ) ).

% iso_tuple_UNIV_I
thf(fact_276_iso__tuple__UNIV__I,axiom,
    ! [X: int] : ( member_int2 @ X @ top_top_set_int ) ).

% iso_tuple_UNIV_I
thf(fact_277_surj__def,axiom,
    ! [F: product_prod_int_int > int] :
      ( ( ( image_5042161079198086560nt_int @ F @ top_to4366644338036079209nt_int )
        = top_top_set_int )
      = ( ! [Y4: int] :
          ? [X3: product_prod_int_int] :
            ( Y4
            = ( F @ X3 ) ) ) ) ).

% surj_def
thf(fact_278_surjI,axiom,
    ! [G: product_prod_int_int > int,F: int > product_prod_int_int] :
      ( ! [X2: int] :
          ( ( G @ ( F @ X2 ) )
          = X2 )
     => ( ( image_5042161079198086560nt_int @ G @ top_to4366644338036079209nt_int )
        = top_top_set_int ) ) ).

% surjI
thf(fact_279_surjE,axiom,
    ! [F: product_prod_int_int > int,Y: int] :
      ( ( ( image_5042161079198086560nt_int @ F @ top_to4366644338036079209nt_int )
        = top_top_set_int )
     => ~ ! [X2: product_prod_int_int] :
            ( Y
           != ( F @ X2 ) ) ) ).

% surjE
thf(fact_280_surjD,axiom,
    ! [F: product_prod_int_int > int,Y: int] :
      ( ( ( image_5042161079198086560nt_int @ F @ top_to4366644338036079209nt_int )
        = top_top_set_int )
     => ? [X2: product_prod_int_int] :
          ( Y
          = ( F @ X2 ) ) ) ).

% surjD
thf(fact_281_top__empty__eq,axiom,
    ( top_to1578927101902068148_int_o
    = ( ^ [X3: product_prod_int_int] : ( member5262025264175285858nt_int @ X3 @ top_to4366644338036079209nt_int ) ) ) ).

% top_empty_eq
thf(fact_282_top__empty__eq,axiom,
    ( top_top_int_o
    = ( ^ [X3: int] : ( member_int2 @ X3 @ top_top_set_int ) ) ) ).

% top_empty_eq
thf(fact_283_map__prod__surj,axiom,
    ! [F: product_prod_int_int > int,G: product_prod_int_int > int] :
      ( ( ( image_5042161079198086560nt_int @ F @ top_to4366644338036079209nt_int )
        = top_top_set_int )
     => ( ( ( image_5042161079198086560nt_int @ G @ top_to4366644338036079209nt_int )
          = top_top_set_int )
       => ( ( image_2053528049745388055nt_int @ ( produc8321826182988070966nt_int @ F @ G ) @ top_to2321149100101803671nt_int )
          = top_to4366644338036079209nt_int ) ) ) ).

% map_prod_surj
thf(fact_284_length__zip,axiom,
    ! [Xs2: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int] :
      ( ( size_s6770063216428074713nt_int @ ( zip_Pr2148635498163357687nt_int @ Xs2 @ Ys ) )
      = ( ord_min_nat @ ( size_s5157815400016825771nt_int @ Xs2 ) @ ( size_s5157815400016825771nt_int @ Ys ) ) ) ).

% length_zip
thf(fact_285_length__zip,axiom,
    ! [Xs2: list_int,Ys: list_int] :
      ( ( size_s5157815400016825771nt_int @ ( zip_int_int @ Xs2 @ Ys ) )
      = ( ord_min_nat @ ( size_size_list_int @ Xs2 ) @ ( size_size_list_int @ Ys ) ) ) ).

% length_zip
thf(fact_286_zip__eq__conv,axiom,
    ! [Xs2: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int,Zs: list_P2336717926344734829nt_int] :
      ( ( ( size_s5157815400016825771nt_int @ Xs2 )
        = ( size_s5157815400016825771nt_int @ Ys ) )
     => ( ( ( zip_Pr2148635498163357687nt_int @ Xs2 @ Ys )
          = Zs )
        = ( ( ( map_Pr815854501217947862nt_int @ produc698254169746827971nt_int @ Zs )
            = Xs2 )
          & ( ( map_Pr815854501217947862nt_int @ produc3892743399831173125nt_int @ Zs )
            = Ys ) ) ) ) ).

% zip_eq_conv
thf(fact_287_zip__eq__conv,axiom,
    ! [Xs2: list_int,Ys: list_int,Zs: list_P5707943133018811711nt_int] :
      ( ( ( size_size_list_int @ Xs2 )
        = ( size_size_list_int @ Ys ) )
     => ( ( ( zip_int_int @ Xs2 @ Ys )
          = Zs )
        = ( ( ( map_Pr6494458893431244577nt_int @ product_fst_int_int @ Zs )
            = Xs2 )
          & ( ( map_Pr6494458893431244577nt_int @ product_snd_int_int @ Zs )
            = Ys ) ) ) ) ).

% zip_eq_conv
thf(fact_288_csquare__fstOp__sndOp,axiom,
    ! [F: ( int > int > $o ) > product_prod_int_int > $o,P: int > int > $o,Q2: int > int > $o] : ( bNF_cs2962985576284809986nt_int @ ( collec213857154873943460nt_int @ ( F @ ( relcompp_int_int_int @ P @ Q2 ) ) ) @ product_snd_int_int @ product_fst_int_int @ ( bNF_fs8167890477030535480nt_int @ P @ Q2 ) @ ( bNF_sn1062102010912252026nt_int @ P @ Q2 ) ) ).

% csquare_fstOp_sndOp
thf(fact_289_map__snd__zip,axiom,
    ! [Xs2: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int] :
      ( ( ( size_s5157815400016825771nt_int @ Xs2 )
        = ( size_s5157815400016825771nt_int @ Ys ) )
     => ( ( map_Pr815854501217947862nt_int @ produc3892743399831173125nt_int @ ( zip_Pr2148635498163357687nt_int @ Xs2 @ Ys ) )
        = Ys ) ) ).

% map_snd_zip
thf(fact_290_map__snd__zip,axiom,
    ! [Xs2: list_int,Ys: list_int] :
      ( ( ( size_size_list_int @ Xs2 )
        = ( size_size_list_int @ Ys ) )
     => ( ( map_Pr6494458893431244577nt_int @ product_snd_int_int @ ( zip_int_int @ Xs2 @ Ys ) )
        = Ys ) ) ).

% map_snd_zip
thf(fact_291_map__fst__zip,axiom,
    ! [Xs2: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int] :
      ( ( ( size_s5157815400016825771nt_int @ Xs2 )
        = ( size_s5157815400016825771nt_int @ Ys ) )
     => ( ( map_Pr815854501217947862nt_int @ produc698254169746827971nt_int @ ( zip_Pr2148635498163357687nt_int @ Xs2 @ Ys ) )
        = Xs2 ) ) ).

% map_fst_zip
thf(fact_292_map__fst__zip,axiom,
    ! [Xs2: list_int,Ys: list_int] :
      ( ( ( size_size_list_int @ Xs2 )
        = ( size_size_list_int @ Ys ) )
     => ( ( map_Pr6494458893431244577nt_int @ product_fst_int_int @ ( zip_int_int @ Xs2 @ Ys ) )
        = Xs2 ) ) ).

% map_fst_zip
thf(fact_293_asymI,axiom,
    ! [R: set_Pr958786334691620121nt_int] :
      ( ! [X2: int,Y3: int] :
          ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y3 ) @ R )
         => ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y3 @ X2 ) @ R ) )
     => ( asym_on_int @ top_top_set_int @ R ) ) ).

% asymI
thf(fact_294_typerep_Osize__neq,axiom,
    ! [X: typerep] :
      ( ( size_size_typerep @ X )
     != zero_zero_nat ) ).

% typerep.size_neq
thf(fact_295_length__map,axiom,
    ! [F: product_prod_int_int > product_prod_int_int,Xs2: list_P5707943133018811711nt_int] :
      ( ( size_s5157815400016825771nt_int @ ( map_Pr5543275981124152452nt_int @ F @ Xs2 ) )
      = ( size_s5157815400016825771nt_int @ Xs2 ) ) ).

% length_map
thf(fact_296_min__0L,axiom,
    ! [N: nat] :
      ( ( ord_min_nat @ zero_zero_nat @ N )
      = zero_zero_nat ) ).

% min_0L
thf(fact_297_min__0R,axiom,
    ! [N: nat] :
      ( ( ord_min_nat @ N @ zero_zero_nat )
      = zero_zero_nat ) ).

% min_0R
thf(fact_298_asym__onI,axiom,
    ! [A: set_Pr958786334691620121nt_int,R: set_Pr2560585780119916871nt_int] :
      ( ! [X2: product_prod_int_int,Y3: product_prod_int_int] :
          ( ( member5262025264175285858nt_int @ X2 @ A )
         => ( ( member5262025264175285858nt_int @ Y3 @ A )
           => ( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ X2 @ Y3 ) @ R )
             => ~ ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ Y3 @ X2 ) @ R ) ) ) )
     => ( asym_o5368087371152427499nt_int @ A @ R ) ) ).

% asym_onI
thf(fact_299_asym__onI,axiom,
    ! [A: set_int,R: set_Pr958786334691620121nt_int] :
      ( ! [X2: int,Y3: int] :
          ( ( member_int2 @ X2 @ A )
         => ( ( member_int2 @ Y3 @ A )
           => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y3 ) @ R )
             => ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y3 @ X2 ) @ R ) ) ) )
     => ( asym_on_int @ A @ R ) ) ).

% asym_onI
thf(fact_300_fst__map__prod,axiom,
    ! [F: int > int,G: int > int,X: product_prod_int_int] :
      ( ( product_fst_int_int @ ( produc6036585564866528938nt_int @ F @ G @ X ) )
      = ( F @ ( product_fst_int_int @ X ) ) ) ).

% fst_map_prod
thf(fact_301_snd__map__prod,axiom,
    ! [F: int > int,G: int > int,X: product_prod_int_int] :
      ( ( product_snd_int_int @ ( produc6036585564866528938nt_int @ F @ G @ X ) )
      = ( G @ ( product_snd_int_int @ X ) ) ) ).

% snd_map_prod
thf(fact_302_list_Oset__map,axiom,
    ! [F: product_prod_int_int > int,V: list_P5707943133018811711nt_int] :
      ( ( set_int2 @ ( map_Pr6494458893431244577nt_int @ F @ V ) )
      = ( image_5042161079198086560nt_int @ F @ ( set_Pr2470121279949933262nt_int @ V ) ) ) ).

% list.set_map
thf(fact_303_list_Oset__map,axiom,
    ! [F: product_prod_int_int > product_prod_int_int,V: list_P5707943133018811711nt_int] :
      ( ( set_Pr2470121279949933262nt_int @ ( map_Pr5543275981124152452nt_int @ F @ V ) )
      = ( image_2653370878348428101nt_int @ F @ ( set_Pr2470121279949933262nt_int @ V ) ) ) ).

% list.set_map
thf(fact_304_map__prod__imageI,axiom,
    ! [A2: int,B: int,R2: set_Pr958786334691620121nt_int,F: int > int,G: int > int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R2 )
     => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( F @ A2 ) @ ( G @ B ) ) @ ( image_2653370878348428101nt_int @ ( produc6036585564866528938nt_int @ F @ G ) @ R2 ) ) ) ).

% map_prod_imageI
thf(fact_305_min__add__distrib__right,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( plus_plus_int @ X @ ( ord_min_int @ Y @ Z ) )
      = ( ord_min_int @ ( plus_plus_int @ X @ Y ) @ ( plus_plus_int @ X @ Z ) ) ) ).

% min_add_distrib_right
thf(fact_306_min__add__distrib__right,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( plus_plus_nat @ X @ ( ord_min_nat @ Y @ Z ) )
      = ( ord_min_nat @ ( plus_plus_nat @ X @ Y ) @ ( plus_plus_nat @ X @ Z ) ) ) ).

% min_add_distrib_right
thf(fact_307_min__add__distrib__left,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( plus_plus_int @ ( ord_min_int @ X @ Y ) @ Z )
      = ( ord_min_int @ ( plus_plus_int @ X @ Z ) @ ( plus_plus_int @ Y @ Z ) ) ) ).

% min_add_distrib_left
thf(fact_308_min__add__distrib__left,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( plus_plus_nat @ ( ord_min_nat @ X @ Y ) @ Z )
      = ( ord_min_nat @ ( plus_plus_nat @ X @ Z ) @ ( plus_plus_nat @ Y @ Z ) ) ) ).

% min_add_distrib_left
thf(fact_309_map__idI,axiom,
    ! [Xs2: list_int,F: int > int] :
      ( ! [X2: int] :
          ( ( member_int2 @ X2 @ ( set_int2 @ Xs2 ) )
         => ( ( F @ X2 )
            = X2 ) )
     => ( ( map_int_int @ F @ Xs2 )
        = Xs2 ) ) ).

% map_idI
thf(fact_310_map__idI,axiom,
    ! [Xs2: list_P5707943133018811711nt_int,F: product_prod_int_int > product_prod_int_int] :
      ( ! [X2: product_prod_int_int] :
          ( ( member5262025264175285858nt_int @ X2 @ ( set_Pr2470121279949933262nt_int @ Xs2 ) )
         => ( ( F @ X2 )
            = X2 ) )
     => ( ( map_Pr5543275981124152452nt_int @ F @ Xs2 )
        = Xs2 ) ) ).

% map_idI
thf(fact_311_list_Omap__ident__strong,axiom,
    ! [T2: list_int,F: int > int] :
      ( ! [Z3: int] :
          ( ( member_int2 @ Z3 @ ( set_int2 @ T2 ) )
         => ( ( F @ Z3 )
            = Z3 ) )
     => ( ( map_int_int @ F @ T2 )
        = T2 ) ) ).

% list.map_ident_strong
thf(fact_312_list_Omap__ident__strong,axiom,
    ! [T2: list_P5707943133018811711nt_int,F: product_prod_int_int > product_prod_int_int] :
      ( ! [Z3: product_prod_int_int] :
          ( ( member5262025264175285858nt_int @ Z3 @ ( set_Pr2470121279949933262nt_int @ T2 ) )
         => ( ( F @ Z3 )
            = Z3 ) )
     => ( ( map_Pr5543275981124152452nt_int @ F @ T2 )
        = T2 ) ) ).

% list.map_ident_strong
thf(fact_313_asym__onD,axiom,
    ! [A: set_Pr958786334691620121nt_int,R: set_Pr2560585780119916871nt_int,X: product_prod_int_int,Y: product_prod_int_int] :
      ( ( asym_o5368087371152427499nt_int @ A @ R )
     => ( ( member5262025264175285858nt_int @ X @ A )
       => ( ( member5262025264175285858nt_int @ Y @ A )
         => ( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ X @ Y ) @ R )
           => ~ ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ Y @ X ) @ R ) ) ) ) ) ).

% asym_onD
thf(fact_314_asym__onD,axiom,
    ! [A: set_int,R: set_Pr958786334691620121nt_int,X: int,Y: int] :
      ( ( asym_on_int @ A @ R )
     => ( ( member_int2 @ X @ A )
       => ( ( member_int2 @ Y @ A )
         => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R )
           => ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y @ X ) @ R ) ) ) ) ) ).

% asym_onD
thf(fact_315_asym__on__def,axiom,
    ( asym_on_int
    = ( ^ [A5: set_int,R3: set_Pr958786334691620121nt_int] :
        ! [X3: int] :
          ( ( member_int2 @ X3 @ A5 )
         => ! [Y4: int] :
              ( ( member_int2 @ Y4 @ A5 )
             => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y4 ) @ R3 )
               => ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y4 @ X3 ) @ R3 ) ) ) ) ) ) ).

% asym_on_def
thf(fact_316_image__set,axiom,
    ! [F: product_prod_int_int > int,Xs2: list_P5707943133018811711nt_int] :
      ( ( image_5042161079198086560nt_int @ F @ ( set_Pr2470121279949933262nt_int @ Xs2 ) )
      = ( set_int2 @ ( map_Pr6494458893431244577nt_int @ F @ Xs2 ) ) ) ).

% image_set
thf(fact_317_image__set,axiom,
    ! [F: product_prod_int_int > product_prod_int_int,Xs2: list_P5707943133018811711nt_int] :
      ( ( image_2653370878348428101nt_int @ F @ ( set_Pr2470121279949933262nt_int @ Xs2 ) )
      = ( set_Pr2470121279949933262nt_int @ ( map_Pr5543275981124152452nt_int @ F @ Xs2 ) ) ) ).

% image_set
thf(fact_318_pair__list__eqI,axiom,
    ! [Xs2: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int] :
      ( ( ( map_Pr6494458893431244577nt_int @ product_fst_int_int @ Xs2 )
        = ( map_Pr6494458893431244577nt_int @ product_fst_int_int @ Ys ) )
     => ( ( ( map_Pr6494458893431244577nt_int @ product_snd_int_int @ Xs2 )
          = ( map_Pr6494458893431244577nt_int @ product_snd_int_int @ Ys ) )
       => ( Xs2 = Ys ) ) ) ).

% pair_list_eqI
thf(fact_319_prod__fun__imageE,axiom,
    ! [C: product_prod_int_int,F: int > int,G: int > int,R2: set_Pr958786334691620121nt_int] :
      ( ( member5262025264175285858nt_int @ C @ ( image_2653370878348428101nt_int @ ( produc6036585564866528938nt_int @ F @ G ) @ R2 ) )
     => ~ ! [X2: int,Y3: int] :
            ( ( C
              = ( product_Pair_int_int @ ( F @ X2 ) @ ( G @ Y3 ) ) )
           => ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y3 ) @ R2 ) ) ) ).

% prod_fun_imageE
thf(fact_320_asymD,axiom,
    ! [R: set_Pr958786334691620121nt_int,X: int,Y: int] :
      ( ( asym_on_int @ top_top_set_int @ R )
     => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R )
       => ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y @ X ) @ R ) ) ) ).

% asymD
thf(fact_321_asym__iff,axiom,
    ! [R: set_Pr958786334691620121nt_int] :
      ( ( asym_on_int @ top_top_set_int @ R )
      = ( ! [X3: int,Y4: int] :
            ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y4 ) @ R )
           => ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y4 @ X3 ) @ R ) ) ) ) ).

% asym_iff
thf(fact_322_zip__map__fst__snd,axiom,
    ! [Zs: list_P5707943133018811711nt_int] :
      ( ( zip_int_int @ ( map_Pr6494458893431244577nt_int @ product_fst_int_int @ Zs ) @ ( map_Pr6494458893431244577nt_int @ product_snd_int_int @ Zs ) )
      = Zs ) ).

% zip_map_fst_snd
thf(fact_323_map__snd__zip__take,axiom,
    ! [Xs2: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int] :
      ( ( map_Pr815854501217947862nt_int @ produc3892743399831173125nt_int @ ( zip_Pr2148635498163357687nt_int @ Xs2 @ Ys ) )
      = ( take_P8218740963776755879nt_int @ ( ord_min_nat @ ( size_s5157815400016825771nt_int @ Xs2 ) @ ( size_s5157815400016825771nt_int @ Ys ) ) @ Ys ) ) ).

% map_snd_zip_take
thf(fact_324_map__snd__zip__take,axiom,
    ! [Xs2: list_int,Ys: list_int] :
      ( ( map_Pr6494458893431244577nt_int @ product_snd_int_int @ ( zip_int_int @ Xs2 @ Ys ) )
      = ( take_int @ ( ord_min_nat @ ( size_size_list_int @ Xs2 ) @ ( size_size_list_int @ Ys ) ) @ Ys ) ) ).

% map_snd_zip_take
thf(fact_325_map__fst__zip__take,axiom,
    ! [Xs2: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int] :
      ( ( map_Pr815854501217947862nt_int @ produc698254169746827971nt_int @ ( zip_Pr2148635498163357687nt_int @ Xs2 @ Ys ) )
      = ( take_P8218740963776755879nt_int @ ( ord_min_nat @ ( size_s5157815400016825771nt_int @ Xs2 ) @ ( size_s5157815400016825771nt_int @ Ys ) ) @ Xs2 ) ) ).

% map_fst_zip_take
thf(fact_326_map__fst__zip__take,axiom,
    ! [Xs2: list_int,Ys: list_int] :
      ( ( map_Pr6494458893431244577nt_int @ product_fst_int_int @ ( zip_int_int @ Xs2 @ Ys ) )
      = ( take_int @ ( ord_min_nat @ ( size_size_list_int @ Xs2 ) @ ( size_size_list_int @ Ys ) ) @ Xs2 ) ) ).

% map_fst_zip_take
thf(fact_327_distinct__insort__key,axiom,
    ! [F: product_prod_int_int > int,X: product_prod_int_int,Xs2: list_P5707943133018811711nt_int] :
      ( ( distinct_int @ ( map_Pr6494458893431244577nt_int @ F @ ( linord5209605980427960106nt_int @ F @ X @ Xs2 ) ) )
      = ( ~ ( member_int2 @ ( F @ X ) @ ( image_5042161079198086560nt_int @ F @ ( set_Pr2470121279949933262nt_int @ Xs2 ) ) )
        & ( distinct_int @ ( map_Pr6494458893431244577nt_int @ F @ Xs2 ) ) ) ) ).

% distinct_insort_key
thf(fact_328_in__lex__prod,axiom,
    ! [A2: int,B: int,A6: int,B5: int,R: set_Pr958786334691620121nt_int,S3: set_Pr958786334691620121nt_int] :
      ( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ ( product_Pair_int_int @ A2 @ B ) @ ( product_Pair_int_int @ A6 @ B5 ) ) @ ( lex_prod_int_int @ R @ S3 ) )
      = ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ A6 ) @ R )
        | ( ( A2 = A6 )
          & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B @ B5 ) @ S3 ) ) ) ) ).

% in_lex_prod
thf(fact_329_in__inv__image,axiom,
    ! [X: int,Y: int,R: set_Pr958786334691620121nt_int,F: int > int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( inv_image_int_int @ R @ F ) )
      = ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( F @ X ) @ ( F @ Y ) ) @ R ) ) ).

% in_inv_image
thf(fact_330_length__take,axiom,
    ! [N: nat,Xs2: list_P5707943133018811711nt_int] :
      ( ( size_s5157815400016825771nt_int @ ( take_P8218740963776755879nt_int @ N @ Xs2 ) )
      = ( ord_min_nat @ ( size_s5157815400016825771nt_int @ Xs2 ) @ N ) ) ).

% length_take
thf(fact_331_in__set__takeD,axiom,
    ! [X: int,N: nat,Xs2: list_int] :
      ( ( member_int2 @ X @ ( set_int2 @ ( take_int @ N @ Xs2 ) ) )
     => ( member_int2 @ X @ ( set_int2 @ Xs2 ) ) ) ).

% in_set_takeD
thf(fact_332_in__set__takeD,axiom,
    ! [X: product_prod_int_int,N: nat,Xs2: list_P5707943133018811711nt_int] :
      ( ( member5262025264175285858nt_int @ X @ ( set_Pr2470121279949933262nt_int @ ( take_P8218740963776755879nt_int @ N @ Xs2 ) ) )
     => ( member5262025264175285858nt_int @ X @ ( set_Pr2470121279949933262nt_int @ Xs2 ) ) ) ).

% in_set_takeD
thf(fact_333_zip__obtain__same__length,axiom,
    ! [Xs2: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int,P: list_P2336717926344734829nt_int > $o] :
      ( ! [Zs2: list_P5707943133018811711nt_int,Ws: list_P5707943133018811711nt_int,N4: nat] :
          ( ( ( size_s5157815400016825771nt_int @ Zs2 )
            = ( size_s5157815400016825771nt_int @ Ws ) )
         => ( ( N4
              = ( ord_min_nat @ ( size_s5157815400016825771nt_int @ Xs2 ) @ ( size_s5157815400016825771nt_int @ Ys ) ) )
           => ( ( Zs2
                = ( take_P8218740963776755879nt_int @ N4 @ Xs2 ) )
             => ( ( Ws
                  = ( take_P8218740963776755879nt_int @ N4 @ Ys ) )
               => ( P @ ( zip_Pr2148635498163357687nt_int @ Zs2 @ Ws ) ) ) ) ) )
     => ( P @ ( zip_Pr2148635498163357687nt_int @ Xs2 @ Ys ) ) ) ).

% zip_obtain_same_length
thf(fact_334_eq__key__imp__eq__value,axiom,
    ! [Xs2: list_P5707943133018811711nt_int,K: int,V1: int,V2: int] :
      ( ( distinct_int @ ( map_Pr6494458893431244577nt_int @ product_fst_int_int @ Xs2 ) )
     => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ K @ V1 ) @ ( set_Pr2470121279949933262nt_int @ Xs2 ) )
       => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ K @ V2 ) @ ( set_Pr2470121279949933262nt_int @ Xs2 ) )
         => ( V1 = V2 ) ) ) ) ).

% eq_key_imp_eq_value
thf(fact_335_distinct__set__subseqs,axiom,
    ! [Xs2: list_P5707943133018811711nt_int] :
      ( ( distin3744728255968310194nt_int @ Xs2 )
     => ( distin8416474371230631186nt_int @ ( map_li8695604081002177118nt_int @ set_Pr2470121279949933262nt_int @ ( subseq1357044202310323342nt_int @ Xs2 ) ) ) ) ).

% distinct_set_subseqs
thf(fact_336_lenlex__irreflexive,axiom,
    ! [R: set_Pr958786334691620121nt_int,Xs2: list_int] :
      ( ! [X2: int] :
          ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ X2 ) @ R )
     => ~ ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs2 @ Xs2 ) @ ( lenlex_int @ R ) ) ) ).

% lenlex_irreflexive
thf(fact_337_Product__Type_OCollect__case__prodD,axiom,
    ! [X: product_prod_int_int,A: int > int > $o] :
      ( ( member5262025264175285858nt_int @ X @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ A ) ) )
     => ( A @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ X ) ) ) ).

% Product_Type.Collect_case_prodD
thf(fact_338_sndOp__in,axiom,
    ! [Ac: product_prod_int_int,P: int > int > $o,Q2: int > int > $o] :
      ( ( member5262025264175285858nt_int @ Ac @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ ( relcompp_int_int_int @ P @ Q2 ) ) ) )
     => ( member5262025264175285858nt_int @ ( bNF_sn1062102010912252026nt_int @ P @ Q2 @ Ac ) @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ Q2 ) ) ) ) ).

% sndOp_in
thf(fact_339_fstOp__in,axiom,
    ! [Ac: product_prod_int_int,P: int > int > $o,Q2: int > int > $o] :
      ( ( member5262025264175285858nt_int @ Ac @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ ( relcompp_int_int_int @ P @ Q2 ) ) ) )
     => ( member5262025264175285858nt_int @ ( bNF_fs8167890477030535480nt_int @ P @ Q2 @ Ac ) @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ P ) ) ) ) ).

% fstOp_in
thf(fact_340_map__of__inject__set,axiom,
    ! [Xs2: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int] :
      ( ( distinct_int @ ( map_Pr6494458893431244577nt_int @ product_fst_int_int @ Xs2 ) )
     => ( ( distinct_int @ ( map_Pr6494458893431244577nt_int @ product_fst_int_int @ Ys ) )
       => ( ( ( map_of_int_int @ Xs2 )
            = ( map_of_int_int @ Ys ) )
          = ( ( set_Pr2470121279949933262nt_int @ Xs2 )
            = ( set_Pr2470121279949933262nt_int @ Ys ) ) ) ) ) ).

% map_of_inject_set
thf(fact_341_ran__distinct,axiom,
    ! [Al: list_P5707943133018811711nt_int] :
      ( ( distinct_int @ ( map_Pr6494458893431244577nt_int @ product_fst_int_int @ Al ) )
     => ( ( ran_int_int @ ( map_of_int_int @ Al ) )
        = ( image_5042161079198086560nt_int @ product_snd_int_int @ ( set_Pr2470121279949933262nt_int @ Al ) ) ) ) ).

% ran_distinct
thf(fact_342_lexord__lex,axiom,
    ! [X: list_P5707943133018811711nt_int,Y: list_P5707943133018811711nt_int,R: set_Pr2560585780119916871nt_int] :
      ( ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ X @ Y ) @ ( lex_Pr5393148144989827363nt_int @ R ) )
      = ( ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ X @ Y ) @ ( lexord8886728381914977324nt_int @ R ) )
        & ( ( size_s5157815400016825771nt_int @ X )
          = ( size_s5157815400016825771nt_int @ Y ) ) ) ) ).

% lexord_lex
thf(fact_343_lexord__linear,axiom,
    ! [R: set_Pr958786334691620121nt_int,X: list_int,Y: list_int] :
      ( ! [A4: int,B4: int] :
          ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A4 @ B4 ) @ R )
          | ( A4 = B4 )
          | ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B4 @ A4 ) @ R ) )
     => ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ X @ Y ) @ ( lexord_int @ R ) )
        | ( X = Y )
        | ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Y @ X ) @ ( lexord_int @ R ) ) ) ) ).

% lexord_linear
thf(fact_344_lexord__irreflexive,axiom,
    ! [R: set_Pr958786334691620121nt_int,Xs2: list_int] :
      ( ! [X2: int] :
          ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ X2 ) @ R )
     => ~ ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs2 @ Xs2 ) @ ( lexord_int @ R ) ) ) ).

% lexord_irreflexive
thf(fact_345_ran__map__of__zip,axiom,
    ! [Xs2: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int] :
      ( ( ( size_s5157815400016825771nt_int @ Xs2 )
        = ( size_s5157815400016825771nt_int @ Ys ) )
     => ( ( distin3744728255968310194nt_int @ Xs2 )
       => ( ( ran_Pr2333488787710362927nt_int @ ( map_of5841684995557474595nt_int @ ( zip_Pr2148635498163357687nt_int @ Xs2 @ Ys ) ) )
          = ( set_Pr2470121279949933262nt_int @ Ys ) ) ) ) ).

% ran_map_of_zip
thf(fact_346_lexord__partial__trans,axiom,
    ! [Xs2: list_int,R: set_Pr958786334691620121nt_int,Ys: list_int,Zs: list_int] :
      ( ! [X2: int,Y3: int,Z3: int] :
          ( ( member_int2 @ X2 @ ( set_int2 @ Xs2 ) )
         => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y3 ) @ R )
           => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y3 @ Z3 ) @ R )
             => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Z3 ) @ R ) ) ) )
     => ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs2 @ Ys ) @ ( lexord_int @ R ) )
       => ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Ys @ Zs ) @ ( lexord_int @ R ) )
         => ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs2 @ Zs ) @ ( lexord_int @ R ) ) ) ) ) ).

% lexord_partial_trans
thf(fact_347_lexord__partial__trans,axiom,
    ! [Xs2: list_P5707943133018811711nt_int,R: set_Pr2560585780119916871nt_int,Ys: list_P5707943133018811711nt_int,Zs: list_P5707943133018811711nt_int] :
      ( ! [X2: product_prod_int_int,Y3: product_prod_int_int,Z3: product_prod_int_int] :
          ( ( member5262025264175285858nt_int @ X2 @ ( set_Pr2470121279949933262nt_int @ Xs2 ) )
         => ( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ X2 @ Y3 ) @ R )
           => ( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ Y3 @ Z3 ) @ R )
             => ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ X2 @ Z3 ) @ R ) ) ) )
     => ( ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ Xs2 @ Ys ) @ ( lexord8886728381914977324nt_int @ R ) )
       => ( ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ Ys @ Zs ) @ ( lexord8886728381914977324nt_int @ R ) )
         => ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ Xs2 @ Zs ) @ ( lexord8886728381914977324nt_int @ R ) ) ) ) ) ).

% lexord_partial_trans
thf(fact_348_map__of__zip__inject,axiom,
    ! [Ys: list_P5707943133018811711nt_int,Xs2: list_P5707943133018811711nt_int,Zs: list_P5707943133018811711nt_int] :
      ( ( ( size_s5157815400016825771nt_int @ Ys )
        = ( size_s5157815400016825771nt_int @ Xs2 ) )
     => ( ( ( size_s5157815400016825771nt_int @ Zs )
          = ( size_s5157815400016825771nt_int @ Xs2 ) )
       => ( ( distin3744728255968310194nt_int @ Xs2 )
         => ( ( ( map_of5841684995557474595nt_int @ ( zip_Pr2148635498163357687nt_int @ Xs2 @ Ys ) )
              = ( map_of5841684995557474595nt_int @ ( zip_Pr2148635498163357687nt_int @ Xs2 @ Zs ) ) )
           => ( Ys = Zs ) ) ) ) ) ).

% map_of_zip_inject
thf(fact_349_map__of__eqI,axiom,
    ! [Xs2: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int] :
      ( ( ( set_int2 @ ( map_Pr6494458893431244577nt_int @ product_fst_int_int @ Xs2 ) )
        = ( set_int2 @ ( map_Pr6494458893431244577nt_int @ product_fst_int_int @ Ys ) ) )
     => ( ! [X2: int] :
            ( ( member_int2 @ X2 @ ( set_int2 @ ( map_Pr6494458893431244577nt_int @ product_fst_int_int @ Xs2 ) ) )
           => ( ( map_of_int_int @ Xs2 @ X2 )
              = ( map_of_int_int @ Ys @ X2 ) ) )
       => ( ( map_of_int_int @ Xs2 )
          = ( map_of_int_int @ Ys ) ) ) ) ).

% map_of_eqI
thf(fact_350_map__of__eq__dom,axiom,
    ! [Xs2: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int] :
      ( ( ( map_of_int_int @ Xs2 )
        = ( map_of_int_int @ Ys ) )
     => ( ( image_5042161079198086560nt_int @ product_fst_int_int @ ( set_Pr2470121279949933262nt_int @ Xs2 ) )
        = ( image_5042161079198086560nt_int @ product_fst_int_int @ ( set_Pr2470121279949933262nt_int @ Ys ) ) ) ) ).

% map_of_eq_dom
thf(fact_351_map__of__is__SomeI,axiom,
    ! [Xys: list_P5707943133018811711nt_int,X: int,Y: int] :
      ( ( distinct_int @ ( map_Pr6494458893431244577nt_int @ product_fst_int_int @ Xys ) )
     => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( set_Pr2470121279949933262nt_int @ Xys ) )
       => ( ( map_of_int_int @ Xys @ X )
          = ( some_int @ Y ) ) ) ) ).

% map_of_is_SomeI
thf(fact_352_Some__eq__map__of__iff,axiom,
    ! [Xys: list_P5707943133018811711nt_int,Y: int,X: int] :
      ( ( distinct_int @ ( map_Pr6494458893431244577nt_int @ product_fst_int_int @ Xys ) )
     => ( ( ( some_int @ Y )
          = ( map_of_int_int @ Xys @ X ) )
        = ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( set_Pr2470121279949933262nt_int @ Xys ) ) ) ) ).

% Some_eq_map_of_iff
thf(fact_353_map__of__eq__Some__iff,axiom,
    ! [Xys: list_P5707943133018811711nt_int,X: int,Y: int] :
      ( ( distinct_int @ ( map_Pr6494458893431244577nt_int @ product_fst_int_int @ Xys ) )
     => ( ( ( map_of_int_int @ Xys @ X )
          = ( some_int @ Y ) )
        = ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( set_Pr2470121279949933262nt_int @ Xys ) ) ) ) ).

% map_of_eq_Some_iff
thf(fact_354_graph__map__of__if__distinct__dom,axiom,
    ! [Al: list_P5707943133018811711nt_int] :
      ( ( distinct_int @ ( map_Pr6494458893431244577nt_int @ product_fst_int_int @ Al ) )
     => ( ( graph_int_int @ ( map_of_int_int @ Al ) )
        = ( set_Pr2470121279949933262nt_int @ Al ) ) ) ).

% graph_map_of_if_distinct_dom
thf(fact_355_Cons__in__lex,axiom,
    ! [X: int,Xs2: list_int,Y: int,Ys: list_int,R: set_Pr958786334691620121nt_int] :
      ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( cons_int @ X @ Xs2 ) @ ( cons_int @ Y @ Ys ) ) @ ( lex_int @ R ) )
      = ( ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R )
          & ( ( size_size_list_int @ Xs2 )
            = ( size_size_list_int @ Ys ) ) )
        | ( ( X = Y )
          & ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs2 @ Ys ) @ ( lex_int @ R ) ) ) ) ) ).

% Cons_in_lex
thf(fact_356_Cons__in__lex,axiom,
    ! [X: product_prod_int_int,Xs2: list_P5707943133018811711nt_int,Y: product_prod_int_int,Ys: list_P5707943133018811711nt_int,R: set_Pr2560585780119916871nt_int] :
      ( ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ ( cons_P3334398858971670639nt_int @ X @ Xs2 ) @ ( cons_P3334398858971670639nt_int @ Y @ Ys ) ) @ ( lex_Pr5393148144989827363nt_int @ R ) )
      = ( ( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ X @ Y ) @ R )
          & ( ( size_s5157815400016825771nt_int @ Xs2 )
            = ( size_s5157815400016825771nt_int @ Ys ) ) )
        | ( ( X = Y )
          & ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ Xs2 @ Ys ) @ ( lex_Pr5393148144989827363nt_int @ R ) ) ) ) ) ).

% Cons_in_lex
thf(fact_357_map__of__zip__is__None,axiom,
    ! [Xs2: list_int,Ys: list_P5707943133018811711nt_int,X: int] :
      ( ( ( size_size_list_int @ Xs2 )
        = ( size_s5157815400016825771nt_int @ Ys ) )
     => ( ( ( map_of7676300617449379688nt_int @ ( zip_in865470750896630164nt_int @ Xs2 @ Ys ) @ X )
          = none_P2377608414092835994nt_int )
        = ( ~ ( member_int2 @ X @ ( set_int2 @ Xs2 ) ) ) ) ) ).

% map_of_zip_is_None
thf(fact_358_map__of__zip__is__None,axiom,
    ! [Xs2: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int,X: product_prod_int_int] :
      ( ( ( size_s5157815400016825771nt_int @ Xs2 )
        = ( size_s5157815400016825771nt_int @ Ys ) )
     => ( ( ( map_of5841684995557474595nt_int @ ( zip_Pr2148635498163357687nt_int @ Xs2 @ Ys ) @ X )
          = none_P2377608414092835994nt_int )
        = ( ~ ( member5262025264175285858nt_int @ X @ ( set_Pr2470121279949933262nt_int @ Xs2 ) ) ) ) ) ).

% map_of_zip_is_None
thf(fact_359_dom__map__of__zip,axiom,
    ! [Xs2: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int] :
      ( ( ( size_s5157815400016825771nt_int @ Xs2 )
        = ( size_s5157815400016825771nt_int @ Ys ) )
     => ( ( dom_Pr634364266132166418nt_int @ ( map_of5841684995557474595nt_int @ ( zip_Pr2148635498163357687nt_int @ Xs2 @ Ys ) ) )
        = ( set_Pr2470121279949933262nt_int @ Xs2 ) ) ) ).

% dom_map_of_zip
thf(fact_360_list_Oinject,axiom,
    ! [X21: int,X222: list_int,Y21: int,Y222: list_int] :
      ( ( ( cons_int @ X21 @ X222 )
        = ( cons_int @ Y21 @ Y222 ) )
      = ( ( X21 = Y21 )
        & ( X222 = Y222 ) ) ) ).

% list.inject
thf(fact_361_zip__Cons__Cons,axiom,
    ! [X: int,Xs2: list_int,Y: int,Ys: list_int] :
      ( ( zip_int_int @ ( cons_int @ X @ Xs2 ) @ ( cons_int @ Y @ Ys ) )
      = ( cons_P3334398858971670639nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( zip_int_int @ Xs2 @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_362_not__in__set__insert,axiom,
    ! [X: product_prod_int_int,Xs2: list_P5707943133018811711nt_int] :
      ( ~ ( member5262025264175285858nt_int @ X @ ( set_Pr2470121279949933262nt_int @ Xs2 ) )
     => ( ( insert5765537519290168021nt_int @ X @ Xs2 )
        = ( cons_P3334398858971670639nt_int @ X @ Xs2 ) ) ) ).

% not_in_set_insert
thf(fact_363_not__in__set__insert,axiom,
    ! [X: int,Xs2: list_int] :
      ( ~ ( member_int2 @ X @ ( set_int2 @ Xs2 ) )
     => ( ( insert_int @ X @ Xs2 )
        = ( cons_int @ X @ Xs2 ) ) ) ).

% not_in_set_insert
thf(fact_364_lexord__cons__cons,axiom,
    ! [A2: int,X: list_int,B: int,Y: list_int,R: set_Pr958786334691620121nt_int] :
      ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( cons_int @ A2 @ X ) @ ( cons_int @ B @ Y ) ) @ ( lexord_int @ R ) )
      = ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R )
        | ( ( A2 = B )
          & ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ X @ Y ) @ ( lexord_int @ R ) ) ) ) ) ).

% lexord_cons_cons
thf(fact_365_in__graphD,axiom,
    ! [K: int,V: int,M2: int > option_int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ K @ V ) @ ( graph_int_int @ M2 ) )
     => ( ( M2 @ K )
        = ( some_int @ V ) ) ) ).

% in_graphD
thf(fact_366_in__graphI,axiom,
    ! [M2: int > option_int,K: int,V: int] :
      ( ( ( M2 @ K )
        = ( some_int @ V ) )
     => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ K @ V ) @ ( graph_int_int @ M2 ) ) ) ).

% in_graphI
thf(fact_367_not__Cons__self2,axiom,
    ! [X: int,Xs2: list_int] :
      ( ( cons_int @ X @ Xs2 )
     != Xs2 ) ).

% not_Cons_self2
thf(fact_368_graph__domD,axiom,
    ! [X: product_prod_int_int,M2: int > option_int] :
      ( ( member5262025264175285858nt_int @ X @ ( graph_int_int @ M2 ) )
     => ( member_int2 @ ( product_fst_int_int @ X ) @ ( dom_int_int @ M2 ) ) ) ).

% graph_domD
thf(fact_369_zip__eq__ConsE,axiom,
    ! [Xs2: list_int,Ys: list_int,Xy: product_prod_int_int,Xys: list_P5707943133018811711nt_int] :
      ( ( ( zip_int_int @ Xs2 @ Ys )
        = ( cons_P3334398858971670639nt_int @ Xy @ Xys ) )
     => ~ ! [X2: int,Xs4: list_int] :
            ( ( Xs2
              = ( cons_int @ X2 @ Xs4 ) )
           => ! [Y3: int,Ys2: list_int] :
                ( ( Ys
                  = ( cons_int @ Y3 @ Ys2 ) )
               => ( ( Xy
                    = ( product_Pair_int_int @ X2 @ Y3 ) )
                 => ( Xys
                   != ( zip_int_int @ Xs4 @ Ys2 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_370_set__ConsD,axiom,
    ! [Y: product_prod_int_int,X: product_prod_int_int,Xs2: list_P5707943133018811711nt_int] :
      ( ( member5262025264175285858nt_int @ Y @ ( set_Pr2470121279949933262nt_int @ ( cons_P3334398858971670639nt_int @ X @ Xs2 ) ) )
     => ( ( Y = X )
        | ( member5262025264175285858nt_int @ Y @ ( set_Pr2470121279949933262nt_int @ Xs2 ) ) ) ) ).

% set_ConsD
thf(fact_371_set__ConsD,axiom,
    ! [Y: int,X: int,Xs2: list_int] :
      ( ( member_int2 @ Y @ ( set_int2 @ ( cons_int @ X @ Xs2 ) ) )
     => ( ( Y = X )
        | ( member_int2 @ Y @ ( set_int2 @ Xs2 ) ) ) ) ).

% set_ConsD
thf(fact_372_list_Oset__cases,axiom,
    ! [E: product_prod_int_int,A2: list_P5707943133018811711nt_int] :
      ( ( member5262025264175285858nt_int @ E @ ( set_Pr2470121279949933262nt_int @ A2 ) )
     => ( ! [Z22: list_P5707943133018811711nt_int] :
            ( A2
           != ( cons_P3334398858971670639nt_int @ E @ Z22 ) )
       => ~ ! [Z1: product_prod_int_int,Z22: list_P5707943133018811711nt_int] :
              ( ( A2
                = ( cons_P3334398858971670639nt_int @ Z1 @ Z22 ) )
             => ~ ( member5262025264175285858nt_int @ E @ ( set_Pr2470121279949933262nt_int @ Z22 ) ) ) ) ) ).

% list.set_cases
thf(fact_373_list_Oset__cases,axiom,
    ! [E: int,A2: list_int] :
      ( ( member_int2 @ E @ ( set_int2 @ A2 ) )
     => ( ! [Z22: list_int] :
            ( A2
           != ( cons_int @ E @ Z22 ) )
       => ~ ! [Z1: int,Z22: list_int] :
              ( ( A2
                = ( cons_int @ Z1 @ Z22 ) )
             => ~ ( member_int2 @ E @ ( set_int2 @ Z22 ) ) ) ) ) ).

% list.set_cases
thf(fact_374_list_Oset__intros_I1_J,axiom,
    ! [X21: product_prod_int_int,X222: list_P5707943133018811711nt_int] : ( member5262025264175285858nt_int @ X21 @ ( set_Pr2470121279949933262nt_int @ ( cons_P3334398858971670639nt_int @ X21 @ X222 ) ) ) ).

% list.set_intros(1)
thf(fact_375_list_Oset__intros_I1_J,axiom,
    ! [X21: int,X222: list_int] : ( member_int2 @ X21 @ ( set_int2 @ ( cons_int @ X21 @ X222 ) ) ) ).

% list.set_intros(1)
thf(fact_376_list_Oset__intros_I2_J,axiom,
    ! [Y: product_prod_int_int,X222: list_P5707943133018811711nt_int,X21: product_prod_int_int] :
      ( ( member5262025264175285858nt_int @ Y @ ( set_Pr2470121279949933262nt_int @ X222 ) )
     => ( member5262025264175285858nt_int @ Y @ ( set_Pr2470121279949933262nt_int @ ( cons_P3334398858971670639nt_int @ X21 @ X222 ) ) ) ) ).

% list.set_intros(2)
thf(fact_377_list_Oset__intros_I2_J,axiom,
    ! [Y: int,X222: list_int,X21: int] :
      ( ( member_int2 @ Y @ ( set_int2 @ X222 ) )
     => ( member_int2 @ Y @ ( set_int2 @ ( cons_int @ X21 @ X222 ) ) ) ) ).

% list.set_intros(2)
thf(fact_378_fst__graph__eq__dom,axiom,
    ! [M2: int > option_int] :
      ( ( image_5042161079198086560nt_int @ product_fst_int_int @ ( graph_int_int @ M2 ) )
      = ( dom_int_int @ M2 ) ) ).

% fst_graph_eq_dom
thf(fact_379_map__eq__Cons__conv,axiom,
    ! [F: int > int,Xs2: list_int,Y: int,Ys: list_int] :
      ( ( ( map_int_int @ F @ Xs2 )
        = ( cons_int @ Y @ Ys ) )
      = ( ? [Z4: int,Zs3: list_int] :
            ( ( Xs2
              = ( cons_int @ Z4 @ Zs3 ) )
            & ( ( F @ Z4 )
              = Y )
            & ( ( map_int_int @ F @ Zs3 )
              = Ys ) ) ) ) ).

% map_eq_Cons_conv
thf(fact_380_Cons__eq__map__conv,axiom,
    ! [X: int,Xs2: list_int,F: int > int,Ys: list_int] :
      ( ( ( cons_int @ X @ Xs2 )
        = ( map_int_int @ F @ Ys ) )
      = ( ? [Z4: int,Zs3: list_int] :
            ( ( Ys
              = ( cons_int @ Z4 @ Zs3 ) )
            & ( X
              = ( F @ Z4 ) )
            & ( Xs2
              = ( map_int_int @ F @ Zs3 ) ) ) ) ) ).

% Cons_eq_map_conv
thf(fact_381_map__eq__Cons__D,axiom,
    ! [F: int > int,Xs2: list_int,Y: int,Ys: list_int] :
      ( ( ( map_int_int @ F @ Xs2 )
        = ( cons_int @ Y @ Ys ) )
     => ? [Z3: int,Zs2: list_int] :
          ( ( Xs2
            = ( cons_int @ Z3 @ Zs2 ) )
          & ( ( F @ Z3 )
            = Y )
          & ( ( map_int_int @ F @ Zs2 )
            = Ys ) ) ) ).

% map_eq_Cons_D
thf(fact_382_Cons__eq__map__D,axiom,
    ! [X: int,Xs2: list_int,F: int > int,Ys: list_int] :
      ( ( ( cons_int @ X @ Xs2 )
        = ( map_int_int @ F @ Ys ) )
     => ? [Z3: int,Zs2: list_int] :
          ( ( Ys
            = ( cons_int @ Z3 @ Zs2 ) )
          & ( X
            = ( F @ Z3 ) )
          & ( Xs2
            = ( map_int_int @ F @ Zs2 ) ) ) ) ).

% Cons_eq_map_D
thf(fact_383_list_Osimps_I9_J,axiom,
    ! [F: int > int,X21: int,X222: list_int] :
      ( ( map_int_int @ F @ ( cons_int @ X21 @ X222 ) )
      = ( cons_int @ ( F @ X21 ) @ ( map_int_int @ F @ X222 ) ) ) ).

% list.simps(9)
thf(fact_384_distinct__length__2__or__more,axiom,
    ! [A2: int,B: int,Xs2: list_int] :
      ( ( distinct_int @ ( cons_int @ A2 @ ( cons_int @ B @ Xs2 ) ) )
      = ( ( A2 != B )
        & ( distinct_int @ ( cons_int @ A2 @ Xs2 ) )
        & ( distinct_int @ ( cons_int @ B @ Xs2 ) ) ) ) ).

% distinct_length_2_or_more
thf(fact_385_splice_Osimps_I2_J,axiom,
    ! [X: int,Xs2: list_int,Ys: list_int] :
      ( ( splice_int @ ( cons_int @ X @ Xs2 ) @ Ys )
      = ( cons_int @ X @ ( splice_int @ Ys @ Xs2 ) ) ) ).

% splice.simps(2)
thf(fact_386_map__of_Osimps_I2_J,axiom,
    ! [P2: product_prod_int_int,Ps2: list_P5707943133018811711nt_int] :
      ( ( map_of_int_int @ ( cons_P3334398858971670639nt_int @ P2 @ Ps2 ) )
      = ( fun_up8666045135305973159on_int @ ( map_of_int_int @ Ps2 ) @ ( product_fst_int_int @ P2 ) @ ( some_int @ ( product_snd_int_int @ P2 ) ) ) ) ).

% map_of.simps(2)
thf(fact_387_member__rec_I1_J,axiom,
    ! [X: int,Xs2: list_int,Y: int] :
      ( ( member_int @ ( cons_int @ X @ Xs2 ) @ Y )
      = ( ( X = Y )
        | ( member_int @ Xs2 @ Y ) ) ) ).

% member_rec(1)
thf(fact_388_distinct_Osimps_I2_J,axiom,
    ! [X: product_prod_int_int,Xs2: list_P5707943133018811711nt_int] :
      ( ( distin3744728255968310194nt_int @ ( cons_P3334398858971670639nt_int @ X @ Xs2 ) )
      = ( ~ ( member5262025264175285858nt_int @ X @ ( set_Pr2470121279949933262nt_int @ Xs2 ) )
        & ( distin3744728255968310194nt_int @ Xs2 ) ) ) ).

% distinct.simps(2)
thf(fact_389_distinct_Osimps_I2_J,axiom,
    ! [X: int,Xs2: list_int] :
      ( ( distinct_int @ ( cons_int @ X @ Xs2 ) )
      = ( ~ ( member_int2 @ X @ ( set_int2 @ Xs2 ) )
        & ( distinct_int @ Xs2 ) ) ) ).

% distinct.simps(2)
thf(fact_390_List_Oinsert__def,axiom,
    ( insert5765537519290168021nt_int
    = ( ^ [X3: product_prod_int_int,Xs3: list_P5707943133018811711nt_int] : ( if_lis8883190402267401221nt_int @ ( member5262025264175285858nt_int @ X3 @ ( set_Pr2470121279949933262nt_int @ Xs3 ) ) @ Xs3 @ ( cons_P3334398858971670639nt_int @ X3 @ Xs3 ) ) ) ) ).

% List.insert_def
thf(fact_391_List_Oinsert__def,axiom,
    ( insert_int
    = ( ^ [X3: int,Xs3: list_int] : ( if_list_int @ ( member_int2 @ X3 @ ( set_int2 @ Xs3 ) ) @ Xs3 @ ( cons_int @ X3 @ Xs3 ) ) ) ) ).

% List.insert_def
thf(fact_392_Cons__in__subseqsD,axiom,
    ! [Y: int,Ys: list_int,Xs2: list_int] :
      ( ( member_list_int2 @ ( cons_int @ Y @ Ys ) @ ( set_list_int2 @ ( subseqs_int @ Xs2 ) ) )
     => ( member_list_int2 @ Ys @ ( set_list_int2 @ ( subseqs_int @ Xs2 ) ) ) ) ).

% Cons_in_subseqsD
thf(fact_393_graph__ranD,axiom,
    ! [X: product_prod_int_int,M2: int > option_int] :
      ( ( member5262025264175285858nt_int @ X @ ( graph_int_int @ M2 ) )
     => ( member_int2 @ ( product_snd_int_int @ X ) @ ( ran_int_int @ M2 ) ) ) ).

% graph_ranD
thf(fact_394_map__of__SomeD,axiom,
    ! [Xs2: list_P5707943133018811711nt_int,K: int,Y: int] :
      ( ( ( map_of_int_int @ Xs2 @ K )
        = ( some_int @ Y ) )
     => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ K @ Y ) @ ( set_Pr2470121279949933262nt_int @ Xs2 ) ) ) ).

% map_of_SomeD
thf(fact_395_weak__map__of__SomeI,axiom,
    ! [K: int,X: int,L: list_P5707943133018811711nt_int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ K @ X ) @ ( set_Pr2470121279949933262nt_int @ L ) )
     => ? [X2: int] :
          ( ( map_of_int_int @ L @ K )
          = ( some_int @ X2 ) ) ) ).

% weak_map_of_SomeI
thf(fact_396_map__of__zip__upd,axiom,
    ! [Ys: list_P5707943133018811711nt_int,Xs2: list_int,Zs: list_P5707943133018811711nt_int,X: int,Y: product_prod_int_int,Z: product_prod_int_int] :
      ( ( ( size_s5157815400016825771nt_int @ Ys )
        = ( size_size_list_int @ Xs2 ) )
     => ( ( ( size_s5157815400016825771nt_int @ Zs )
          = ( size_size_list_int @ Xs2 ) )
       => ( ~ ( member_int2 @ X @ ( set_int2 @ Xs2 ) )
         => ( ( ( fun_up1437153861220271380nt_int @ ( map_of7676300617449379688nt_int @ ( zip_in865470750896630164nt_int @ Xs2 @ Ys ) ) @ X @ ( some_P4184893108420464158nt_int @ Y ) )
              = ( fun_up1437153861220271380nt_int @ ( map_of7676300617449379688nt_int @ ( zip_in865470750896630164nt_int @ Xs2 @ Zs ) ) @ X @ ( some_P4184893108420464158nt_int @ Z ) ) )
           => ( ( map_of7676300617449379688nt_int @ ( zip_in865470750896630164nt_int @ Xs2 @ Ys ) )
              = ( map_of7676300617449379688nt_int @ ( zip_in865470750896630164nt_int @ Xs2 @ Zs ) ) ) ) ) ) ) ).

% map_of_zip_upd
thf(fact_397_map__of__zip__upd,axiom,
    ! [Ys: list_P5707943133018811711nt_int,Xs2: list_P5707943133018811711nt_int,Zs: list_P5707943133018811711nt_int,X: product_prod_int_int,Y: product_prod_int_int,Z: product_prod_int_int] :
      ( ( ( size_s5157815400016825771nt_int @ Ys )
        = ( size_s5157815400016825771nt_int @ Xs2 ) )
     => ( ( ( size_s5157815400016825771nt_int @ Zs )
          = ( size_s5157815400016825771nt_int @ Xs2 ) )
       => ( ~ ( member5262025264175285858nt_int @ X @ ( set_Pr2470121279949933262nt_int @ Xs2 ) )
         => ( ( ( fun_up7774278882650222851nt_int @ ( map_of5841684995557474595nt_int @ ( zip_Pr2148635498163357687nt_int @ Xs2 @ Ys ) ) @ X @ ( some_P4184893108420464158nt_int @ Y ) )
              = ( fun_up7774278882650222851nt_int @ ( map_of5841684995557474595nt_int @ ( zip_Pr2148635498163357687nt_int @ Xs2 @ Zs ) ) @ X @ ( some_P4184893108420464158nt_int @ Z ) ) )
           => ( ( map_of5841684995557474595nt_int @ ( zip_Pr2148635498163357687nt_int @ Xs2 @ Ys ) )
              = ( map_of5841684995557474595nt_int @ ( zip_Pr2148635498163357687nt_int @ Xs2 @ Zs ) ) ) ) ) ) ) ).

% map_of_zip_upd
thf(fact_398_map__of__zip__is__Some,axiom,
    ! [Xs2: list_int,Ys: list_P5707943133018811711nt_int,X: int] :
      ( ( ( size_size_list_int @ Xs2 )
        = ( size_s5157815400016825771nt_int @ Ys ) )
     => ( ( member_int2 @ X @ ( set_int2 @ Xs2 ) )
        = ( ? [Y4: product_prod_int_int] :
              ( ( map_of7676300617449379688nt_int @ ( zip_in865470750896630164nt_int @ Xs2 @ Ys ) @ X )
              = ( some_P4184893108420464158nt_int @ Y4 ) ) ) ) ) ).

% map_of_zip_is_Some
thf(fact_399_map__of__zip__is__Some,axiom,
    ! [Xs2: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int,X: product_prod_int_int] :
      ( ( ( size_s5157815400016825771nt_int @ Xs2 )
        = ( size_s5157815400016825771nt_int @ Ys ) )
     => ( ( member5262025264175285858nt_int @ X @ ( set_Pr2470121279949933262nt_int @ Xs2 ) )
        = ( ? [Y4: product_prod_int_int] :
              ( ( map_of5841684995557474595nt_int @ ( zip_Pr2148635498163357687nt_int @ Xs2 @ Ys ) @ X )
              = ( some_P4184893108420464158nt_int @ Y4 ) ) ) ) ) ).

% map_of_zip_is_Some
thf(fact_400_dom__map__of__conv__image__fst,axiom,
    ! [Xys: list_P5707943133018811711nt_int] :
      ( ( dom_int_int @ ( map_of_int_int @ Xys ) )
      = ( image_5042161079198086560nt_int @ product_fst_int_int @ ( set_Pr2470121279949933262nt_int @ Xys ) ) ) ).

% dom_map_of_conv_image_fst
thf(fact_401_snd__graph__ran,axiom,
    ! [M2: int > option_int] :
      ( ( image_5042161079198086560nt_int @ product_snd_int_int @ ( graph_int_int @ M2 ) )
      = ( ran_int_int @ M2 ) ) ).

% snd_graph_ran
thf(fact_402_map__of__eq__None__iff,axiom,
    ! [Xys: list_P5707943133018811711nt_int,X: int] :
      ( ( ( map_of_int_int @ Xys @ X )
        = none_int )
      = ( ~ ( member_int2 @ X @ ( image_5042161079198086560nt_int @ product_fst_int_int @ ( set_Pr2470121279949933262nt_int @ Xys ) ) ) ) ) ).

% map_of_eq_None_iff
thf(fact_403_Cons__lenlex__iff,axiom,
    ! [M2: int,Ms: list_int,N: int,Ns: list_int,R: set_Pr958786334691620121nt_int] :
      ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( cons_int @ M2 @ Ms ) @ ( cons_int @ N @ Ns ) ) @ ( lenlex_int @ R ) )
      = ( ( ord_less_nat @ ( size_size_list_int @ Ms ) @ ( size_size_list_int @ Ns ) )
        | ( ( ( size_size_list_int @ Ms )
            = ( size_size_list_int @ Ns ) )
          & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ M2 @ N ) @ R ) )
        | ( ( M2 = N )
          & ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Ms @ Ns ) @ ( lenlex_int @ R ) ) ) ) ) ).

% Cons_lenlex_iff
thf(fact_404_Cons__lenlex__iff,axiom,
    ! [M2: product_prod_int_int,Ms: list_P5707943133018811711nt_int,N: product_prod_int_int,Ns: list_P5707943133018811711nt_int,R: set_Pr2560585780119916871nt_int] :
      ( ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ ( cons_P3334398858971670639nt_int @ M2 @ Ms ) @ ( cons_P3334398858971670639nt_int @ N @ Ns ) ) @ ( lenlex6370358691973319492nt_int @ R ) )
      = ( ( ord_less_nat @ ( size_s5157815400016825771nt_int @ Ms ) @ ( size_s5157815400016825771nt_int @ Ns ) )
        | ( ( ( size_s5157815400016825771nt_int @ Ms )
            = ( size_s5157815400016825771nt_int @ Ns ) )
          & ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ M2 @ N ) @ R ) )
        | ( ( M2 = N )
          & ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ Ms @ Ns ) @ ( lenlex6370358691973319492nt_int @ R ) ) ) ) ) ).

% Cons_lenlex_iff
thf(fact_405_map__upd__upds__conv__if,axiom,
    ! [X: int,Ys: list_P5707943133018811711nt_int,Xs2: list_int,F: int > option4624381673175914239nt_int,Y: product_prod_int_int] :
      ( ( ( member_int2 @ X @ ( set_int2 @ ( take_int @ ( size_s5157815400016825771nt_int @ Ys ) @ Xs2 ) ) )
       => ( ( map_up1450196046888760469nt_int @ ( fun_up1437153861220271380nt_int @ F @ X @ ( some_P4184893108420464158nt_int @ Y ) ) @ Xs2 @ Ys )
          = ( map_up1450196046888760469nt_int @ F @ Xs2 @ Ys ) ) )
      & ( ~ ( member_int2 @ X @ ( set_int2 @ ( take_int @ ( size_s5157815400016825771nt_int @ Ys ) @ Xs2 ) ) )
       => ( ( map_up1450196046888760469nt_int @ ( fun_up1437153861220271380nt_int @ F @ X @ ( some_P4184893108420464158nt_int @ Y ) ) @ Xs2 @ Ys )
          = ( fun_up1437153861220271380nt_int @ ( map_up1450196046888760469nt_int @ F @ Xs2 @ Ys ) @ X @ ( some_P4184893108420464158nt_int @ Y ) ) ) ) ) ).

% map_upd_upds_conv_if
thf(fact_406_map__upd__upds__conv__if,axiom,
    ! [X: product_prod_int_int,Ys: list_P5707943133018811711nt_int,Xs2: list_P5707943133018811711nt_int,F: product_prod_int_int > option4624381673175914239nt_int,Y: product_prod_int_int] :
      ( ( ( member5262025264175285858nt_int @ X @ ( set_Pr2470121279949933262nt_int @ ( take_P8218740963776755879nt_int @ ( size_s5157815400016825771nt_int @ Ys ) @ Xs2 ) ) )
       => ( ( map_up6499465237031507894nt_int @ ( fun_up7774278882650222851nt_int @ F @ X @ ( some_P4184893108420464158nt_int @ Y ) ) @ Xs2 @ Ys )
          = ( map_up6499465237031507894nt_int @ F @ Xs2 @ Ys ) ) )
      & ( ~ ( member5262025264175285858nt_int @ X @ ( set_Pr2470121279949933262nt_int @ ( take_P8218740963776755879nt_int @ ( size_s5157815400016825771nt_int @ Ys ) @ Xs2 ) ) )
       => ( ( map_up6499465237031507894nt_int @ ( fun_up7774278882650222851nt_int @ F @ X @ ( some_P4184893108420464158nt_int @ Y ) ) @ Xs2 @ Ys )
          = ( fun_up7774278882650222851nt_int @ ( map_up6499465237031507894nt_int @ F @ Xs2 @ Ys ) @ X @ ( some_P4184893108420464158nt_int @ Y ) ) ) ) ) ).

% map_upd_upds_conv_if
thf(fact_407_length__product,axiom,
    ! [Xs2: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int] :
      ( ( size_s6770063216428074713nt_int @ ( produc1028813369992947845nt_int @ Xs2 @ Ys ) )
      = ( times_times_nat @ ( size_s5157815400016825771nt_int @ Xs2 ) @ ( size_s5157815400016825771nt_int @ Ys ) ) ) ).

% length_product
thf(fact_408_length__product,axiom,
    ! [Xs2: list_int,Ys: list_int] :
      ( ( size_s5157815400016825771nt_int @ ( product_int_int @ Xs2 @ Ys ) )
      = ( times_times_nat @ ( size_size_list_int @ Xs2 ) @ ( size_size_list_int @ Ys ) ) ) ).

% length_product
thf(fact_409_length__n__lists__elem,axiom,
    ! [Ys: list_P5707943133018811711nt_int,N: nat,Xs2: list_P5707943133018811711nt_int] :
      ( ( member2764346250752101224nt_int @ Ys @ ( set_li2659200638379878868nt_int @ ( n_list2671089462871817415nt_int @ N @ Xs2 ) ) )
     => ( ( size_s5157815400016825771nt_int @ Ys )
        = N ) ) ).

% length_n_lists_elem
thf(fact_410_insertCI,axiom,
    ! [A2: product_prod_int_int,B2: set_Pr958786334691620121nt_int,B: product_prod_int_int] :
      ( ( ~ ( member5262025264175285858nt_int @ A2 @ B2 )
       => ( A2 = B ) )
     => ( member5262025264175285858nt_int @ A2 @ ( insert5033312907999012233nt_int @ B @ B2 ) ) ) ).

% insertCI
thf(fact_411_insertCI,axiom,
    ! [A2: int,B2: set_int,B: int] :
      ( ( ~ ( member_int2 @ A2 @ B2 )
       => ( A2 = B ) )
     => ( member_int2 @ A2 @ ( insert_int2 @ B @ B2 ) ) ) ).

% insertCI
thf(fact_412_insert__iff,axiom,
    ! [A2: product_prod_int_int,B: product_prod_int_int,A: set_Pr958786334691620121nt_int] :
      ( ( member5262025264175285858nt_int @ A2 @ ( insert5033312907999012233nt_int @ B @ A ) )
      = ( ( A2 = B )
        | ( member5262025264175285858nt_int @ A2 @ A ) ) ) ).

% insert_iff
thf(fact_413_insert__iff,axiom,
    ! [A2: int,B: int,A: set_int] :
      ( ( member_int2 @ A2 @ ( insert_int2 @ B @ A ) )
      = ( ( A2 = B )
        | ( member_int2 @ A2 @ A ) ) ) ).

% insert_iff
thf(fact_414_not__gr__zero,axiom,
    ! [N: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
      = ( N = zero_zero_nat ) ) ).

% not_gr_zero
thf(fact_415_add__less__cancel__left,axiom,
    ! [C: nat,A2: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) )
      = ( ord_less_nat @ A2 @ B ) ) ).

% add_less_cancel_left
thf(fact_416_add__less__cancel__left,axiom,
    ! [C: int,A2: int,B: int] :
      ( ( ord_less_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B ) )
      = ( ord_less_int @ A2 @ B ) ) ).

% add_less_cancel_left
thf(fact_417_add__less__cancel__right,axiom,
    ! [A2: nat,C: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) )
      = ( ord_less_nat @ A2 @ B ) ) ).

% add_less_cancel_right
thf(fact_418_add__less__cancel__right,axiom,
    ! [A2: int,C: int,B: int] :
      ( ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B @ C ) )
      = ( ord_less_int @ A2 @ B ) ) ).

% add_less_cancel_right
thf(fact_419_image__insert,axiom,
    ! [F: product_prod_int_int > int,A2: product_prod_int_int,B2: set_Pr958786334691620121nt_int] :
      ( ( image_5042161079198086560nt_int @ F @ ( insert5033312907999012233nt_int @ A2 @ B2 ) )
      = ( insert_int2 @ ( F @ A2 ) @ ( image_5042161079198086560nt_int @ F @ B2 ) ) ) ).

% image_insert
thf(fact_420_insert__image,axiom,
    ! [X: product_prod_int_int,A: set_Pr958786334691620121nt_int,F: product_prod_int_int > int] :
      ( ( member5262025264175285858nt_int @ X @ A )
     => ( ( insert_int2 @ ( F @ X ) @ ( image_5042161079198086560nt_int @ F @ A ) )
        = ( image_5042161079198086560nt_int @ F @ A ) ) ) ).

% insert_image
thf(fact_421_less__nat__zero__code,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% less_nat_zero_code
thf(fact_422_neq0__conv,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
      = ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% neq0_conv
thf(fact_423_bot__nat__0_Onot__eq__extremum,axiom,
    ! [A2: nat] :
      ( ( A2 != zero_zero_nat )
      = ( ord_less_nat @ zero_zero_nat @ A2 ) ) ).

% bot_nat_0.not_eq_extremum
thf(fact_424_nat__add__left__cancel__less,axiom,
    ! [K: nat,M2: nat,N: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N ) )
      = ( ord_less_nat @ M2 @ N ) ) ).

% nat_add_left_cancel_less
thf(fact_425_mult__cancel2,axiom,
    ! [M2: nat,K: nat,N: nat] :
      ( ( ( times_times_nat @ M2 @ K )
        = ( times_times_nat @ N @ K ) )
      = ( ( M2 = N )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel2
thf(fact_426_mult__cancel1,axiom,
    ! [K: nat,M2: nat,N: nat] :
      ( ( ( times_times_nat @ K @ M2 )
        = ( times_times_nat @ K @ N ) )
      = ( ( M2 = N )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel1
thf(fact_427_mult__0__right,axiom,
    ! [M2: nat] :
      ( ( times_times_nat @ M2 @ zero_zero_nat )
      = zero_zero_nat ) ).

% mult_0_right
thf(fact_428_mult__is__0,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( times_times_nat @ M2 @ N )
        = zero_zero_nat )
      = ( ( M2 = zero_zero_nat )
        | ( N = zero_zero_nat ) ) ) ).

% mult_is_0
thf(fact_429_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A2: int] :
      ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A2 @ A2 ) )
      = ( ord_less_int @ zero_zero_int @ A2 ) ) ).

% zero_less_double_add_iff_zero_less_single_add
thf(fact_430_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A2: int] :
      ( ( ord_less_int @ ( plus_plus_int @ A2 @ A2 ) @ zero_zero_int )
      = ( ord_less_int @ A2 @ zero_zero_int ) ) ).

% double_add_less_zero_iff_single_add_less_zero
thf(fact_431_less__add__same__cancel2,axiom,
    ! [A2: nat,B: nat] :
      ( ( ord_less_nat @ A2 @ ( plus_plus_nat @ B @ A2 ) )
      = ( ord_less_nat @ zero_zero_nat @ B ) ) ).

% less_add_same_cancel2
thf(fact_432_less__add__same__cancel2,axiom,
    ! [A2: int,B: int] :
      ( ( ord_less_int @ A2 @ ( plus_plus_int @ B @ A2 ) )
      = ( ord_less_int @ zero_zero_int @ B ) ) ).

% less_add_same_cancel2
thf(fact_433_less__add__same__cancel1,axiom,
    ! [A2: nat,B: nat] :
      ( ( ord_less_nat @ A2 @ ( plus_plus_nat @ A2 @ B ) )
      = ( ord_less_nat @ zero_zero_nat @ B ) ) ).

% less_add_same_cancel1
thf(fact_434_less__add__same__cancel1,axiom,
    ! [A2: int,B: int] :
      ( ( ord_less_int @ A2 @ ( plus_plus_int @ A2 @ B ) )
      = ( ord_less_int @ zero_zero_int @ B ) ) ).

% less_add_same_cancel1
thf(fact_435_add__less__same__cancel2,axiom,
    ! [A2: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A2 @ B ) @ B )
      = ( ord_less_nat @ A2 @ zero_zero_nat ) ) ).

% add_less_same_cancel2
thf(fact_436_add__less__same__cancel2,axiom,
    ! [A2: int,B: int] :
      ( ( ord_less_int @ ( plus_plus_int @ A2 @ B ) @ B )
      = ( ord_less_int @ A2 @ zero_zero_int ) ) ).

% add_less_same_cancel2
thf(fact_437_add__less__same__cancel1,axiom,
    ! [B: nat,A2: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ B @ A2 ) @ B )
      = ( ord_less_nat @ A2 @ zero_zero_nat ) ) ).

% add_less_same_cancel1
thf(fact_438_add__less__same__cancel1,axiom,
    ! [B: int,A2: int] :
      ( ( ord_less_int @ ( plus_plus_int @ B @ A2 ) @ B )
      = ( ord_less_int @ A2 @ zero_zero_int ) ) ).

% add_less_same_cancel1
thf(fact_439_list_Osimps_I15_J,axiom,
    ! [X21: product_prod_int_int,X222: list_P5707943133018811711nt_int] :
      ( ( set_Pr2470121279949933262nt_int @ ( cons_P3334398858971670639nt_int @ X21 @ X222 ) )
      = ( insert5033312907999012233nt_int @ X21 @ ( set_Pr2470121279949933262nt_int @ X222 ) ) ) ).

% list.simps(15)
thf(fact_440_list_Osimps_I15_J,axiom,
    ! [X21: int,X222: list_int] :
      ( ( set_int2 @ ( cons_int @ X21 @ X222 ) )
      = ( insert_int2 @ X21 @ ( set_int2 @ X222 ) ) ) ).

% list.simps(15)
thf(fact_441_add__gr__0,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M2 @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ M2 )
        | ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% add_gr_0
thf(fact_442_mult__less__cancel2,axiom,
    ! [M2: nat,K: nat,N: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N @ K ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K )
        & ( ord_less_nat @ M2 @ N ) ) ) ).

% mult_less_cancel2
thf(fact_443_nat__0__less__mult__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M2 @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ M2 )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% nat_0_less_mult_iff
thf(fact_444_List_Oset__insert,axiom,
    ! [X: product_prod_int_int,Xs2: list_P5707943133018811711nt_int] :
      ( ( set_Pr2470121279949933262nt_int @ ( insert5765537519290168021nt_int @ X @ Xs2 ) )
      = ( insert5033312907999012233nt_int @ X @ ( set_Pr2470121279949933262nt_int @ Xs2 ) ) ) ).

% List.set_insert
thf(fact_445_verit__comp__simplify1_I1_J,axiom,
    ! [A2: nat] :
      ~ ( ord_less_nat @ A2 @ A2 ) ).

% verit_comp_simplify1(1)
thf(fact_446_verit__comp__simplify1_I1_J,axiom,
    ! [A2: int] :
      ~ ( ord_less_int @ A2 @ A2 ) ).

% verit_comp_simplify1(1)
thf(fact_447_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( ( times_times_nat @ ( times_times_nat @ A2 @ B ) @ C )
      = ( times_times_nat @ A2 @ ( times_times_nat @ B @ C ) ) ) ).

% ab_semigroup_mult_class.mult_ac(1)
thf(fact_448_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A2: int,B: int,C: int] :
      ( ( times_times_int @ ( times_times_int @ A2 @ B ) @ C )
      = ( times_times_int @ A2 @ ( times_times_int @ B @ C ) ) ) ).

% ab_semigroup_mult_class.mult_ac(1)
thf(fact_449_mult_Oassoc,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( ( times_times_nat @ ( times_times_nat @ A2 @ B ) @ C )
      = ( times_times_nat @ A2 @ ( times_times_nat @ B @ C ) ) ) ).

% mult.assoc
thf(fact_450_mult_Oassoc,axiom,
    ! [A2: int,B: int,C: int] :
      ( ( times_times_int @ ( times_times_int @ A2 @ B ) @ C )
      = ( times_times_int @ A2 @ ( times_times_int @ B @ C ) ) ) ).

% mult.assoc
thf(fact_451_mult_Ocommute,axiom,
    ( times_times_nat
    = ( ^ [A3: nat,B3: nat] : ( times_times_nat @ B3 @ A3 ) ) ) ).

% mult.commute
thf(fact_452_mult_Ocommute,axiom,
    ( times_times_int
    = ( ^ [A3: int,B3: int] : ( times_times_int @ B3 @ A3 ) ) ) ).

% mult.commute
thf(fact_453_mult_Oleft__commute,axiom,
    ! [B: nat,A2: nat,C: nat] :
      ( ( times_times_nat @ B @ ( times_times_nat @ A2 @ C ) )
      = ( times_times_nat @ A2 @ ( times_times_nat @ B @ C ) ) ) ).

% mult.left_commute
thf(fact_454_mult_Oleft__commute,axiom,
    ! [B: int,A2: int,C: int] :
      ( ( times_times_int @ B @ ( times_times_int @ A2 @ C ) )
      = ( times_times_int @ A2 @ ( times_times_int @ B @ C ) ) ) ).

% mult.left_commute
thf(fact_455_insertE,axiom,
    ! [A2: product_prod_int_int,B: product_prod_int_int,A: set_Pr958786334691620121nt_int] :
      ( ( member5262025264175285858nt_int @ A2 @ ( insert5033312907999012233nt_int @ B @ A ) )
     => ( ( A2 != B )
       => ( member5262025264175285858nt_int @ A2 @ A ) ) ) ).

% insertE
thf(fact_456_insertE,axiom,
    ! [A2: int,B: int,A: set_int] :
      ( ( member_int2 @ A2 @ ( insert_int2 @ B @ A ) )
     => ( ( A2 != B )
       => ( member_int2 @ A2 @ A ) ) ) ).

% insertE
thf(fact_457_insertI1,axiom,
    ! [A2: product_prod_int_int,B2: set_Pr958786334691620121nt_int] : ( member5262025264175285858nt_int @ A2 @ ( insert5033312907999012233nt_int @ A2 @ B2 ) ) ).

% insertI1
thf(fact_458_insertI1,axiom,
    ! [A2: int,B2: set_int] : ( member_int2 @ A2 @ ( insert_int2 @ A2 @ B2 ) ) ).

% insertI1
thf(fact_459_insertI2,axiom,
    ! [A2: product_prod_int_int,B2: set_Pr958786334691620121nt_int,B: product_prod_int_int] :
      ( ( member5262025264175285858nt_int @ A2 @ B2 )
     => ( member5262025264175285858nt_int @ A2 @ ( insert5033312907999012233nt_int @ B @ B2 ) ) ) ).

% insertI2
thf(fact_460_insertI2,axiom,
    ! [A2: int,B2: set_int,B: int] :
      ( ( member_int2 @ A2 @ B2 )
     => ( member_int2 @ A2 @ ( insert_int2 @ B @ B2 ) ) ) ).

% insertI2
thf(fact_461_Set_Oset__insert,axiom,
    ! [X: product_prod_int_int,A: set_Pr958786334691620121nt_int] :
      ( ( member5262025264175285858nt_int @ X @ A )
     => ~ ! [B6: set_Pr958786334691620121nt_int] :
            ( ( A
              = ( insert5033312907999012233nt_int @ X @ B6 ) )
           => ( member5262025264175285858nt_int @ X @ B6 ) ) ) ).

% Set.set_insert
thf(fact_462_Set_Oset__insert,axiom,
    ! [X: int,A: set_int] :
      ( ( member_int2 @ X @ A )
     => ~ ! [B6: set_int] :
            ( ( A
              = ( insert_int2 @ X @ B6 ) )
           => ( member_int2 @ X @ B6 ) ) ) ).

% Set.set_insert
thf(fact_463_nat__neq__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( M2 != N )
      = ( ( ord_less_nat @ M2 @ N )
        | ( ord_less_nat @ N @ M2 ) ) ) ).

% nat_neq_iff
thf(fact_464_insert__ident,axiom,
    ! [X: product_prod_int_int,A: set_Pr958786334691620121nt_int,B2: set_Pr958786334691620121nt_int] :
      ( ~ ( member5262025264175285858nt_int @ X @ A )
     => ( ~ ( member5262025264175285858nt_int @ X @ B2 )
       => ( ( ( insert5033312907999012233nt_int @ X @ A )
            = ( insert5033312907999012233nt_int @ X @ B2 ) )
          = ( A = B2 ) ) ) ) ).

% insert_ident
thf(fact_465_insert__ident,axiom,
    ! [X: int,A: set_int,B2: set_int] :
      ( ~ ( member_int2 @ X @ A )
     => ( ~ ( member_int2 @ X @ B2 )
       => ( ( ( insert_int2 @ X @ A )
            = ( insert_int2 @ X @ B2 ) )
          = ( A = B2 ) ) ) ) ).

% insert_ident
thf(fact_466_less__not__refl,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ N ) ).

% less_not_refl
thf(fact_467_insert__absorb,axiom,
    ! [A2: product_prod_int_int,A: set_Pr958786334691620121nt_int] :
      ( ( member5262025264175285858nt_int @ A2 @ A )
     => ( ( insert5033312907999012233nt_int @ A2 @ A )
        = A ) ) ).

% insert_absorb
thf(fact_468_insert__absorb,axiom,
    ! [A2: int,A: set_int] :
      ( ( member_int2 @ A2 @ A )
     => ( ( insert_int2 @ A2 @ A )
        = A ) ) ).

% insert_absorb
thf(fact_469_insert__eq__iff,axiom,
    ! [A2: product_prod_int_int,A: set_Pr958786334691620121nt_int,B: product_prod_int_int,B2: set_Pr958786334691620121nt_int] :
      ( ~ ( member5262025264175285858nt_int @ A2 @ A )
     => ( ~ ( member5262025264175285858nt_int @ B @ B2 )
       => ( ( ( insert5033312907999012233nt_int @ A2 @ A )
            = ( insert5033312907999012233nt_int @ B @ B2 ) )
          = ( ( ( A2 = B )
             => ( A = B2 ) )
            & ( ( A2 != B )
             => ? [C4: set_Pr958786334691620121nt_int] :
                  ( ( A
                    = ( insert5033312907999012233nt_int @ B @ C4 ) )
                  & ~ ( member5262025264175285858nt_int @ B @ C4 )
                  & ( B2
                    = ( insert5033312907999012233nt_int @ A2 @ C4 ) )
                  & ~ ( member5262025264175285858nt_int @ A2 @ C4 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_470_insert__eq__iff,axiom,
    ! [A2: int,A: set_int,B: int,B2: set_int] :
      ( ~ ( member_int2 @ A2 @ A )
     => ( ~ ( member_int2 @ B @ B2 )
       => ( ( ( insert_int2 @ A2 @ A )
            = ( insert_int2 @ B @ B2 ) )
          = ( ( ( A2 = B )
             => ( A = B2 ) )
            & ( ( A2 != B )
             => ? [C4: set_int] :
                  ( ( A
                    = ( insert_int2 @ B @ C4 ) )
                  & ~ ( member_int2 @ B @ C4 )
                  & ( B2
                    = ( insert_int2 @ A2 @ C4 ) )
                  & ~ ( member_int2 @ A2 @ C4 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_471_less__not__refl2,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ N @ M2 )
     => ( M2 != N ) ) ).

% less_not_refl2
thf(fact_472_less__not__refl3,axiom,
    ! [S3: nat,T2: nat] :
      ( ( ord_less_nat @ S3 @ T2 )
     => ( S3 != T2 ) ) ).

% less_not_refl3
thf(fact_473_less__irrefl__nat,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ N ) ).

% less_irrefl_nat
thf(fact_474_nat__less__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ! [N4: nat] :
          ( ! [M3: nat] :
              ( ( ord_less_nat @ M3 @ N4 )
             => ( P @ M3 ) )
         => ( P @ N4 ) )
     => ( P @ N ) ) ).

% nat_less_induct
thf(fact_475_infinite__descent,axiom,
    ! [P: nat > $o,N: nat] :
      ( ! [N4: nat] :
          ( ~ ( P @ N4 )
         => ? [M3: nat] :
              ( ( ord_less_nat @ M3 @ N4 )
              & ~ ( P @ M3 ) ) )
     => ( P @ N ) ) ).

% infinite_descent
thf(fact_476_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_477_mk__disjoint__insert,axiom,
    ! [A2: product_prod_int_int,A: set_Pr958786334691620121nt_int] :
      ( ( member5262025264175285858nt_int @ A2 @ A )
     => ? [B6: set_Pr958786334691620121nt_int] :
          ( ( A
            = ( insert5033312907999012233nt_int @ A2 @ B6 ) )
          & ~ ( member5262025264175285858nt_int @ A2 @ B6 ) ) ) ).

% mk_disjoint_insert
thf(fact_478_mk__disjoint__insert,axiom,
    ! [A2: int,A: set_int] :
      ( ( member_int2 @ A2 @ A )
     => ? [B6: set_int] :
          ( ( A
            = ( insert_int2 @ A2 @ B6 ) )
          & ~ ( member_int2 @ A2 @ B6 ) ) ) ).

% mk_disjoint_insert
thf(fact_479_mult__less__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ord_less_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).

% mult_less_mono1
thf(fact_480_mult__less__mono2,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ord_less_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ) ).

% mult_less_mono2
thf(fact_481_gr__zeroI,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% gr_zeroI
thf(fact_482_not__less__zero,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% not_less_zero
thf(fact_483_gr__implies__not__zero,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not_zero
thf(fact_484_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_485_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).

% less_numeral_extra(3)
thf(fact_486_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).

% less_numeral_extra(3)
thf(fact_487_crossproduct__eq,axiom,
    ! [W: nat,Y: nat,X: nat,Z: nat] :
      ( ( ( plus_plus_nat @ ( times_times_nat @ W @ Y ) @ ( times_times_nat @ X @ Z ) )
        = ( plus_plus_nat @ ( times_times_nat @ W @ Z ) @ ( times_times_nat @ X @ Y ) ) )
      = ( ( W = X )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
thf(fact_488_crossproduct__eq,axiom,
    ! [W: int,Y: int,X: int,Z: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ W @ Y ) @ ( times_times_int @ X @ Z ) )
        = ( plus_plus_int @ ( times_times_int @ W @ Z ) @ ( times_times_int @ X @ Y ) ) )
      = ( ( W = X )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
thf(fact_489_crossproduct__noteq,axiom,
    ! [A2: nat,B: nat,C: nat,D2: nat] :
      ( ( ( A2 != B )
        & ( C != D2 ) )
      = ( ( plus_plus_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ D2 ) )
       != ( plus_plus_nat @ ( times_times_nat @ A2 @ D2 ) @ ( times_times_nat @ B @ C ) ) ) ) ).

% crossproduct_noteq
thf(fact_490_crossproduct__noteq,axiom,
    ! [A2: int,B: int,C: int,D2: int] :
      ( ( ( A2 != B )
        & ( C != D2 ) )
      = ( ( plus_plus_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ D2 ) )
       != ( plus_plus_int @ ( times_times_int @ A2 @ D2 ) @ ( times_times_int @ B @ C ) ) ) ) ).

% crossproduct_noteq
thf(fact_491_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_nat @ I @ J )
        & ( ord_less_nat @ K @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_492_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( ord_less_int @ I @ J )
        & ( ord_less_int @ K @ L ) )
     => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_493_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( I = J )
        & ( ord_less_nat @ K @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_494_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( I = J )
        & ( ord_less_int @ K @ L ) )
     => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_495_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_nat @ I @ J )
        & ( K = L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_496_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( ord_less_int @ I @ J )
        & ( K = L ) )
     => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_497_add__strict__mono,axiom,
    ! [A2: nat,B: nat,C: nat,D2: nat] :
      ( ( ord_less_nat @ A2 @ B )
     => ( ( ord_less_nat @ C @ D2 )
       => ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).

% add_strict_mono
thf(fact_498_add__strict__mono,axiom,
    ! [A2: int,B: int,C: int,D2: int] :
      ( ( ord_less_int @ A2 @ B )
     => ( ( ord_less_int @ C @ D2 )
       => ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B @ D2 ) ) ) ) ).

% add_strict_mono
thf(fact_499_add__strict__left__mono,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A2 @ B )
     => ( ord_less_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) ) ) ).

% add_strict_left_mono
thf(fact_500_add__strict__left__mono,axiom,
    ! [A2: int,B: int,C: int] :
      ( ( ord_less_int @ A2 @ B )
     => ( ord_less_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B ) ) ) ).

% add_strict_left_mono
thf(fact_501_add__strict__right__mono,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A2 @ B )
     => ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).

% add_strict_right_mono
thf(fact_502_add__strict__right__mono,axiom,
    ! [A2: int,B: int,C: int] :
      ( ( ord_less_int @ A2 @ B )
     => ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).

% add_strict_right_mono
thf(fact_503_add__less__imp__less__left,axiom,
    ! [C: nat,A2: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) )
     => ( ord_less_nat @ A2 @ B ) ) ).

% add_less_imp_less_left
thf(fact_504_add__less__imp__less__left,axiom,
    ! [C: int,A2: int,B: int] :
      ( ( ord_less_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B ) )
     => ( ord_less_int @ A2 @ B ) ) ).

% add_less_imp_less_left
thf(fact_505_add__less__imp__less__right,axiom,
    ! [A2: nat,C: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) )
     => ( ord_less_nat @ A2 @ B ) ) ).

% add_less_imp_less_right
thf(fact_506_add__less__imp__less__right,axiom,
    ! [A2: int,C: int,B: int] :
      ( ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B @ C ) )
     => ( ord_less_int @ A2 @ B ) ) ).

% add_less_imp_less_right
thf(fact_507_mult__0,axiom,
    ! [N: nat] :
      ( ( times_times_nat @ zero_zero_nat @ N )
      = zero_zero_nat ) ).

% mult_0
thf(fact_508_infinite__descent0,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N4: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N4 )
           => ( ~ ( P @ N4 )
             => ? [M3: nat] :
                  ( ( ord_less_nat @ M3 @ N4 )
                  & ~ ( P @ M3 ) ) ) )
       => ( P @ N ) ) ) ).

% infinite_descent0
thf(fact_509_gr__implies__not0,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not0
thf(fact_510_less__zeroE,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% less_zeroE
thf(fact_511_not__less0,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% not_less0
thf(fact_512_not__gr0,axiom,
    ! [N: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
      = ( N = zero_zero_nat ) ) ).

% not_gr0
thf(fact_513_gr0I,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% gr0I
thf(fact_514_bot__nat__0_Oextremum__strict,axiom,
    ! [A2: nat] :
      ~ ( ord_less_nat @ A2 @ zero_zero_nat ) ).

% bot_nat_0.extremum_strict
thf(fact_515_add__mult__distrib,axiom,
    ! [M2: nat,N: nat,K: nat] :
      ( ( times_times_nat @ ( plus_plus_nat @ M2 @ N ) @ K )
      = ( plus_plus_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N @ K ) ) ) ).

% add_mult_distrib
thf(fact_516_add__mult__distrib2,axiom,
    ! [K: nat,M2: nat,N: nat] :
      ( ( times_times_nat @ K @ ( plus_plus_nat @ M2 @ N ) )
      = ( plus_plus_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) ) ) ).

% add_mult_distrib2
thf(fact_517_length__induct,axiom,
    ! [P: list_P5707943133018811711nt_int > $o,Xs2: list_P5707943133018811711nt_int] :
      ( ! [Xs: list_P5707943133018811711nt_int] :
          ( ! [Ys3: list_P5707943133018811711nt_int] :
              ( ( ord_less_nat @ ( size_s5157815400016825771nt_int @ Ys3 ) @ ( size_s5157815400016825771nt_int @ Xs ) )
             => ( P @ Ys3 ) )
         => ( P @ Xs ) )
     => ( P @ Xs2 ) ) ).

% length_induct
thf(fact_518_add__lessD1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
     => ( ord_less_nat @ I @ K ) ) ).

% add_lessD1
thf(fact_519_add__less__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( ord_less_nat @ K @ L )
       => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).

% add_less_mono
thf(fact_520_not__add__less1,axiom,
    ! [I: nat,J: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).

% not_add_less1
thf(fact_521_not__add__less2,axiom,
    ! [J: nat,I: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).

% not_add_less2
thf(fact_522_add__less__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).

% add_less_mono1
thf(fact_523_trans__less__add1,axiom,
    ! [I: nat,J: nat,M2: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M2 ) ) ) ).

% trans_less_add1
thf(fact_524_trans__less__add2,axiom,
    ! [I: nat,J: nat,M2: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ord_less_nat @ I @ ( plus_plus_nat @ M2 @ J ) ) ) ).

% trans_less_add2
thf(fact_525_less__add__eq__less,axiom,
    ! [K: nat,L: nat,M2: nat,N: nat] :
      ( ( ord_less_nat @ K @ L )
     => ( ( ( plus_plus_nat @ M2 @ L )
          = ( plus_plus_nat @ K @ N ) )
       => ( ord_less_nat @ M2 @ N ) ) ) ).

% less_add_eq_less
thf(fact_526_nat__mult__min__left,axiom,
    ! [M2: nat,N: nat,Q: nat] :
      ( ( times_times_nat @ ( ord_min_nat @ M2 @ N ) @ Q )
      = ( ord_min_nat @ ( times_times_nat @ M2 @ Q ) @ ( times_times_nat @ N @ Q ) ) ) ).

% nat_mult_min_left
thf(fact_527_nat__mult__min__right,axiom,
    ! [M2: nat,N: nat,Q: nat] :
      ( ( times_times_nat @ M2 @ ( ord_min_nat @ N @ Q ) )
      = ( ord_min_nat @ ( times_times_nat @ M2 @ N ) @ ( times_times_nat @ M2 @ Q ) ) ) ).

% nat_mult_min_right
thf(fact_528_add__scale__eq__noteq,axiom,
    ! [R: nat,A2: nat,B: nat,C: nat,D2: nat] :
      ( ( R != zero_zero_nat )
     => ( ( ( A2 = B )
          & ( C != D2 ) )
       => ( ( plus_plus_nat @ A2 @ ( times_times_nat @ R @ C ) )
         != ( plus_plus_nat @ B @ ( times_times_nat @ R @ D2 ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_529_add__scale__eq__noteq,axiom,
    ! [R: int,A2: int,B: int,C: int,D2: int] :
      ( ( R != zero_zero_int )
     => ( ( ( A2 = B )
          & ( C != D2 ) )
       => ( ( plus_plus_int @ A2 @ ( times_times_int @ R @ C ) )
         != ( plus_plus_int @ B @ ( times_times_int @ R @ D2 ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_530_pos__add__strict,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_nat @ B @ C )
       => ( ord_less_nat @ B @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).

% pos_add_strict
thf(fact_531_pos__add__strict,axiom,
    ! [A2: int,B: int,C: int] :
      ( ( ord_less_int @ zero_zero_int @ A2 )
     => ( ( ord_less_int @ B @ C )
       => ( ord_less_int @ B @ ( plus_plus_int @ A2 @ C ) ) ) ) ).

% pos_add_strict
thf(fact_532_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A2: nat,B: nat] :
      ( ( ord_less_nat @ A2 @ B )
     => ~ ! [C5: nat] :
            ( ( B
              = ( plus_plus_nat @ A2 @ C5 ) )
           => ( C5 = zero_zero_nat ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
thf(fact_533_add__pos__pos,axiom,
    ! [A2: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ B )
       => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).

% add_pos_pos
thf(fact_534_add__pos__pos,axiom,
    ! [A2: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ A2 )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A2 @ B ) ) ) ) ).

% add_pos_pos
thf(fact_535_add__neg__neg,axiom,
    ! [A2: nat,B: nat] :
      ( ( ord_less_nat @ A2 @ zero_zero_nat )
     => ( ( ord_less_nat @ B @ zero_zero_nat )
       => ( ord_less_nat @ ( plus_plus_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).

% add_neg_neg
thf(fact_536_add__neg__neg,axiom,
    ! [A2: int,B: int] :
      ( ( ord_less_int @ A2 @ zero_zero_int )
     => ( ( ord_less_int @ B @ zero_zero_int )
       => ( ord_less_int @ ( plus_plus_int @ A2 @ B ) @ zero_zero_int ) ) ) ).

% add_neg_neg
thf(fact_537_less__imp__add__positive,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_nat @ I @ J )
     => ? [K2: nat] :
          ( ( ord_less_nat @ zero_zero_nat @ K2 )
          & ( ( plus_plus_nat @ I @ K2 )
            = J ) ) ) ).

% less_imp_add_positive
thf(fact_538_length__pos__if__in__set,axiom,
    ! [X: int,Xs2: list_int] :
      ( ( member_int2 @ X @ ( set_int2 @ Xs2 ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_int @ Xs2 ) ) ) ).

% length_pos_if_in_set
thf(fact_539_length__pos__if__in__set,axiom,
    ! [X: product_prod_int_int,Xs2: list_P5707943133018811711nt_int] :
      ( ( member5262025264175285858nt_int @ X @ ( set_Pr2470121279949933262nt_int @ Xs2 ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s5157815400016825771nt_int @ Xs2 ) ) ) ).

% length_pos_if_in_set
thf(fact_540_in__set__product__lists__length,axiom,
    ! [Xs2: list_P5707943133018811711nt_int,Xss: list_l1670014477004246597nt_int] :
      ( ( member2764346250752101224nt_int @ Xs2 @ ( set_li2659200638379878868nt_int @ ( produc5568053154996169768nt_int @ Xss ) ) )
     => ( ( size_s5157815400016825771nt_int @ Xs2 )
        = ( size_s2969076144586574001nt_int @ Xss ) ) ) ).

% in_set_product_lists_length
thf(fact_541_nat__mult__less__cancel__disj,axiom,
    ! [K: nat,M2: nat,N: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K )
        & ( ord_less_nat @ M2 @ N ) ) ) ).

% nat_mult_less_cancel_disj
thf(fact_542_mult__zero__left,axiom,
    ! [A2: nat] :
      ( ( times_times_nat @ zero_zero_nat @ A2 )
      = zero_zero_nat ) ).

% mult_zero_left
thf(fact_543_mult__zero__left,axiom,
    ! [A2: int] :
      ( ( times_times_int @ zero_zero_int @ A2 )
      = zero_zero_int ) ).

% mult_zero_left
thf(fact_544_mult__zero__right,axiom,
    ! [A2: nat] :
      ( ( times_times_nat @ A2 @ zero_zero_nat )
      = zero_zero_nat ) ).

% mult_zero_right
thf(fact_545_mult__zero__right,axiom,
    ! [A2: int] :
      ( ( times_times_int @ A2 @ zero_zero_int )
      = zero_zero_int ) ).

% mult_zero_right
thf(fact_546_mult__eq__0__iff,axiom,
    ! [A2: nat,B: nat] :
      ( ( ( times_times_nat @ A2 @ B )
        = zero_zero_nat )
      = ( ( A2 = zero_zero_nat )
        | ( B = zero_zero_nat ) ) ) ).

% mult_eq_0_iff
thf(fact_547_mult__eq__0__iff,axiom,
    ! [A2: int,B: int] :
      ( ( ( times_times_int @ A2 @ B )
        = zero_zero_int )
      = ( ( A2 = zero_zero_int )
        | ( B = zero_zero_int ) ) ) ).

% mult_eq_0_iff
thf(fact_548_mult__cancel__left,axiom,
    ! [C: nat,A2: nat,B: nat] :
      ( ( ( times_times_nat @ C @ A2 )
        = ( times_times_nat @ C @ B ) )
      = ( ( C = zero_zero_nat )
        | ( A2 = B ) ) ) ).

% mult_cancel_left
thf(fact_549_mult__cancel__left,axiom,
    ! [C: int,A2: int,B: int] :
      ( ( ( times_times_int @ C @ A2 )
        = ( times_times_int @ C @ B ) )
      = ( ( C = zero_zero_int )
        | ( A2 = B ) ) ) ).

% mult_cancel_left
thf(fact_550_mult__cancel__right,axiom,
    ! [A2: nat,C: nat,B: nat] :
      ( ( ( times_times_nat @ A2 @ C )
        = ( times_times_nat @ B @ C ) )
      = ( ( C = zero_zero_nat )
        | ( A2 = B ) ) ) ).

% mult_cancel_right
thf(fact_551_mult__cancel__right,axiom,
    ! [A2: int,C: int,B: int] :
      ( ( ( times_times_int @ A2 @ C )
        = ( times_times_int @ B @ C ) )
      = ( ( C = zero_zero_int )
        | ( A2 = B ) ) ) ).

% mult_cancel_right
thf(fact_552_zmult__zless__mono2,axiom,
    ! [I: int,J: int,K: int] :
      ( ( ord_less_int @ I @ J )
     => ( ( ord_less_int @ zero_zero_int @ K )
       => ( ord_less_int @ ( times_times_int @ K @ I ) @ ( times_times_int @ K @ J ) ) ) ) ).

% zmult_zless_mono2
thf(fact_553_times__int__code_I1_J,axiom,
    ! [K: int] :
      ( ( times_times_int @ K @ zero_zero_int )
      = zero_zero_int ) ).

% times_int_code(1)
thf(fact_554_times__int__code_I2_J,axiom,
    ! [L: int] :
      ( ( times_times_int @ zero_zero_int @ L )
      = zero_zero_int ) ).

% times_int_code(2)
thf(fact_555_less__int__code_I1_J,axiom,
    ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).

% less_int_code(1)
thf(fact_556_int__distrib_I1_J,axiom,
    ! [Z12: int,Z23: int,W: int] :
      ( ( times_times_int @ ( plus_plus_int @ Z12 @ Z23 ) @ W )
      = ( plus_plus_int @ ( times_times_int @ Z12 @ W ) @ ( times_times_int @ Z23 @ W ) ) ) ).

% int_distrib(1)
thf(fact_557_int__distrib_I2_J,axiom,
    ! [W: int,Z12: int,Z23: int] :
      ( ( times_times_int @ W @ ( plus_plus_int @ Z12 @ Z23 ) )
      = ( plus_plus_int @ ( times_times_int @ W @ Z12 ) @ ( times_times_int @ W @ Z23 ) ) ) ).

% int_distrib(2)
thf(fact_558_mult__right__cancel,axiom,
    ! [C: nat,A2: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( ( ( times_times_nat @ A2 @ C )
          = ( times_times_nat @ B @ C ) )
        = ( A2 = B ) ) ) ).

% mult_right_cancel
thf(fact_559_mult__right__cancel,axiom,
    ! [C: int,A2: int,B: int] :
      ( ( C != zero_zero_int )
     => ( ( ( times_times_int @ A2 @ C )
          = ( times_times_int @ B @ C ) )
        = ( A2 = B ) ) ) ).

% mult_right_cancel
thf(fact_560_mult__left__cancel,axiom,
    ! [C: nat,A2: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( ( ( times_times_nat @ C @ A2 )
          = ( times_times_nat @ C @ B ) )
        = ( A2 = B ) ) ) ).

% mult_left_cancel
thf(fact_561_mult__left__cancel,axiom,
    ! [C: int,A2: int,B: int] :
      ( ( C != zero_zero_int )
     => ( ( ( times_times_int @ C @ A2 )
          = ( times_times_int @ C @ B ) )
        = ( A2 = B ) ) ) ).

% mult_left_cancel
thf(fact_562_no__zero__divisors,axiom,
    ! [A2: nat,B: nat] :
      ( ( A2 != zero_zero_nat )
     => ( ( B != zero_zero_nat )
       => ( ( times_times_nat @ A2 @ B )
         != zero_zero_nat ) ) ) ).

% no_zero_divisors
thf(fact_563_no__zero__divisors,axiom,
    ! [A2: int,B: int] :
      ( ( A2 != zero_zero_int )
     => ( ( B != zero_zero_int )
       => ( ( times_times_int @ A2 @ B )
         != zero_zero_int ) ) ) ).

% no_zero_divisors
thf(fact_564_divisors__zero,axiom,
    ! [A2: nat,B: nat] :
      ( ( ( times_times_nat @ A2 @ B )
        = zero_zero_nat )
     => ( ( A2 = zero_zero_nat )
        | ( B = zero_zero_nat ) ) ) ).

% divisors_zero
thf(fact_565_divisors__zero,axiom,
    ! [A2: int,B: int] :
      ( ( ( times_times_int @ A2 @ B )
        = zero_zero_int )
     => ( ( A2 = zero_zero_int )
        | ( B = zero_zero_int ) ) ) ).

% divisors_zero
thf(fact_566_mult__not__zero,axiom,
    ! [A2: nat,B: nat] :
      ( ( ( times_times_nat @ A2 @ B )
       != zero_zero_nat )
     => ( ( A2 != zero_zero_nat )
        & ( B != zero_zero_nat ) ) ) ).

% mult_not_zero
thf(fact_567_mult__not__zero,axiom,
    ! [A2: int,B: int] :
      ( ( ( times_times_int @ A2 @ B )
       != zero_zero_int )
     => ( ( A2 != zero_zero_int )
        & ( B != zero_zero_int ) ) ) ).

% mult_not_zero
thf(fact_568_ring__class_Oring__distribs_I2_J,axiom,
    ! [A2: int,B: int,C: int] :
      ( ( times_times_int @ ( plus_plus_int @ A2 @ B ) @ C )
      = ( plus_plus_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) ) ) ).

% ring_class.ring_distribs(2)
thf(fact_569_ring__class_Oring__distribs_I1_J,axiom,
    ! [A2: int,B: int,C: int] :
      ( ( times_times_int @ A2 @ ( plus_plus_int @ B @ C ) )
      = ( plus_plus_int @ ( times_times_int @ A2 @ B ) @ ( times_times_int @ A2 @ C ) ) ) ).

% ring_class.ring_distribs(1)
thf(fact_570_comm__semiring__class_Odistrib,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( ( times_times_nat @ ( plus_plus_nat @ A2 @ B ) @ C )
      = ( plus_plus_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ C ) ) ) ).

% comm_semiring_class.distrib
thf(fact_571_comm__semiring__class_Odistrib,axiom,
    ! [A2: int,B: int,C: int] :
      ( ( times_times_int @ ( plus_plus_int @ A2 @ B ) @ C )
      = ( plus_plus_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) ) ) ).

% comm_semiring_class.distrib
thf(fact_572_distrib__left,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( ( times_times_nat @ A2 @ ( plus_plus_nat @ B @ C ) )
      = ( plus_plus_nat @ ( times_times_nat @ A2 @ B ) @ ( times_times_nat @ A2 @ C ) ) ) ).

% distrib_left
thf(fact_573_distrib__left,axiom,
    ! [A2: int,B: int,C: int] :
      ( ( times_times_int @ A2 @ ( plus_plus_int @ B @ C ) )
      = ( plus_plus_int @ ( times_times_int @ A2 @ B ) @ ( times_times_int @ A2 @ C ) ) ) ).

% distrib_left
thf(fact_574_distrib__right,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( ( times_times_nat @ ( plus_plus_nat @ A2 @ B ) @ C )
      = ( plus_plus_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ C ) ) ) ).

% distrib_right
thf(fact_575_distrib__right,axiom,
    ! [A2: int,B: int,C: int] :
      ( ( times_times_int @ ( plus_plus_int @ A2 @ B ) @ C )
      = ( plus_plus_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) ) ) ).

% distrib_right
thf(fact_576_combine__common__factor,axiom,
    ! [A2: nat,E: nat,B: nat,C: nat] :
      ( ( plus_plus_nat @ ( times_times_nat @ A2 @ E ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E ) @ C ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A2 @ B ) @ E ) @ C ) ) ).

% combine_common_factor
thf(fact_577_combine__common__factor,axiom,
    ! [A2: int,E: int,B: int,C: int] :
      ( ( plus_plus_int @ ( times_times_int @ A2 @ E ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ C ) )
      = ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A2 @ B ) @ E ) @ C ) ) ).

% combine_common_factor
thf(fact_578_nat__mult__eq__cancel__disj,axiom,
    ! [K: nat,M2: nat,N: nat] :
      ( ( ( times_times_nat @ K @ M2 )
        = ( times_times_nat @ K @ N ) )
      = ( ( K = zero_zero_nat )
        | ( M2 = N ) ) ) ).

% nat_mult_eq_cancel_disj
thf(fact_579_left__add__mult__distrib,axiom,
    ! [I: nat,U: nat,J: nat,K: nat] :
      ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ K ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I @ J ) @ U ) @ K ) ) ).

% left_add_mult_distrib
thf(fact_580_mult__neg__neg,axiom,
    ! [A2: int,B: int] :
      ( ( ord_less_int @ A2 @ zero_zero_int )
     => ( ( ord_less_int @ B @ zero_zero_int )
       => ( ord_less_int @ zero_zero_int @ ( times_times_int @ A2 @ B ) ) ) ) ).

% mult_neg_neg
thf(fact_581_not__square__less__zero,axiom,
    ! [A2: int] :
      ~ ( ord_less_int @ ( times_times_int @ A2 @ A2 ) @ zero_zero_int ) ).

% not_square_less_zero
thf(fact_582_mult__less__0__iff,axiom,
    ! [A2: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ A2 @ B ) @ zero_zero_int )
      = ( ( ( ord_less_int @ zero_zero_int @ A2 )
          & ( ord_less_int @ B @ zero_zero_int ) )
        | ( ( ord_less_int @ A2 @ zero_zero_int )
          & ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).

% mult_less_0_iff
thf(fact_583_mult__neg__pos,axiom,
    ! [A2: nat,B: nat] :
      ( ( ord_less_nat @ A2 @ zero_zero_nat )
     => ( ( ord_less_nat @ zero_zero_nat @ B )
       => ( ord_less_nat @ ( times_times_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).

% mult_neg_pos
thf(fact_584_mult__neg__pos,axiom,
    ! [A2: int,B: int] :
      ( ( ord_less_int @ A2 @ zero_zero_int )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ord_less_int @ ( times_times_int @ A2 @ B ) @ zero_zero_int ) ) ) ).

% mult_neg_pos
thf(fact_585_mult__pos__neg,axiom,
    ! [A2: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_nat @ B @ zero_zero_nat )
       => ( ord_less_nat @ ( times_times_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).

% mult_pos_neg
thf(fact_586_mult__pos__neg,axiom,
    ! [A2: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ A2 )
     => ( ( ord_less_int @ B @ zero_zero_int )
       => ( ord_less_int @ ( times_times_int @ A2 @ B ) @ zero_zero_int ) ) ) ).

% mult_pos_neg
thf(fact_587_mult__pos__pos,axiom,
    ! [A2: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ B )
       => ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A2 @ B ) ) ) ) ).

% mult_pos_pos
thf(fact_588_mult__pos__pos,axiom,
    ! [A2: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ A2 )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ord_less_int @ zero_zero_int @ ( times_times_int @ A2 @ B ) ) ) ) ).

% mult_pos_pos
thf(fact_589_mult__pos__neg2,axiom,
    ! [A2: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_nat @ B @ zero_zero_nat )
       => ( ord_less_nat @ ( times_times_nat @ B @ A2 ) @ zero_zero_nat ) ) ) ).

% mult_pos_neg2
thf(fact_590_mult__pos__neg2,axiom,
    ! [A2: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ A2 )
     => ( ( ord_less_int @ B @ zero_zero_int )
       => ( ord_less_int @ ( times_times_int @ B @ A2 ) @ zero_zero_int ) ) ) ).

% mult_pos_neg2
thf(fact_591_zero__less__mult__iff,axiom,
    ! [A2: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A2 @ B ) )
      = ( ( ( ord_less_int @ zero_zero_int @ A2 )
          & ( ord_less_int @ zero_zero_int @ B ) )
        | ( ( ord_less_int @ A2 @ zero_zero_int )
          & ( ord_less_int @ B @ zero_zero_int ) ) ) ) ).

% zero_less_mult_iff
thf(fact_592_zero__less__mult__pos,axiom,
    ! [A2: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A2 @ B ) )
     => ( ( ord_less_nat @ zero_zero_nat @ A2 )
       => ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).

% zero_less_mult_pos
thf(fact_593_zero__less__mult__pos,axiom,
    ! [A2: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A2 @ B ) )
     => ( ( ord_less_int @ zero_zero_int @ A2 )
       => ( ord_less_int @ zero_zero_int @ B ) ) ) ).

% zero_less_mult_pos
thf(fact_594_zero__less__mult__pos2,axiom,
    ! [B: nat,A2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B @ A2 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ A2 )
       => ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).

% zero_less_mult_pos2
thf(fact_595_zero__less__mult__pos2,axiom,
    ! [B: int,A2: int] :
      ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B @ A2 ) )
     => ( ( ord_less_int @ zero_zero_int @ A2 )
       => ( ord_less_int @ zero_zero_int @ B ) ) ) ).

% zero_less_mult_pos2
thf(fact_596_mult__less__cancel__left__neg,axiom,
    ! [C: int,A2: int,B: int] :
      ( ( ord_less_int @ C @ zero_zero_int )
     => ( ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) )
        = ( ord_less_int @ B @ A2 ) ) ) ).

% mult_less_cancel_left_neg
thf(fact_597_mult__less__cancel__left__pos,axiom,
    ! [C: int,A2: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ C )
     => ( ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) )
        = ( ord_less_int @ A2 @ B ) ) ) ).

% mult_less_cancel_left_pos
thf(fact_598_mult__strict__left__mono__neg,axiom,
    ! [B: int,A2: int,C: int] :
      ( ( ord_less_int @ B @ A2 )
     => ( ( ord_less_int @ C @ zero_zero_int )
       => ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) ) ) ) ).

% mult_strict_left_mono_neg
thf(fact_599_mult__strict__left__mono,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A2 @ B )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B ) ) ) ) ).

% mult_strict_left_mono
thf(fact_600_mult__strict__left__mono,axiom,
    ! [A2: int,B: int,C: int] :
      ( ( ord_less_int @ A2 @ B )
     => ( ( ord_less_int @ zero_zero_int @ C )
       => ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) ) ) ) ).

% mult_strict_left_mono
thf(fact_601_mult__less__cancel__left__disj,axiom,
    ! [C: int,A2: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
          & ( ord_less_int @ A2 @ B ) )
        | ( ( ord_less_int @ C @ zero_zero_int )
          & ( ord_less_int @ B @ A2 ) ) ) ) ).

% mult_less_cancel_left_disj
thf(fact_602_mult__strict__right__mono__neg,axiom,
    ! [B: int,A2: int,C: int] :
      ( ( ord_less_int @ B @ A2 )
     => ( ( ord_less_int @ C @ zero_zero_int )
       => ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).

% mult_strict_right_mono_neg
thf(fact_603_mult__strict__right__mono,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A2 @ B )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).

% mult_strict_right_mono
thf(fact_604_mult__strict__right__mono,axiom,
    ! [A2: int,B: int,C: int] :
      ( ( ord_less_int @ A2 @ B )
     => ( ( ord_less_int @ zero_zero_int @ C )
       => ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).

% mult_strict_right_mono
thf(fact_605_mult__less__cancel__right__disj,axiom,
    ! [A2: int,C: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
          & ( ord_less_int @ A2 @ B ) )
        | ( ( ord_less_int @ C @ zero_zero_int )
          & ( ord_less_int @ B @ A2 ) ) ) ) ).

% mult_less_cancel_right_disj
thf(fact_606_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A2 @ B )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_607_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A2: int,B: int,C: int] :
      ( ( ord_less_int @ A2 @ B )
     => ( ( ord_less_int @ zero_zero_int @ C )
       => ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_608_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_609_nat__mult__eq__cancel1,axiom,
    ! [K: nat,M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( ( times_times_nat @ K @ M2 )
          = ( times_times_nat @ K @ N ) )
        = ( M2 = N ) ) ) ).

% nat_mult_eq_cancel1
thf(fact_610_nat__mult__less__cancel1,axiom,
    ! [K: nat,M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( ord_less_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) )
        = ( ord_less_nat @ M2 @ N ) ) ) ).

% nat_mult_less_cancel1
thf(fact_611_not__sum__squares__lt__zero,axiom,
    ! [X: int,Y: int] :
      ~ ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int ) ).

% not_sum_squares_lt_zero
thf(fact_612_sum__squares__gt__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) )
      = ( ( X != zero_zero_int )
        | ( Y != zero_zero_int ) ) ) ).

% sum_squares_gt_zero_iff
thf(fact_613_in__measure,axiom,
    ! [X: int,Y: int,F: int > nat] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( measure_int @ F ) )
      = ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) ) ).

% in_measure
thf(fact_614_sum__squares__eq__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) )
        = zero_zero_int )
      = ( ( X = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_squares_eq_zero_iff
thf(fact_615_mult__less__iff1,axiom,
    ! [Z: int,X: int,Y: int] :
      ( ( ord_less_int @ zero_zero_int @ Z )
     => ( ( ord_less_int @ ( times_times_int @ X @ Z ) @ ( times_times_int @ Y @ Z ) )
        = ( ord_less_int @ X @ Y ) ) ) ).

% mult_less_iff1
thf(fact_616_lexord__take__index__conv,axiom,
    ! [X: list_int,Y: list_int,R: set_Pr958786334691620121nt_int] :
      ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ X @ Y ) @ ( lexord_int @ R ) )
      = ( ( ( ord_less_nat @ ( size_size_list_int @ X ) @ ( size_size_list_int @ Y ) )
          & ( ( take_int @ ( size_size_list_int @ X ) @ Y )
            = X ) )
        | ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( ord_min_nat @ ( size_size_list_int @ X ) @ ( size_size_list_int @ Y ) ) )
            & ( ( take_int @ I2 @ X )
              = ( take_int @ I2 @ Y ) )
            & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( nth_int @ X @ I2 ) @ ( nth_int @ Y @ I2 ) ) @ R ) ) ) ) ).

% lexord_take_index_conv
thf(fact_617_lexord__take__index__conv,axiom,
    ! [X: list_P5707943133018811711nt_int,Y: list_P5707943133018811711nt_int,R: set_Pr2560585780119916871nt_int] :
      ( ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ X @ Y ) @ ( lexord8886728381914977324nt_int @ R ) )
      = ( ( ( ord_less_nat @ ( size_s5157815400016825771nt_int @ X ) @ ( size_s5157815400016825771nt_int @ Y ) )
          & ( ( take_P8218740963776755879nt_int @ ( size_s5157815400016825771nt_int @ X ) @ Y )
            = X ) )
        | ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( ord_min_nat @ ( size_s5157815400016825771nt_int @ X ) @ ( size_s5157815400016825771nt_int @ Y ) ) )
            & ( ( take_P8218740963776755879nt_int @ I2 @ X )
              = ( take_P8218740963776755879nt_int @ I2 @ Y ) )
            & ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ ( nth_Pr4439495888332055232nt_int @ X @ I2 ) @ ( nth_Pr4439495888332055232nt_int @ Y @ I2 ) ) @ R ) ) ) ) ).

% lexord_take_index_conv
thf(fact_618_nth__Cons__0,axiom,
    ! [X: int,Xs2: list_int] :
      ( ( nth_int @ ( cons_int @ X @ Xs2 ) @ zero_zero_nat )
      = X ) ).

% nth_Cons_0
thf(fact_619_nth__zip,axiom,
    ! [I: nat,Xs2: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int] :
      ( ( ord_less_nat @ I @ ( size_s5157815400016825771nt_int @ Xs2 ) )
     => ( ( ord_less_nat @ I @ ( size_s5157815400016825771nt_int @ Ys ) )
       => ( ( nth_Pr7104135640663552750nt_int @ ( zip_Pr2148635498163357687nt_int @ Xs2 @ Ys ) @ I )
          = ( produc3646306378393792727nt_int @ ( nth_Pr4439495888332055232nt_int @ Xs2 @ I ) @ ( nth_Pr4439495888332055232nt_int @ Ys @ I ) ) ) ) ) ).

% nth_zip
thf(fact_620_nth__equalityI,axiom,
    ! [Xs2: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int] :
      ( ( ( size_s5157815400016825771nt_int @ Xs2 )
        = ( size_s5157815400016825771nt_int @ Ys ) )
     => ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s5157815400016825771nt_int @ Xs2 ) )
           => ( ( nth_Pr4439495888332055232nt_int @ Xs2 @ I3 )
              = ( nth_Pr4439495888332055232nt_int @ Ys @ I3 ) ) )
       => ( Xs2 = Ys ) ) ) ).

% nth_equalityI
thf(fact_621_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > product_prod_int_int > $o] :
      ( ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ K )
           => ? [X9: product_prod_int_int] : ( P @ I2 @ X9 ) ) )
      = ( ? [Xs3: list_P5707943133018811711nt_int] :
            ( ( ( size_s5157815400016825771nt_int @ Xs3 )
              = K )
            & ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ K )
               => ( P @ I2 @ ( nth_Pr4439495888332055232nt_int @ Xs3 @ I2 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_622_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y2: list_P5707943133018811711nt_int,Z2: list_P5707943133018811711nt_int] : ( Y2 = Z2 ) )
    = ( ^ [Xs3: list_P5707943133018811711nt_int,Ys4: list_P5707943133018811711nt_int] :
          ( ( ( size_s5157815400016825771nt_int @ Xs3 )
            = ( size_s5157815400016825771nt_int @ Ys4 ) )
          & ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_s5157815400016825771nt_int @ Xs3 ) )
             => ( ( nth_Pr4439495888332055232nt_int @ Xs3 @ I2 )
                = ( nth_Pr4439495888332055232nt_int @ Ys4 @ I2 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_623_nth__mem,axiom,
    ! [N: nat,Xs2: list_int] :
      ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs2 ) )
     => ( member_int2 @ ( nth_int @ Xs2 @ N ) @ ( set_int2 @ Xs2 ) ) ) ).

% nth_mem
thf(fact_624_nth__mem,axiom,
    ! [N: nat,Xs2: list_P5707943133018811711nt_int] :
      ( ( ord_less_nat @ N @ ( size_s5157815400016825771nt_int @ Xs2 ) )
     => ( member5262025264175285858nt_int @ ( nth_Pr4439495888332055232nt_int @ Xs2 @ N ) @ ( set_Pr2470121279949933262nt_int @ Xs2 ) ) ) ).

% nth_mem
thf(fact_625_list__ball__nth,axiom,
    ! [N: nat,Xs2: list_P5707943133018811711nt_int,P: product_prod_int_int > $o] :
      ( ( ord_less_nat @ N @ ( size_s5157815400016825771nt_int @ Xs2 ) )
     => ( ! [X2: product_prod_int_int] :
            ( ( member5262025264175285858nt_int @ X2 @ ( set_Pr2470121279949933262nt_int @ Xs2 ) )
           => ( P @ X2 ) )
       => ( P @ ( nth_Pr4439495888332055232nt_int @ Xs2 @ N ) ) ) ) ).

% list_ball_nth
thf(fact_626_in__set__conv__nth,axiom,
    ! [X: int,Xs2: list_int] :
      ( ( member_int2 @ X @ ( set_int2 @ Xs2 ) )
      = ( ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs2 ) )
            & ( ( nth_int @ Xs2 @ I2 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_627_in__set__conv__nth,axiom,
    ! [X: product_prod_int_int,Xs2: list_P5707943133018811711nt_int] :
      ( ( member5262025264175285858nt_int @ X @ ( set_Pr2470121279949933262nt_int @ Xs2 ) )
      = ( ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s5157815400016825771nt_int @ Xs2 ) )
            & ( ( nth_Pr4439495888332055232nt_int @ Xs2 @ I2 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_628_all__nth__imp__all__set,axiom,
    ! [Xs2: list_int,P: int > $o,X: int] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs2 ) )
         => ( P @ ( nth_int @ Xs2 @ I3 ) ) )
     => ( ( member_int2 @ X @ ( set_int2 @ Xs2 ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_629_all__nth__imp__all__set,axiom,
    ! [Xs2: list_P5707943133018811711nt_int,P: product_prod_int_int > $o,X: product_prod_int_int] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ ( size_s5157815400016825771nt_int @ Xs2 ) )
         => ( P @ ( nth_Pr4439495888332055232nt_int @ Xs2 @ I3 ) ) )
     => ( ( member5262025264175285858nt_int @ X @ ( set_Pr2470121279949933262nt_int @ Xs2 ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_630_all__set__conv__all__nth,axiom,
    ! [Xs2: list_P5707943133018811711nt_int,P: product_prod_int_int > $o] :
      ( ( ! [X3: product_prod_int_int] :
            ( ( member5262025264175285858nt_int @ X3 @ ( set_Pr2470121279949933262nt_int @ Xs2 ) )
           => ( P @ X3 ) ) )
      = ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s5157815400016825771nt_int @ Xs2 ) )
           => ( P @ ( nth_Pr4439495888332055232nt_int @ Xs2 @ I2 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_631_nth__eq__iff__index__eq,axiom,
    ! [Xs2: list_P5707943133018811711nt_int,I: nat,J: nat] :
      ( ( distin3744728255968310194nt_int @ Xs2 )
     => ( ( ord_less_nat @ I @ ( size_s5157815400016825771nt_int @ Xs2 ) )
       => ( ( ord_less_nat @ J @ ( size_s5157815400016825771nt_int @ Xs2 ) )
         => ( ( ( nth_Pr4439495888332055232nt_int @ Xs2 @ I )
              = ( nth_Pr4439495888332055232nt_int @ Xs2 @ J ) )
            = ( I = J ) ) ) ) ) ).

% nth_eq_iff_index_eq
thf(fact_632_distinct__conv__nth,axiom,
    ( distin3744728255968310194nt_int
    = ( ^ [Xs3: list_P5707943133018811711nt_int] :
        ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_s5157815400016825771nt_int @ Xs3 ) )
         => ! [J2: nat] :
              ( ( ord_less_nat @ J2 @ ( size_s5157815400016825771nt_int @ Xs3 ) )
             => ( ( I2 != J2 )
               => ( ( nth_Pr4439495888332055232nt_int @ Xs3 @ I2 )
                 != ( nth_Pr4439495888332055232nt_int @ Xs3 @ J2 ) ) ) ) ) ) ) ).

% distinct_conv_nth
thf(fact_633_distinct__Ex1,axiom,
    ! [Xs2: list_int,X: int] :
      ( ( distinct_int @ Xs2 )
     => ( ( member_int2 @ X @ ( set_int2 @ Xs2 ) )
       => ? [X2: nat] :
            ( ( ord_less_nat @ X2 @ ( size_size_list_int @ Xs2 ) )
            & ( ( nth_int @ Xs2 @ X2 )
              = X )
            & ! [Y9: nat] :
                ( ( ( ord_less_nat @ Y9 @ ( size_size_list_int @ Xs2 ) )
                  & ( ( nth_int @ Xs2 @ Y9 )
                    = X ) )
               => ( Y9 = X2 ) ) ) ) ) ).

% distinct_Ex1
thf(fact_634_distinct__Ex1,axiom,
    ! [Xs2: list_P5707943133018811711nt_int,X: product_prod_int_int] :
      ( ( distin3744728255968310194nt_int @ Xs2 )
     => ( ( member5262025264175285858nt_int @ X @ ( set_Pr2470121279949933262nt_int @ Xs2 ) )
       => ? [X2: nat] :
            ( ( ord_less_nat @ X2 @ ( size_s5157815400016825771nt_int @ Xs2 ) )
            & ( ( nth_Pr4439495888332055232nt_int @ Xs2 @ X2 )
              = X )
            & ! [Y9: nat] :
                ( ( ( ord_less_nat @ Y9 @ ( size_s5157815400016825771nt_int @ Xs2 ) )
                  & ( ( nth_Pr4439495888332055232nt_int @ Xs2 @ Y9 )
                    = X ) )
               => ( Y9 = X2 ) ) ) ) ) ).

% distinct_Ex1
thf(fact_635_in__set__zip,axiom,
    ! [P2: produc1219242969750017639nt_int,Xs2: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int] :
      ( ( member8566619992076573584nt_int @ P2 @ ( set_Pr5878228222108503548nt_int @ ( zip_Pr2148635498163357687nt_int @ Xs2 @ Ys ) ) )
      = ( ? [N3: nat] :
            ( ( ( nth_Pr4439495888332055232nt_int @ Xs2 @ N3 )
              = ( produc698254169746827971nt_int @ P2 ) )
            & ( ( nth_Pr4439495888332055232nt_int @ Ys @ N3 )
              = ( produc3892743399831173125nt_int @ P2 ) )
            & ( ord_less_nat @ N3 @ ( size_s5157815400016825771nt_int @ Xs2 ) )
            & ( ord_less_nat @ N3 @ ( size_s5157815400016825771nt_int @ Ys ) ) ) ) ) ).

% in_set_zip
thf(fact_636_in__set__zip,axiom,
    ! [P2: product_prod_int_int,Xs2: list_int,Ys: list_int] :
      ( ( member5262025264175285858nt_int @ P2 @ ( set_Pr2470121279949933262nt_int @ ( zip_int_int @ Xs2 @ Ys ) ) )
      = ( ? [N3: nat] :
            ( ( ( nth_int @ Xs2 @ N3 )
              = ( product_fst_int_int @ P2 ) )
            & ( ( nth_int @ Ys @ N3 )
              = ( product_snd_int_int @ P2 ) )
            & ( ord_less_nat @ N3 @ ( size_size_list_int @ Xs2 ) )
            & ( ord_less_nat @ N3 @ ( size_size_list_int @ Ys ) ) ) ) ) ).

% in_set_zip
thf(fact_637_map__of__zip__nth,axiom,
    ! [Xs2: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int,I: nat] :
      ( ( ( size_s5157815400016825771nt_int @ Xs2 )
        = ( size_s5157815400016825771nt_int @ Ys ) )
     => ( ( distin3744728255968310194nt_int @ Xs2 )
       => ( ( ord_less_nat @ I @ ( size_s5157815400016825771nt_int @ Ys ) )
         => ( ( map_of5841684995557474595nt_int @ ( zip_Pr2148635498163357687nt_int @ Xs2 @ Ys ) @ ( nth_Pr4439495888332055232nt_int @ Xs2 @ I ) )
            = ( some_P4184893108420464158nt_int @ ( nth_Pr4439495888332055232nt_int @ Ys @ I ) ) ) ) ) ) ).

% map_of_zip_nth
thf(fact_638_lex__take__index,axiom,
    ! [Xs2: list_int,Ys: list_int,R: set_Pr958786334691620121nt_int] :
      ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs2 @ Ys ) @ ( lex_int @ R ) )
     => ~ ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs2 ) )
           => ( ( ord_less_nat @ I3 @ ( size_size_list_int @ Ys ) )
             => ( ( ( take_int @ I3 @ Xs2 )
                  = ( take_int @ I3 @ Ys ) )
               => ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( nth_int @ Xs2 @ I3 ) @ ( nth_int @ Ys @ I3 ) ) @ R ) ) ) ) ) ).

% lex_take_index
thf(fact_639_lex__take__index,axiom,
    ! [Xs2: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int,R: set_Pr2560585780119916871nt_int] :
      ( ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ Xs2 @ Ys ) @ ( lex_Pr5393148144989827363nt_int @ R ) )
     => ~ ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s5157815400016825771nt_int @ Xs2 ) )
           => ( ( ord_less_nat @ I3 @ ( size_s5157815400016825771nt_int @ Ys ) )
             => ( ( ( take_P8218740963776755879nt_int @ I3 @ Xs2 )
                  = ( take_P8218740963776755879nt_int @ I3 @ Ys ) )
               => ~ ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ ( nth_Pr4439495888332055232nt_int @ Xs2 @ I3 ) @ ( nth_Pr4439495888332055232nt_int @ Ys @ I3 ) ) @ R ) ) ) ) ) ).

% lex_take_index
thf(fact_640_dom__map__upds,axiom,
    ! [M2: product_prod_int_int > option4624381673175914239nt_int,Xs2: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int] :
      ( ( dom_Pr634364266132166418nt_int @ ( map_up6499465237031507894nt_int @ M2 @ Xs2 @ Ys ) )
      = ( sup_su6024340866399070445nt_int @ ( set_Pr2470121279949933262nt_int @ ( take_P8218740963776755879nt_int @ ( size_s5157815400016825771nt_int @ Ys ) @ Xs2 ) ) @ ( dom_Pr634364266132166418nt_int @ M2 ) ) ) ).

% dom_map_upds
thf(fact_641_dom__map__upds,axiom,
    ! [M2: nat > option4624381673175914239nt_int,Xs2: list_nat,Ys: list_P5707943133018811711nt_int] :
      ( ( dom_na3440251890004644245nt_int @ ( map_up7635725999279402481nt_int @ M2 @ Xs2 @ Ys ) )
      = ( sup_sup_set_nat @ ( set_nat2 @ ( take_nat @ ( size_s5157815400016825771nt_int @ Ys ) @ Xs2 ) ) @ ( dom_na3440251890004644245nt_int @ M2 ) ) ) ).

% dom_map_upds
thf(fact_642_find__Some__iff2,axiom,
    ! [X: product_prod_int_int,P: product_prod_int_int > $o,Xs2: list_P5707943133018811711nt_int] :
      ( ( ( some_P4184893108420464158nt_int @ X )
        = ( find_P5021385047576723413nt_int @ P @ Xs2 ) )
      = ( ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s5157815400016825771nt_int @ Xs2 ) )
            & ( P @ ( nth_Pr4439495888332055232nt_int @ Xs2 @ I2 ) )
            & ( X
              = ( nth_Pr4439495888332055232nt_int @ Xs2 @ I2 ) )
            & ! [J2: nat] :
                ( ( ord_less_nat @ J2 @ I2 )
               => ~ ( P @ ( nth_Pr4439495888332055232nt_int @ Xs2 @ J2 ) ) ) ) ) ) ).

% find_Some_iff2
thf(fact_643_find__Some__iff,axiom,
    ! [P: product_prod_int_int > $o,Xs2: list_P5707943133018811711nt_int,X: product_prod_int_int] :
      ( ( ( find_P5021385047576723413nt_int @ P @ Xs2 )
        = ( some_P4184893108420464158nt_int @ X ) )
      = ( ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s5157815400016825771nt_int @ Xs2 ) )
            & ( P @ ( nth_Pr4439495888332055232nt_int @ Xs2 @ I2 ) )
            & ( X
              = ( nth_Pr4439495888332055232nt_int @ Xs2 @ I2 ) )
            & ! [J2: nat] :
                ( ( ord_less_nat @ J2 @ I2 )
               => ~ ( P @ ( nth_Pr4439495888332055232nt_int @ Xs2 @ J2 ) ) ) ) ) ) ).

% find_Some_iff
thf(fact_644_nth__enumerate__eq,axiom,
    ! [M2: nat,Xs2: list_P5707943133018811711nt_int,N: nat] :
      ( ( ord_less_nat @ M2 @ ( size_s5157815400016825771nt_int @ Xs2 ) )
     => ( ( nth_Pr7109764839600559639nt_int @ ( enumer5233495138702836510nt_int @ N @ Xs2 ) @ M2 )
        = ( produc6532261156004778512nt_int @ ( plus_plus_nat @ N @ M2 ) @ ( nth_Pr4439495888332055232nt_int @ Xs2 @ M2 ) ) ) ) ).

% nth_enumerate_eq
thf(fact_645_Un__iff,axiom,
    ! [C: product_prod_int_int,A: set_Pr958786334691620121nt_int,B2: set_Pr958786334691620121nt_int] :
      ( ( member5262025264175285858nt_int @ C @ ( sup_su6024340866399070445nt_int @ A @ B2 ) )
      = ( ( member5262025264175285858nt_int @ C @ A )
        | ( member5262025264175285858nt_int @ C @ B2 ) ) ) ).

% Un_iff
thf(fact_646_Un__iff,axiom,
    ! [C: int,A: set_int,B2: set_int] :
      ( ( member_int2 @ C @ ( sup_sup_set_int @ A @ B2 ) )
      = ( ( member_int2 @ C @ A )
        | ( member_int2 @ C @ B2 ) ) ) ).

% Un_iff
thf(fact_647_Un__iff,axiom,
    ! [C: nat,A: set_nat,B2: set_nat] :
      ( ( member_nat2 @ C @ ( sup_sup_set_nat @ A @ B2 ) )
      = ( ( member_nat2 @ C @ A )
        | ( member_nat2 @ C @ B2 ) ) ) ).

% Un_iff
thf(fact_648_UnCI,axiom,
    ! [C: product_prod_int_int,B2: set_Pr958786334691620121nt_int,A: set_Pr958786334691620121nt_int] :
      ( ( ~ ( member5262025264175285858nt_int @ C @ B2 )
       => ( member5262025264175285858nt_int @ C @ A ) )
     => ( member5262025264175285858nt_int @ C @ ( sup_su6024340866399070445nt_int @ A @ B2 ) ) ) ).

% UnCI
thf(fact_649_UnCI,axiom,
    ! [C: int,B2: set_int,A: set_int] :
      ( ( ~ ( member_int2 @ C @ B2 )
       => ( member_int2 @ C @ A ) )
     => ( member_int2 @ C @ ( sup_sup_set_int @ A @ B2 ) ) ) ).

% UnCI
thf(fact_650_UnCI,axiom,
    ! [C: nat,B2: set_nat,A: set_nat] :
      ( ( ~ ( member_nat2 @ C @ B2 )
       => ( member_nat2 @ C @ A ) )
     => ( member_nat2 @ C @ ( sup_sup_set_nat @ A @ B2 ) ) ) ).

% UnCI
thf(fact_651_Un__insert__right,axiom,
    ! [A: set_nat,A2: nat,B2: set_nat] :
      ( ( sup_sup_set_nat @ A @ ( insert_nat2 @ A2 @ B2 ) )
      = ( insert_nat2 @ A2 @ ( sup_sup_set_nat @ A @ B2 ) ) ) ).

% Un_insert_right
thf(fact_652_Un__insert__left,axiom,
    ! [A2: nat,B2: set_nat,C2: set_nat] :
      ( ( sup_sup_set_nat @ ( insert_nat2 @ A2 @ B2 ) @ C2 )
      = ( insert_nat2 @ A2 @ ( sup_sup_set_nat @ B2 @ C2 ) ) ) ).

% Un_insert_left
thf(fact_653_length__enumerate,axiom,
    ! [N: nat,Xs2: list_P5707943133018811711nt_int] :
      ( ( size_s2001693051472072450nt_int @ ( enumer5233495138702836510nt_int @ N @ Xs2 ) )
      = ( size_s5157815400016825771nt_int @ Xs2 ) ) ).

% length_enumerate
thf(fact_654_set__union,axiom,
    ! [Xs2: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int] :
      ( ( set_Pr2470121279949933262nt_int @ ( union_56799373549498035nt_int @ Xs2 @ Ys ) )
      = ( sup_su6024340866399070445nt_int @ ( set_Pr2470121279949933262nt_int @ Xs2 ) @ ( set_Pr2470121279949933262nt_int @ Ys ) ) ) ).

% set_union
thf(fact_655_set__union,axiom,
    ! [Xs2: list_nat,Ys: list_nat] :
      ( ( set_nat2 @ ( union_nat @ Xs2 @ Ys ) )
      = ( sup_sup_set_nat @ ( set_nat2 @ Xs2 ) @ ( set_nat2 @ Ys ) ) ) ).

% set_union
thf(fact_656_Un__UNIV__right,axiom,
    ! [A: set_nat] :
      ( ( sup_sup_set_nat @ A @ top_top_set_nat )
      = top_top_set_nat ) ).

% Un_UNIV_right
thf(fact_657_Un__UNIV__left,axiom,
    ! [B2: set_nat] :
      ( ( sup_sup_set_nat @ top_top_set_nat @ B2 )
      = top_top_set_nat ) ).

% Un_UNIV_left
thf(fact_658_Un__left__commute,axiom,
    ! [A: set_nat,B2: set_nat,C2: set_nat] :
      ( ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ B2 @ C2 ) )
      = ( sup_sup_set_nat @ B2 @ ( sup_sup_set_nat @ A @ C2 ) ) ) ).

% Un_left_commute
thf(fact_659_Un__left__absorb,axiom,
    ! [A: set_nat,B2: set_nat] :
      ( ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ A @ B2 ) )
      = ( sup_sup_set_nat @ A @ B2 ) ) ).

% Un_left_absorb
thf(fact_660_Un__commute,axiom,
    ( sup_sup_set_nat
    = ( ^ [A5: set_nat,B7: set_nat] : ( sup_sup_set_nat @ B7 @ A5 ) ) ) ).

% Un_commute
thf(fact_661_Un__absorb,axiom,
    ! [A: set_nat] :
      ( ( sup_sup_set_nat @ A @ A )
      = A ) ).

% Un_absorb
thf(fact_662_Un__assoc,axiom,
    ! [A: set_nat,B2: set_nat,C2: set_nat] :
      ( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A @ B2 ) @ C2 )
      = ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ B2 @ C2 ) ) ) ).

% Un_assoc
thf(fact_663_ball__Un,axiom,
    ! [A: set_nat,B2: set_nat,P: nat > $o] :
      ( ( ! [X3: nat] :
            ( ( member_nat2 @ X3 @ ( sup_sup_set_nat @ A @ B2 ) )
           => ( P @ X3 ) ) )
      = ( ! [X3: nat] :
            ( ( member_nat2 @ X3 @ A )
           => ( P @ X3 ) )
        & ! [X3: nat] :
            ( ( member_nat2 @ X3 @ B2 )
           => ( P @ X3 ) ) ) ) ).

% ball_Un
thf(fact_664_bex__Un,axiom,
    ! [A: set_nat,B2: set_nat,P: nat > $o] :
      ( ( ? [X3: nat] :
            ( ( member_nat2 @ X3 @ ( sup_sup_set_nat @ A @ B2 ) )
            & ( P @ X3 ) ) )
      = ( ? [X3: nat] :
            ( ( member_nat2 @ X3 @ A )
            & ( P @ X3 ) )
        | ? [X3: nat] :
            ( ( member_nat2 @ X3 @ B2 )
            & ( P @ X3 ) ) ) ) ).

% bex_Un
thf(fact_665_UnI2,axiom,
    ! [C: product_prod_int_int,B2: set_Pr958786334691620121nt_int,A: set_Pr958786334691620121nt_int] :
      ( ( member5262025264175285858nt_int @ C @ B2 )
     => ( member5262025264175285858nt_int @ C @ ( sup_su6024340866399070445nt_int @ A @ B2 ) ) ) ).

% UnI2
thf(fact_666_UnI2,axiom,
    ! [C: int,B2: set_int,A: set_int] :
      ( ( member_int2 @ C @ B2 )
     => ( member_int2 @ C @ ( sup_sup_set_int @ A @ B2 ) ) ) ).

% UnI2
thf(fact_667_UnI2,axiom,
    ! [C: nat,B2: set_nat,A: set_nat] :
      ( ( member_nat2 @ C @ B2 )
     => ( member_nat2 @ C @ ( sup_sup_set_nat @ A @ B2 ) ) ) ).

% UnI2
thf(fact_668_UnI1,axiom,
    ! [C: product_prod_int_int,A: set_Pr958786334691620121nt_int,B2: set_Pr958786334691620121nt_int] :
      ( ( member5262025264175285858nt_int @ C @ A )
     => ( member5262025264175285858nt_int @ C @ ( sup_su6024340866399070445nt_int @ A @ B2 ) ) ) ).

% UnI1
thf(fact_669_UnI1,axiom,
    ! [C: int,A: set_int,B2: set_int] :
      ( ( member_int2 @ C @ A )
     => ( member_int2 @ C @ ( sup_sup_set_int @ A @ B2 ) ) ) ).

% UnI1
thf(fact_670_UnI1,axiom,
    ! [C: nat,A: set_nat,B2: set_nat] :
      ( ( member_nat2 @ C @ A )
     => ( member_nat2 @ C @ ( sup_sup_set_nat @ A @ B2 ) ) ) ).

% UnI1
thf(fact_671_UnE,axiom,
    ! [C: product_prod_int_int,A: set_Pr958786334691620121nt_int,B2: set_Pr958786334691620121nt_int] :
      ( ( member5262025264175285858nt_int @ C @ ( sup_su6024340866399070445nt_int @ A @ B2 ) )
     => ( ~ ( member5262025264175285858nt_int @ C @ A )
       => ( member5262025264175285858nt_int @ C @ B2 ) ) ) ).

% UnE
thf(fact_672_UnE,axiom,
    ! [C: int,A: set_int,B2: set_int] :
      ( ( member_int2 @ C @ ( sup_sup_set_int @ A @ B2 ) )
     => ( ~ ( member_int2 @ C @ A )
       => ( member_int2 @ C @ B2 ) ) ) ).

% UnE
thf(fact_673_UnE,axiom,
    ! [C: nat,A: set_nat,B2: set_nat] :
      ( ( member_nat2 @ C @ ( sup_sup_set_nat @ A @ B2 ) )
     => ( ~ ( member_nat2 @ C @ A )
       => ( member_nat2 @ C @ B2 ) ) ) ).

% UnE
thf(fact_674_image__Un,axiom,
    ! [F: product_prod_int_int > int,A: set_Pr958786334691620121nt_int,B2: set_Pr958786334691620121nt_int] :
      ( ( image_5042161079198086560nt_int @ F @ ( sup_su6024340866399070445nt_int @ A @ B2 ) )
      = ( sup_sup_set_int @ ( image_5042161079198086560nt_int @ F @ A ) @ ( image_5042161079198086560nt_int @ F @ B2 ) ) ) ).

% image_Un
thf(fact_675_image__Un,axiom,
    ! [F: nat > nat,A: set_nat,B2: set_nat] :
      ( ( image_nat_nat @ F @ ( sup_sup_set_nat @ A @ B2 ) )
      = ( sup_sup_set_nat @ ( image_nat_nat @ F @ A ) @ ( image_nat_nat @ F @ B2 ) ) ) ).

% image_Un
thf(fact_676_find__cong,axiom,
    ! [Xs2: list_int,Ys: list_int,P: int > $o,Q2: int > $o] :
      ( ( Xs2 = Ys )
     => ( ! [X2: int] :
            ( ( member_int2 @ X2 @ ( set_int2 @ Ys ) )
           => ( ( P @ X2 )
              = ( Q2 @ X2 ) ) )
       => ( ( find_int @ P @ Xs2 )
          = ( find_int @ Q2 @ Ys ) ) ) ) ).

% find_cong
thf(fact_677_find__cong,axiom,
    ! [Xs2: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int,P: product_prod_int_int > $o,Q2: product_prod_int_int > $o] :
      ( ( Xs2 = Ys )
     => ( ! [X2: product_prod_int_int] :
            ( ( member5262025264175285858nt_int @ X2 @ ( set_Pr2470121279949933262nt_int @ Ys ) )
           => ( ( P @ X2 )
              = ( Q2 @ X2 ) ) )
       => ( ( find_P5021385047576723413nt_int @ P @ Xs2 )
          = ( find_P5021385047576723413nt_int @ Q2 @ Ys ) ) ) ) ).

% find_cong
thf(fact_678_find_Osimps_I2_J,axiom,
    ! [P: int > $o,X: int,Xs2: list_int] :
      ( ( ( P @ X )
       => ( ( find_int @ P @ ( cons_int @ X @ Xs2 ) )
          = ( some_int @ X ) ) )
      & ( ~ ( P @ X )
       => ( ( find_int @ P @ ( cons_int @ X @ Xs2 ) )
          = ( find_int @ P @ Xs2 ) ) ) ) ).

% find.simps(2)
thf(fact_679_find__None__iff,axiom,
    ! [P: int > $o,Xs2: list_int] :
      ( ( ( find_int @ P @ Xs2 )
        = none_int )
      = ( ~ ? [X3: int] :
              ( ( member_int2 @ X3 @ ( set_int2 @ Xs2 ) )
              & ( P @ X3 ) ) ) ) ).

% find_None_iff
thf(fact_680_find__None__iff,axiom,
    ! [P: product_prod_int_int > $o,Xs2: list_P5707943133018811711nt_int] :
      ( ( ( find_P5021385047576723413nt_int @ P @ Xs2 )
        = none_P2377608414092835994nt_int )
      = ( ~ ? [X3: product_prod_int_int] :
              ( ( member5262025264175285858nt_int @ X3 @ ( set_Pr2470121279949933262nt_int @ Xs2 ) )
              & ( P @ X3 ) ) ) ) ).

% find_None_iff
thf(fact_681_find__None__iff2,axiom,
    ! [P: int > $o,Xs2: list_int] :
      ( ( none_int
        = ( find_int @ P @ Xs2 ) )
      = ( ~ ? [X3: int] :
              ( ( member_int2 @ X3 @ ( set_int2 @ Xs2 ) )
              & ( P @ X3 ) ) ) ) ).

% find_None_iff2
thf(fact_682_find__None__iff2,axiom,
    ! [P: product_prod_int_int > $o,Xs2: list_P5707943133018811711nt_int] :
      ( ( none_P2377608414092835994nt_int
        = ( find_P5021385047576723413nt_int @ P @ Xs2 ) )
      = ( ~ ? [X3: product_prod_int_int] :
              ( ( member5262025264175285858nt_int @ X3 @ ( set_Pr2470121279949933262nt_int @ Xs2 ) )
              & ( P @ X3 ) ) ) ) ).

% find_None_iff2
thf(fact_683_boolean__algebra_Odisj__one__right,axiom,
    ! [X: set_nat] :
      ( ( sup_sup_set_nat @ X @ top_top_set_nat )
      = top_top_set_nat ) ).

% boolean_algebra.disj_one_right
thf(fact_684_boolean__algebra_Odisj__one__left,axiom,
    ! [X: set_nat] :
      ( ( sup_sup_set_nat @ top_top_set_nat @ X )
      = top_top_set_nat ) ).

% boolean_algebra.disj_one_left
thf(fact_685_sup__top__right,axiom,
    ! [X: set_nat] :
      ( ( sup_sup_set_nat @ X @ top_top_set_nat )
      = top_top_set_nat ) ).

% sup_top_right
thf(fact_686_sup__top__left,axiom,
    ! [X: set_nat] :
      ( ( sup_sup_set_nat @ top_top_set_nat @ X )
      = top_top_set_nat ) ).

% sup_top_left
thf(fact_687_listrel__iff__nth,axiom,
    ! [Xs2: list_int,Ys: list_int,R: set_Pr958786334691620121nt_int] :
      ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs2 @ Ys ) @ ( listrel_int_int @ R ) )
      = ( ( ( size_size_list_int @ Xs2 )
          = ( size_size_list_int @ Ys ) )
        & ! [N3: nat] :
            ( ( ord_less_nat @ N3 @ ( size_size_list_int @ Xs2 ) )
           => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( nth_int @ Xs2 @ N3 ) @ ( nth_int @ Ys @ N3 ) ) @ R ) ) ) ) ).

% listrel_iff_nth
thf(fact_688_listrel__iff__nth,axiom,
    ! [Xs2: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int,R: set_Pr2560585780119916871nt_int] :
      ( ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ Xs2 @ Ys ) @ ( listre7525836092462517041nt_int @ R ) )
      = ( ( ( size_s5157815400016825771nt_int @ Xs2 )
          = ( size_s5157815400016825771nt_int @ Ys ) )
        & ! [N3: nat] :
            ( ( ord_less_nat @ N3 @ ( size_s5157815400016825771nt_int @ Xs2 ) )
           => ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ ( nth_Pr4439495888332055232nt_int @ Xs2 @ N3 ) @ ( nth_Pr4439495888332055232nt_int @ Ys @ N3 ) ) @ R ) ) ) ) ).

% listrel_iff_nth
thf(fact_689_in__measures_I2_J,axiom,
    ! [X: int,Y: int,F: int > nat,Fs: list_int_nat] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( measures_int @ ( cons_int_nat @ F @ Fs ) ) )
      = ( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
        | ( ( ( F @ X )
            = ( F @ Y ) )
          & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( measures_int @ Fs ) ) ) ) ) ).

% in_measures(2)
thf(fact_690_nth__equal__first__eq,axiom,
    ! [X: int,Xs2: list_int,N: nat] :
      ( ~ ( member_int2 @ X @ ( set_int2 @ Xs2 ) )
     => ( ( ord_less_eq_nat @ N @ ( size_size_list_int @ Xs2 ) )
       => ( ( ( nth_int @ ( cons_int @ X @ Xs2 ) @ N )
            = X )
          = ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
thf(fact_691_nth__equal__first__eq,axiom,
    ! [X: product_prod_int_int,Xs2: list_P5707943133018811711nt_int,N: nat] :
      ( ~ ( member5262025264175285858nt_int @ X @ ( set_Pr2470121279949933262nt_int @ Xs2 ) )
     => ( ( ord_less_eq_nat @ N @ ( size_s5157815400016825771nt_int @ Xs2 ) )
       => ( ( ( nth_Pr4439495888332055232nt_int @ ( cons_P3334398858971670639nt_int @ X @ Xs2 ) @ N )
            = X )
          = ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
thf(fact_692_partition__set,axiom,
    ! [P: product_prod_int_int > $o,Xs2: list_P5707943133018811711nt_int,Yes: list_P5707943133018811711nt_int,No: list_P5707943133018811711nt_int] :
      ( ( ( partit6659055674896272056nt_int @ P @ Xs2 )
        = ( produc1932183703851549015nt_int @ Yes @ No ) )
     => ( ( sup_su6024340866399070445nt_int @ ( set_Pr2470121279949933262nt_int @ Yes ) @ ( set_Pr2470121279949933262nt_int @ No ) )
        = ( set_Pr2470121279949933262nt_int @ Xs2 ) ) ) ).

% partition_set
thf(fact_693_partition__set,axiom,
    ! [P: nat > $o,Xs2: list_nat,Yes: list_nat,No: list_nat] :
      ( ( ( partition_nat @ P @ Xs2 )
        = ( produc2694037385005941721st_nat @ Yes @ No ) )
     => ( ( sup_sup_set_nat @ ( set_nat2 @ Yes ) @ ( set_nat2 @ No ) )
        = ( set_nat2 @ Xs2 ) ) ) ).

% partition_set
thf(fact_694_le__zero__eq,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ N @ zero_zero_nat )
      = ( N = zero_zero_nat ) ) ).

% le_zero_eq
thf(fact_695_add__le__cancel__left,axiom,
    ! [C: nat,A2: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) )
      = ( ord_less_eq_nat @ A2 @ B ) ) ).

% add_le_cancel_left
thf(fact_696_add__le__cancel__left,axiom,
    ! [C: int,A2: int,B: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B ) )
      = ( ord_less_eq_int @ A2 @ B ) ) ).

% add_le_cancel_left
thf(fact_697_add__le__cancel__right,axiom,
    ! [A2: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) )
      = ( ord_less_eq_nat @ A2 @ B ) ) ).

% add_le_cancel_right
thf(fact_698_add__le__cancel__right,axiom,
    ! [A2: int,C: int,B: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B @ C ) )
      = ( ord_less_eq_int @ A2 @ B ) ) ).

% add_le_cancel_right
thf(fact_699_bot__nat__0_Oextremum,axiom,
    ! [A2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A2 ) ).

% bot_nat_0.extremum
thf(fact_700_le0,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).

% le0
thf(fact_701_nat__add__left__cancel__le,axiom,
    ! [K: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N ) )
      = ( ord_less_eq_nat @ M2 @ N ) ) ).

% nat_add_left_cancel_le
thf(fact_702_add__le__same__cancel1,axiom,
    ! [B: nat,A2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A2 ) @ B )
      = ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).

% add_le_same_cancel1
thf(fact_703_add__le__same__cancel1,axiom,
    ! [B: int,A2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ B @ A2 ) @ B )
      = ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).

% add_le_same_cancel1
thf(fact_704_add__le__same__cancel2,axiom,
    ! [A2: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ B ) @ B )
      = ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).

% add_le_same_cancel2
thf(fact_705_add__le__same__cancel2,axiom,
    ! [A2: int,B: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ B ) @ B )
      = ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).

% add_le_same_cancel2
thf(fact_706_le__add__same__cancel1,axiom,
    ! [A2: nat,B: nat] :
      ( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ A2 @ B ) )
      = ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).

% le_add_same_cancel1
thf(fact_707_le__add__same__cancel1,axiom,
    ! [A2: int,B: int] :
      ( ( ord_less_eq_int @ A2 @ ( plus_plus_int @ A2 @ B ) )
      = ( ord_less_eq_int @ zero_zero_int @ B ) ) ).

% le_add_same_cancel1
thf(fact_708_le__add__same__cancel2,axiom,
    ! [A2: nat,B: nat] :
      ( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ B @ A2 ) )
      = ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).

% le_add_same_cancel2
thf(fact_709_le__add__same__cancel2,axiom,
    ! [A2: int,B: int] :
      ( ( ord_less_eq_int @ A2 @ ( plus_plus_int @ B @ A2 ) )
      = ( ord_less_eq_int @ zero_zero_int @ B ) ) ).

% le_add_same_cancel2
thf(fact_710_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ A2 ) @ zero_zero_int )
      = ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).

% double_add_le_zero_iff_single_add_le_zero
thf(fact_711_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A2 @ A2 ) )
      = ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ).

% zero_le_double_add_iff_zero_le_single_add
thf(fact_712_take__all__iff,axiom,
    ! [N: nat,Xs2: list_P5707943133018811711nt_int] :
      ( ( ( take_P8218740963776755879nt_int @ N @ Xs2 )
        = Xs2 )
      = ( ord_less_eq_nat @ ( size_s5157815400016825771nt_int @ Xs2 ) @ N ) ) ).

% take_all_iff
thf(fact_713_take__all,axiom,
    ! [Xs2: list_P5707943133018811711nt_int,N: nat] :
      ( ( ord_less_eq_nat @ ( size_s5157815400016825771nt_int @ Xs2 ) @ N )
     => ( ( take_P8218740963776755879nt_int @ N @ Xs2 )
        = Xs2 ) ) ).

% take_all
thf(fact_714_nat__mult__le__cancel__disj,axiom,
    ! [K: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ord_less_eq_nat @ M2 @ N ) ) ) ).

% nat_mult_le_cancel_disj
thf(fact_715_mult__le__cancel2,axiom,
    ! [M2: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N @ K ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ord_less_eq_nat @ M2 @ N ) ) ) ).

% mult_le_cancel2
thf(fact_716_verit__comp__simplify1_I3_J,axiom,
    ! [B5: nat,A6: nat] :
      ( ( ~ ( ord_less_eq_nat @ B5 @ A6 ) )
      = ( ord_less_nat @ A6 @ B5 ) ) ).

% verit_comp_simplify1(3)
thf(fact_717_verit__comp__simplify1_I3_J,axiom,
    ! [B5: int,A6: int] :
      ( ( ~ ( ord_less_eq_int @ B5 @ A6 ) )
      = ( ord_less_int @ A6 @ B5 ) ) ).

% verit_comp_simplify1(3)
thf(fact_718_nat__less__le,axiom,
    ( ord_less_nat
    = ( ^ [M4: nat,N3: nat] :
          ( ( ord_less_eq_nat @ M4 @ N3 )
          & ( M4 != N3 ) ) ) ) ).

% nat_less_le
thf(fact_719_less__imp__le__nat,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( ord_less_eq_nat @ M2 @ N ) ) ).

% less_imp_le_nat
thf(fact_720_le__eq__less__or__eq,axiom,
    ( ord_less_eq_nat
    = ( ^ [M4: nat,N3: nat] :
          ( ( ord_less_nat @ M4 @ N3 )
          | ( M4 = N3 ) ) ) ) ).

% le_eq_less_or_eq
thf(fact_721_less__or__eq__imp__le,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( ord_less_nat @ M2 @ N )
        | ( M2 = N ) )
     => ( ord_less_eq_nat @ M2 @ N ) ) ).

% less_or_eq_imp_le
thf(fact_722_le__neq__implies__less,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( M2 != N )
       => ( ord_less_nat @ M2 @ N ) ) ) ).

% le_neq_implies_less
thf(fact_723_less__mono__imp__le__mono,axiom,
    ! [F: nat > nat,I: nat,J: nat] :
      ( ! [I3: nat,J3: nat] :
          ( ( ord_less_nat @ I3 @ J3 )
         => ( ord_less_nat @ ( F @ I3 ) @ ( F @ J3 ) ) )
     => ( ( ord_less_eq_nat @ I @ J )
       => ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).

% less_mono_imp_le_mono
thf(fact_724_add__leE,axiom,
    ! [M2: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N )
     => ~ ( ( ord_less_eq_nat @ M2 @ N )
         => ~ ( ord_less_eq_nat @ K @ N ) ) ) ).

% add_leE
thf(fact_725_le__add1,axiom,
    ! [N: nat,M2: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M2 ) ) ).

% le_add1
thf(fact_726_le__add2,axiom,
    ! [N: nat,M2: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M2 @ N ) ) ).

% le_add2
thf(fact_727_add__leD1,axiom,
    ! [M2: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N )
     => ( ord_less_eq_nat @ M2 @ N ) ) ).

% add_leD1
thf(fact_728_add__leD2,axiom,
    ! [M2: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N )
     => ( ord_less_eq_nat @ K @ N ) ) ).

% add_leD2
thf(fact_729_le__Suc__ex,axiom,
    ! [K: nat,L: nat] :
      ( ( ord_less_eq_nat @ K @ L )
     => ? [N4: nat] :
          ( L
          = ( plus_plus_nat @ K @ N4 ) ) ) ).

% le_Suc_ex
thf(fact_730_add__le__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_eq_nat @ K @ L )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).

% add_le_mono
thf(fact_731_add__le__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).

% add_le_mono1
thf(fact_732_trans__le__add1,axiom,
    ! [I: nat,J: nat,M2: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M2 ) ) ) ).

% trans_le_add1
thf(fact_733_trans__le__add2,axiom,
    ! [I: nat,J: nat,M2: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M2 @ J ) ) ) ).

% trans_le_add2
thf(fact_734_nat__le__iff__add,axiom,
    ( ord_less_eq_nat
    = ( ^ [M4: nat,N3: nat] :
        ? [K3: nat] :
          ( N3
          = ( plus_plus_nat @ M4 @ K3 ) ) ) ) ).

% nat_le_iff_add
thf(fact_735_le__cube,axiom,
    ! [M2: nat] : ( ord_less_eq_nat @ M2 @ ( times_times_nat @ M2 @ ( times_times_nat @ M2 @ M2 ) ) ) ).

% le_cube
thf(fact_736_le__square,axiom,
    ! [M2: nat] : ( ord_less_eq_nat @ M2 @ ( times_times_nat @ M2 @ M2 ) ) ).

% le_square
thf(fact_737_mult__le__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_eq_nat @ K @ L )
       => ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).

% mult_le_mono
thf(fact_738_mult__le__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ).

% mult_le_mono1
thf(fact_739_mult__le__mono2,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ).

% mult_le_mono2
thf(fact_740_Nat_Oex__has__greatest__nat,axiom,
    ! [P: nat > $o,K: nat,B: nat] :
      ( ( P @ K )
     => ( ! [Y3: nat] :
            ( ( P @ Y3 )
           => ( ord_less_eq_nat @ Y3 @ B ) )
       => ? [X2: nat] :
            ( ( P @ X2 )
            & ! [Y9: nat] :
                ( ( P @ Y9 )
               => ( ord_less_eq_nat @ Y9 @ X2 ) ) ) ) ) ).

% Nat.ex_has_greatest_nat
thf(fact_741_nat__le__linear,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
      | ( ord_less_eq_nat @ N @ M2 ) ) ).

% nat_le_linear
thf(fact_742_le__antisym,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( ord_less_eq_nat @ N @ M2 )
       => ( M2 = N ) ) ) ).

% le_antisym
thf(fact_743_eq__imp__le,axiom,
    ! [M2: nat,N: nat] :
      ( ( M2 = N )
     => ( ord_less_eq_nat @ M2 @ N ) ) ).

% eq_imp_le
thf(fact_744_le__trans,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_eq_nat @ J @ K )
       => ( ord_less_eq_nat @ I @ K ) ) ) ).

% le_trans
thf(fact_745_le__refl,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).

% le_refl
thf(fact_746_verit__la__disequality,axiom,
    ! [A2: nat,B: nat] :
      ( ( A2 = B )
      | ~ ( ord_less_eq_nat @ A2 @ B )
      | ~ ( ord_less_eq_nat @ B @ A2 ) ) ).

% verit_la_disequality
thf(fact_747_verit__la__disequality,axiom,
    ! [A2: int,B: int] :
      ( ( A2 = B )
      | ~ ( ord_less_eq_int @ A2 @ B )
      | ~ ( ord_less_eq_int @ B @ A2 ) ) ).

% verit_la_disequality
thf(fact_748_verit__comp__simplify1_I2_J,axiom,
    ! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).

% verit_comp_simplify1(2)
thf(fact_749_verit__comp__simplify1_I2_J,axiom,
    ! [A2: int] : ( ord_less_eq_int @ A2 @ A2 ) ).

% verit_comp_simplify1(2)
thf(fact_750_le__0__eq,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ N @ zero_zero_nat )
      = ( N = zero_zero_nat ) ) ).

% le_0_eq
thf(fact_751_bot__nat__0_Oextremum__uniqueI,axiom,
    ! [A2: nat] :
      ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
     => ( A2 = zero_zero_nat ) ) ).

% bot_nat_0.extremum_uniqueI
thf(fact_752_bot__nat__0_Oextremum__unique,axiom,
    ! [A2: nat] :
      ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
      = ( A2 = zero_zero_nat ) ) ).

% bot_nat_0.extremum_unique
thf(fact_753_less__eq__nat_Osimps_I1_J,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).

% less_eq_nat.simps(1)
thf(fact_754_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_eq_nat @ I @ J )
        & ( K = L ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_755_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( ord_less_eq_int @ I @ J )
        & ( K = L ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_756_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( I = J )
        & ( ord_less_eq_nat @ K @ L ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_757_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( I = J )
        & ( ord_less_eq_int @ K @ L ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_758_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_eq_nat @ I @ J )
        & ( ord_less_eq_nat @ K @ L ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_759_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( ord_less_eq_int @ I @ J )
        & ( ord_less_eq_int @ K @ L ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_760_add__mono,axiom,
    ! [A2: nat,B: nat,C: nat,D2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B )
     => ( ( ord_less_eq_nat @ C @ D2 )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).

% add_mono
thf(fact_761_add__mono,axiom,
    ! [A2: int,B: int,C: int,D2: int] :
      ( ( ord_less_eq_int @ A2 @ B )
     => ( ( ord_less_eq_int @ C @ D2 )
       => ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B @ D2 ) ) ) ) ).

% add_mono
thf(fact_762_add__left__mono,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) ) ) ).

% add_left_mono
thf(fact_763_add__left__mono,axiom,
    ! [A2: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A2 @ B )
     => ( ord_less_eq_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B ) ) ) ).

% add_left_mono
thf(fact_764_less__eqE,axiom,
    ! [A2: nat,B: nat] :
      ( ( ord_less_eq_nat @ A2 @ B )
     => ~ ! [C5: nat] :
            ( B
           != ( plus_plus_nat @ A2 @ C5 ) ) ) ).

% less_eqE
thf(fact_765_add__right__mono,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).

% add_right_mono
thf(fact_766_add__right__mono,axiom,
    ! [A2: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A2 @ B )
     => ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).

% add_right_mono
thf(fact_767_le__iff__add,axiom,
    ( ord_less_eq_nat
    = ( ^ [A3: nat,B3: nat] :
        ? [C3: nat] :
          ( B3
          = ( plus_plus_nat @ A3 @ C3 ) ) ) ) ).

% le_iff_add
thf(fact_768_add__le__imp__le__left,axiom,
    ! [C: nat,A2: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) )
     => ( ord_less_eq_nat @ A2 @ B ) ) ).

% add_le_imp_le_left
thf(fact_769_add__le__imp__le__left,axiom,
    ! [C: int,A2: int,B: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B ) )
     => ( ord_less_eq_int @ A2 @ B ) ) ).

% add_le_imp_le_left
thf(fact_770_add__le__imp__le__right,axiom,
    ! [A2: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) )
     => ( ord_less_eq_nat @ A2 @ B ) ) ).

% add_le_imp_le_right
thf(fact_771_add__le__imp__le__right,axiom,
    ! [A2: int,C: int,B: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B @ C ) )
     => ( ord_less_eq_int @ A2 @ B ) ) ).

% add_le_imp_le_right
thf(fact_772_zero__le,axiom,
    ! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).

% zero_le
thf(fact_773_le__numeral__extra_I3_J,axiom,
    ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).

% le_numeral_extra(3)
thf(fact_774_le__numeral__extra_I3_J,axiom,
    ord_less_eq_int @ zero_zero_int @ zero_zero_int ).

% le_numeral_extra(3)
thf(fact_775_measures__lesseq,axiom,
    ! [F: int > nat,X: int,Y: int,Fs: list_int_nat] :
      ( ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) )
     => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( measures_int @ Fs ) )
       => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( measures_int @ ( cons_int_nat @ F @ Fs ) ) ) ) ) ).

% measures_lesseq
thf(fact_776_mult__mono,axiom,
    ! [A2: nat,B: nat,C: nat,D2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B )
     => ( ( ord_less_eq_nat @ C @ D2 )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).

% mult_mono
thf(fact_777_mult__mono,axiom,
    ! [A2: int,B: int,C: int,D2: int] :
      ( ( ord_less_eq_int @ A2 @ B )
     => ( ( ord_less_eq_int @ C @ D2 )
       => ( ( ord_less_eq_int @ zero_zero_int @ B )
         => ( ( ord_less_eq_int @ zero_zero_int @ C )
           => ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).

% mult_mono
thf(fact_778_mult__mono_H,axiom,
    ! [A2: nat,B: nat,C: nat,D2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B )
     => ( ( ord_less_eq_nat @ C @ D2 )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).

% mult_mono'
thf(fact_779_mult__mono_H,axiom,
    ! [A2: int,B: int,C: int,D2: int] :
      ( ( ord_less_eq_int @ A2 @ B )
     => ( ( ord_less_eq_int @ C @ D2 )
       => ( ( ord_less_eq_int @ zero_zero_int @ A2 )
         => ( ( ord_less_eq_int @ zero_zero_int @ C )
           => ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).

% mult_mono'
thf(fact_780_zero__le__square,axiom,
    ! [A2: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A2 @ A2 ) ) ).

% zero_le_square
thf(fact_781_split__mult__pos__le,axiom,
    ! [A2: int,B: int] :
      ( ( ( ( ord_less_eq_int @ zero_zero_int @ A2 )
          & ( ord_less_eq_int @ zero_zero_int @ B ) )
        | ( ( ord_less_eq_int @ A2 @ zero_zero_int )
          & ( ord_less_eq_int @ B @ zero_zero_int ) ) )
     => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A2 @ B ) ) ) ).

% split_mult_pos_le
thf(fact_782_mult__left__mono__neg,axiom,
    ! [B: int,A2: int,C: int] :
      ( ( ord_less_eq_int @ B @ A2 )
     => ( ( ord_less_eq_int @ C @ zero_zero_int )
       => ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) ) ) ) ).

% mult_left_mono_neg
thf(fact_783_mult__nonpos__nonpos,axiom,
    ! [A2: int,B: int] :
      ( ( ord_less_eq_int @ A2 @ zero_zero_int )
     => ( ( ord_less_eq_int @ B @ zero_zero_int )
       => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A2 @ B ) ) ) ) ).

% mult_nonpos_nonpos
thf(fact_784_mult__left__mono,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B ) ) ) ) ).

% mult_left_mono
thf(fact_785_mult__left__mono,axiom,
    ! [A2: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A2 @ B )
     => ( ( ord_less_eq_int @ zero_zero_int @ C )
       => ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) ) ) ) ).

% mult_left_mono
thf(fact_786_mult__right__mono__neg,axiom,
    ! [B: int,A2: int,C: int] :
      ( ( ord_less_eq_int @ B @ A2 )
     => ( ( ord_less_eq_int @ C @ zero_zero_int )
       => ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).

% mult_right_mono_neg
thf(fact_787_mult__right__mono,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).

% mult_right_mono
thf(fact_788_mult__right__mono,axiom,
    ! [A2: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A2 @ B )
     => ( ( ord_less_eq_int @ zero_zero_int @ C )
       => ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).

% mult_right_mono
thf(fact_789_mult__le__0__iff,axiom,
    ! [A2: int,B: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ A2 @ B ) @ zero_zero_int )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ A2 )
          & ( ord_less_eq_int @ B @ zero_zero_int ) )
        | ( ( ord_less_eq_int @ A2 @ zero_zero_int )
          & ( ord_less_eq_int @ zero_zero_int @ B ) ) ) ) ).

% mult_le_0_iff
thf(fact_790_split__mult__neg__le,axiom,
    ! [A2: nat,B: nat] :
      ( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
          & ( ord_less_eq_nat @ B @ zero_zero_nat ) )
        | ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
          & ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
     => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B ) @ zero_zero_nat ) ) ).

% split_mult_neg_le
thf(fact_791_split__mult__neg__le,axiom,
    ! [A2: int,B: int] :
      ( ( ( ( ord_less_eq_int @ zero_zero_int @ A2 )
          & ( ord_less_eq_int @ B @ zero_zero_int ) )
        | ( ( ord_less_eq_int @ A2 @ zero_zero_int )
          & ( ord_less_eq_int @ zero_zero_int @ B ) ) )
     => ( ord_less_eq_int @ ( times_times_int @ A2 @ B ) @ zero_zero_int ) ) ).

% split_mult_neg_le
thf(fact_792_mult__nonneg__nonneg,axiom,
    ! [A2: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A2 @ B ) ) ) ) ).

% mult_nonneg_nonneg
thf(fact_793_mult__nonneg__nonneg,axiom,
    ! [A2: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A2 @ B ) ) ) ) ).

% mult_nonneg_nonneg
thf(fact_794_mult__nonneg__nonpos,axiom,
    ! [A2: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
       => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).

% mult_nonneg_nonpos
thf(fact_795_mult__nonneg__nonpos,axiom,
    ! [A2: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A2 )
     => ( ( ord_less_eq_int @ B @ zero_zero_int )
       => ( ord_less_eq_int @ ( times_times_int @ A2 @ B ) @ zero_zero_int ) ) ) ).

% mult_nonneg_nonpos
thf(fact_796_mult__nonpos__nonneg,axiom,
    ! [A2: nat,B: nat] :
      ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).

% mult_nonpos_nonneg
thf(fact_797_mult__nonpos__nonneg,axiom,
    ! [A2: int,B: int] :
      ( ( ord_less_eq_int @ A2 @ zero_zero_int )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ord_less_eq_int @ ( times_times_int @ A2 @ B ) @ zero_zero_int ) ) ) ).

% mult_nonpos_nonneg
thf(fact_798_mult__nonneg__nonpos2,axiom,
    ! [A2: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
       => ( ord_less_eq_nat @ ( times_times_nat @ B @ A2 ) @ zero_zero_nat ) ) ) ).

% mult_nonneg_nonpos2
thf(fact_799_mult__nonneg__nonpos2,axiom,
    ! [A2: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A2 )
     => ( ( ord_less_eq_int @ B @ zero_zero_int )
       => ( ord_less_eq_int @ ( times_times_int @ B @ A2 ) @ zero_zero_int ) ) ) ).

% mult_nonneg_nonpos2
thf(fact_800_zero__le__mult__iff,axiom,
    ! [A2: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A2 @ B ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ A2 )
          & ( ord_less_eq_int @ zero_zero_int @ B ) )
        | ( ( ord_less_eq_int @ A2 @ zero_zero_int )
          & ( ord_less_eq_int @ B @ zero_zero_int ) ) ) ) ).

% zero_le_mult_iff
thf(fact_801_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_802_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A2: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A2 @ B )
     => ( ( ord_less_eq_int @ zero_zero_int @ C )
       => ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_803_add__decreasing,axiom,
    ! [A2: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ C @ B )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ B ) ) ) ).

% add_decreasing
thf(fact_804_add__decreasing,axiom,
    ! [A2: int,C: int,B: int] :
      ( ( ord_less_eq_int @ A2 @ zero_zero_int )
     => ( ( ord_less_eq_int @ C @ B )
       => ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ B ) ) ) ).

% add_decreasing
thf(fact_805_add__increasing,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).

% add_increasing
thf(fact_806_add__increasing,axiom,
    ! [A2: int,B: int,C: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A2 )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ord_less_eq_int @ B @ ( plus_plus_int @ A2 @ C ) ) ) ) ).

% add_increasing
thf(fact_807_add__decreasing2,axiom,
    ! [C: nat,A2: nat,B: nat] :
      ( ( ord_less_eq_nat @ C @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ A2 @ B )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ B ) ) ) ).

% add_decreasing2
thf(fact_808_add__decreasing2,axiom,
    ! [C: int,A2: int,B: int] :
      ( ( ord_less_eq_int @ C @ zero_zero_int )
     => ( ( ord_less_eq_int @ A2 @ B )
       => ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ B ) ) ) ).

% add_decreasing2
thf(fact_809_add__increasing2,axiom,
    ! [C: nat,B: nat,A2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ C )
     => ( ( ord_less_eq_nat @ B @ A2 )
       => ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).

% add_increasing2
thf(fact_810_add__increasing2,axiom,
    ! [C: int,B: int,A2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C )
     => ( ( ord_less_eq_int @ B @ A2 )
       => ( ord_less_eq_int @ B @ ( plus_plus_int @ A2 @ C ) ) ) ) ).

% add_increasing2
thf(fact_811_add__nonneg__nonneg,axiom,
    ! [A2: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_812_add__nonneg__nonneg,axiom,
    ! [A2: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A2 @ B ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_813_add__nonpos__nonpos,axiom,
    ! [A2: nat,B: nat] :
      ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).

% add_nonpos_nonpos
thf(fact_814_add__nonpos__nonpos,axiom,
    ! [A2: int,B: int] :
      ( ( ord_less_eq_int @ A2 @ zero_zero_int )
     => ( ( ord_less_eq_int @ B @ zero_zero_int )
       => ( ord_less_eq_int @ ( plus_plus_int @ A2 @ B ) @ zero_zero_int ) ) ) ).

% add_nonpos_nonpos
thf(fact_815_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_816_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_817_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_818_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_819_add__less__le__mono,axiom,
    ! [A2: nat,B: nat,C: nat,D2: nat] :
      ( ( ord_less_nat @ A2 @ B )
     => ( ( ord_less_eq_nat @ C @ D2 )
       => ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).

% add_less_le_mono
thf(fact_820_add__less__le__mono,axiom,
    ! [A2: int,B: int,C: int,D2: int] :
      ( ( ord_less_int @ A2 @ B )
     => ( ( ord_less_eq_int @ C @ D2 )
       => ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B @ D2 ) ) ) ) ).

% add_less_le_mono
thf(fact_821_add__le__less__mono,axiom,
    ! [A2: nat,B: nat,C: nat,D2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B )
     => ( ( ord_less_nat @ C @ D2 )
       => ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).

% add_le_less_mono
thf(fact_822_add__le__less__mono,axiom,
    ! [A2: int,B: int,C: int,D2: int] :
      ( ( ord_less_eq_int @ A2 @ B )
     => ( ( ord_less_int @ C @ D2 )
       => ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B @ D2 ) ) ) ) ).

% add_le_less_mono
thf(fact_823_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_nat @ I @ J )
        & ( ord_less_eq_nat @ K @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_824_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( ord_less_int @ I @ J )
        & ( ord_less_eq_int @ K @ L ) )
     => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_825_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_eq_nat @ I @ J )
        & ( ord_less_nat @ K @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_826_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( ord_less_eq_int @ I @ J )
        & ( ord_less_int @ K @ L ) )
     => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_827_ex__least__nat__le,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ N )
     => ( ~ ( P @ zero_zero_nat )
       => ? [K2: nat] :
            ( ( ord_less_eq_nat @ K2 @ N )
            & ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ K2 )
               => ~ ( P @ I4 ) )
            & ( P @ K2 ) ) ) ) ).

% ex_least_nat_le
thf(fact_828_impossible__Cons,axiom,
    ! [Xs2: list_int,Ys: list_int,X: int] :
      ( ( ord_less_eq_nat @ ( size_size_list_int @ Xs2 ) @ ( size_size_list_int @ Ys ) )
     => ( Xs2
       != ( cons_int @ X @ Ys ) ) ) ).

% impossible_Cons
thf(fact_829_impossible__Cons,axiom,
    ! [Xs2: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int,X: product_prod_int_int] :
      ( ( ord_less_eq_nat @ ( size_s5157815400016825771nt_int @ Xs2 ) @ ( size_s5157815400016825771nt_int @ Ys ) )
     => ( Xs2
       != ( cons_P3334398858971670639nt_int @ X @ Ys ) ) ) ).

% impossible_Cons
thf(fact_830_mono__nat__linear__lb,axiom,
    ! [F: nat > nat,M2: nat,K: nat] :
      ( ! [M5: nat,N4: nat] :
          ( ( ord_less_nat @ M5 @ N4 )
         => ( ord_less_nat @ ( F @ M5 ) @ ( F @ N4 ) ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M2 ) @ K ) @ ( F @ ( plus_plus_nat @ M2 @ K ) ) ) ) ).

% mono_nat_linear_lb
thf(fact_831_insort__key_Osimps_I2_J,axiom,
    ! [F: int > nat,X: int,Y: int,Ys: list_int] :
      ( ( ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) )
       => ( ( linord737317855127579385nt_nat @ F @ X @ ( cons_int @ Y @ Ys ) )
          = ( cons_int @ X @ ( cons_int @ Y @ Ys ) ) ) )
      & ( ~ ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) )
       => ( ( linord737317855127579385nt_nat @ F @ X @ ( cons_int @ Y @ Ys ) )
          = ( cons_int @ Y @ ( linord737317855127579385nt_nat @ F @ X @ Ys ) ) ) ) ) ).

% insort_key.simps(2)
thf(fact_832_insort__key_Osimps_I2_J,axiom,
    ! [F: int > int,X: int,Y: int,Ys: list_int] :
      ( ( ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y ) )
       => ( ( linord734827384618529109nt_int @ F @ X @ ( cons_int @ Y @ Ys ) )
          = ( cons_int @ X @ ( cons_int @ Y @ Ys ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y ) )
       => ( ( linord734827384618529109nt_int @ F @ X @ ( cons_int @ Y @ Ys ) )
          = ( cons_int @ Y @ ( linord734827384618529109nt_int @ F @ X @ Ys ) ) ) ) ) ).

% insort_key.simps(2)
thf(fact_833_count__le__length,axiom,
    ! [Xs2: list_P5707943133018811711nt_int,X: product_prod_int_int] : ( ord_less_eq_nat @ ( count_1024995598469094197nt_int @ Xs2 @ X ) @ ( size_s5157815400016825771nt_int @ Xs2 ) ) ).

% count_le_length
thf(fact_834_listrel__eq__len,axiom,
    ! [Xs2: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int,R: set_Pr2560585780119916871nt_int] :
      ( ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ Xs2 @ Ys ) @ ( listre7525836092462517041nt_int @ R ) )
     => ( ( size_s5157815400016825771nt_int @ Xs2 )
        = ( size_s5157815400016825771nt_int @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_835_mult__le__cancel__left,axiom,
    ! [C: int,A2: int,B: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ A2 @ B ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ B @ A2 ) ) ) ) ).

% mult_le_cancel_left
thf(fact_836_mult__le__cancel__right,axiom,
    ! [A2: int,C: int,B: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ A2 @ B ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ B @ A2 ) ) ) ) ).

% mult_le_cancel_right
thf(fact_837_mult__left__less__imp__less,axiom,
    ! [C: nat,A2: nat,B: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ A2 @ B ) ) ) ).

% mult_left_less_imp_less
thf(fact_838_mult__left__less__imp__less,axiom,
    ! [C: int,A2: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ C )
       => ( ord_less_int @ A2 @ B ) ) ) ).

% mult_left_less_imp_less
thf(fact_839_mult__strict__mono,axiom,
    ! [A2: nat,B: nat,C: nat,D2: nat] :
      ( ( ord_less_nat @ A2 @ B )
     => ( ( ord_less_nat @ C @ D2 )
       => ( ( ord_less_nat @ zero_zero_nat @ B )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_840_mult__strict__mono,axiom,
    ! [A2: int,B: int,C: int,D2: int] :
      ( ( ord_less_int @ A2 @ B )
     => ( ( ord_less_int @ C @ D2 )
       => ( ( ord_less_int @ zero_zero_int @ B )
         => ( ( ord_less_eq_int @ zero_zero_int @ C )
           => ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_841_mult__less__cancel__left,axiom,
    ! [C: int,A2: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ A2 @ B ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ B @ A2 ) ) ) ) ).

% mult_less_cancel_left
thf(fact_842_mult__right__less__imp__less,axiom,
    ! [A2: nat,C: nat,B: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ C ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ A2 @ B ) ) ) ).

% mult_right_less_imp_less
thf(fact_843_mult__right__less__imp__less,axiom,
    ! [A2: int,C: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ C )
       => ( ord_less_int @ A2 @ B ) ) ) ).

% mult_right_less_imp_less
thf(fact_844_mult__strict__mono_H,axiom,
    ! [A2: nat,B: nat,C: nat,D2: nat] :
      ( ( ord_less_nat @ A2 @ B )
     => ( ( ord_less_nat @ C @ D2 )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_845_mult__strict__mono_H,axiom,
    ! [A2: int,B: int,C: int,D2: int] :
      ( ( ord_less_int @ A2 @ B )
     => ( ( ord_less_int @ C @ D2 )
       => ( ( ord_less_eq_int @ zero_zero_int @ A2 )
         => ( ( ord_less_eq_int @ zero_zero_int @ C )
           => ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_846_mult__less__cancel__right,axiom,
    ! [A2: int,C: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ A2 @ B ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ B @ A2 ) ) ) ) ).

% mult_less_cancel_right
thf(fact_847_mult__le__cancel__left__neg,axiom,
    ! [C: int,A2: int,B: int] :
      ( ( ord_less_int @ C @ zero_zero_int )
     => ( ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) )
        = ( ord_less_eq_int @ B @ A2 ) ) ) ).

% mult_le_cancel_left_neg
thf(fact_848_mult__le__cancel__left__pos,axiom,
    ! [C: int,A2: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ C )
     => ( ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) )
        = ( ord_less_eq_int @ A2 @ B ) ) ) ).

% mult_le_cancel_left_pos
thf(fact_849_mult__left__le__imp__le,axiom,
    ! [C: nat,A2: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B ) )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ A2 @ B ) ) ) ).

% mult_left_le_imp_le
thf(fact_850_mult__left__le__imp__le,axiom,
    ! [C: int,A2: int,B: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) )
     => ( ( ord_less_int @ zero_zero_int @ C )
       => ( ord_less_eq_int @ A2 @ B ) ) ) ).

% mult_left_le_imp_le
thf(fact_851_mult__right__le__imp__le,axiom,
    ! [A2: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ C ) )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ A2 @ B ) ) ) ).

% mult_right_le_imp_le
thf(fact_852_mult__right__le__imp__le,axiom,
    ! [A2: int,C: int,B: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) )
     => ( ( ord_less_int @ zero_zero_int @ C )
       => ( ord_less_eq_int @ A2 @ B ) ) ) ).

% mult_right_le_imp_le
thf(fact_853_mult__le__less__imp__less,axiom,
    ! [A2: nat,B: nat,C: nat,D2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B )
     => ( ( ord_less_nat @ C @ D2 )
       => ( ( ord_less_nat @ zero_zero_nat @ A2 )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_854_mult__le__less__imp__less,axiom,
    ! [A2: int,B: int,C: int,D2: int] :
      ( ( ord_less_eq_int @ A2 @ B )
     => ( ( ord_less_int @ C @ D2 )
       => ( ( ord_less_int @ zero_zero_int @ A2 )
         => ( ( ord_less_eq_int @ zero_zero_int @ C )
           => ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_855_mult__less__le__imp__less,axiom,
    ! [A2: nat,B: nat,C: nat,D2: nat] :
      ( ( ord_less_nat @ A2 @ B )
     => ( ( ord_less_eq_nat @ C @ D2 )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
         => ( ( ord_less_nat @ zero_zero_nat @ C )
           => ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_856_mult__less__le__imp__less,axiom,
    ! [A2: int,B: int,C: int,D2: int] :
      ( ( ord_less_int @ A2 @ B )
     => ( ( ord_less_eq_int @ C @ D2 )
       => ( ( ord_less_eq_int @ zero_zero_int @ A2 )
         => ( ( ord_less_int @ zero_zero_int @ C )
           => ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_857_mult__le__cancel__iff1,axiom,
    ! [Z: int,X: int,Y: int] :
      ( ( ord_less_int @ zero_zero_int @ Z )
     => ( ( ord_less_eq_int @ ( times_times_int @ X @ Z ) @ ( times_times_int @ Y @ Z ) )
        = ( ord_less_eq_int @ X @ Y ) ) ) ).

% mult_le_cancel_iff1
thf(fact_858_mult__le__cancel__iff2,axiom,
    ! [Z: int,X: int,Y: int] :
      ( ( ord_less_int @ zero_zero_int @ Z )
     => ( ( ord_less_eq_int @ ( times_times_int @ Z @ X ) @ ( times_times_int @ Z @ Y ) )
        = ( ord_less_eq_int @ X @ Y ) ) ) ).

% mult_le_cancel_iff2
thf(fact_859_add__strict__increasing2,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_nat @ B @ C )
       => ( ord_less_nat @ B @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).

% add_strict_increasing2
thf(fact_860_add__strict__increasing2,axiom,
    ! [A2: int,B: int,C: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A2 )
     => ( ( ord_less_int @ B @ C )
       => ( ord_less_int @ B @ ( plus_plus_int @ A2 @ C ) ) ) ) ).

% add_strict_increasing2
thf(fact_861_add__strict__increasing,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_nat @ B @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).

% add_strict_increasing
thf(fact_862_add__strict__increasing,axiom,
    ! [A2: int,B: int,C: int] :
      ( ( ord_less_int @ zero_zero_int @ A2 )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ord_less_int @ B @ ( plus_plus_int @ A2 @ C ) ) ) ) ).

% add_strict_increasing
thf(fact_863_add__pos__nonneg,axiom,
    ! [A2: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).

% add_pos_nonneg
thf(fact_864_add__pos__nonneg,axiom,
    ! [A2: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ A2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A2 @ B ) ) ) ) ).

% add_pos_nonneg
thf(fact_865_add__nonpos__neg,axiom,
    ! [A2: nat,B: nat] :
      ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
     => ( ( ord_less_nat @ B @ zero_zero_nat )
       => ( ord_less_nat @ ( plus_plus_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).

% add_nonpos_neg
thf(fact_866_add__nonpos__neg,axiom,
    ! [A2: int,B: int] :
      ( ( ord_less_eq_int @ A2 @ zero_zero_int )
     => ( ( ord_less_int @ B @ zero_zero_int )
       => ( ord_less_int @ ( plus_plus_int @ A2 @ B ) @ zero_zero_int ) ) ) ).

% add_nonpos_neg
thf(fact_867_add__nonneg__pos,axiom,
    ! [A2: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ B )
       => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).

% add_nonneg_pos
thf(fact_868_add__nonneg__pos,axiom,
    ! [A2: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A2 )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A2 @ B ) ) ) ) ).

% add_nonneg_pos
thf(fact_869_add__neg__nonpos,axiom,
    ! [A2: nat,B: nat] :
      ( ( ord_less_nat @ A2 @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
       => ( ord_less_nat @ ( plus_plus_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).

% add_neg_nonpos
thf(fact_870_add__neg__nonpos,axiom,
    ! [A2: int,B: int] :
      ( ( ord_less_int @ A2 @ zero_zero_int )
     => ( ( ord_less_eq_int @ B @ zero_zero_int )
       => ( ord_less_int @ ( plus_plus_int @ A2 @ B ) @ zero_zero_int ) ) ) ).

% add_neg_nonpos
thf(fact_871_sum__squares__ge__zero,axiom,
    ! [X: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) ) ).

% sum_squares_ge_zero
thf(fact_872_sum__squares__le__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int )
      = ( ( X = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_squares_le_zero_iff
thf(fact_873_nat__mult__le__cancel1,axiom,
    ! [K: nat,M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) )
        = ( ord_less_eq_nat @ M2 @ N ) ) ) ).

% nat_mult_le_cancel1
thf(fact_874_insort__is__Cons,axiom,
    ! [Xs2: list_P5707943133018811711nt_int,F: product_prod_int_int > nat,A2: product_prod_int_int] :
      ( ! [X2: product_prod_int_int] :
          ( ( member5262025264175285858nt_int @ X2 @ ( set_Pr2470121279949933262nt_int @ Xs2 ) )
         => ( ord_less_eq_nat @ ( F @ A2 ) @ ( F @ X2 ) ) )
     => ( ( linord5212096450937010382nt_nat @ F @ A2 @ Xs2 )
        = ( cons_P3334398858971670639nt_int @ A2 @ Xs2 ) ) ) ).

% insort_is_Cons
thf(fact_875_insort__is__Cons,axiom,
    ! [Xs2: list_int,F: int > nat,A2: int] :
      ( ! [X2: int] :
          ( ( member_int2 @ X2 @ ( set_int2 @ Xs2 ) )
         => ( ord_less_eq_nat @ ( F @ A2 ) @ ( F @ X2 ) ) )
     => ( ( linord737317855127579385nt_nat @ F @ A2 @ Xs2 )
        = ( cons_int @ A2 @ Xs2 ) ) ) ).

% insort_is_Cons
thf(fact_876_insort__is__Cons,axiom,
    ! [Xs2: list_P5707943133018811711nt_int,F: product_prod_int_int > int,A2: product_prod_int_int] :
      ( ! [X2: product_prod_int_int] :
          ( ( member5262025264175285858nt_int @ X2 @ ( set_Pr2470121279949933262nt_int @ Xs2 ) )
         => ( ord_less_eq_int @ ( F @ A2 ) @ ( F @ X2 ) ) )
     => ( ( linord5209605980427960106nt_int @ F @ A2 @ Xs2 )
        = ( cons_P3334398858971670639nt_int @ A2 @ Xs2 ) ) ) ).

% insort_is_Cons
thf(fact_877_insort__is__Cons,axiom,
    ! [Xs2: list_int,F: int > int,A2: int] :
      ( ! [X2: int] :
          ( ( member_int2 @ X2 @ ( set_int2 @ Xs2 ) )
         => ( ord_less_eq_int @ ( F @ A2 ) @ ( F @ X2 ) ) )
     => ( ( linord734827384618529109nt_int @ F @ A2 @ Xs2 )
        = ( cons_int @ A2 @ Xs2 ) ) ) ).

% insort_is_Cons
thf(fact_878_listrel__Cons2,axiom,
    ! [Xs2: list_int,Y: int,Ys: list_int,R: set_Pr958786334691620121nt_int] :
      ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs2 @ ( cons_int @ Y @ Ys ) ) @ ( listrel_int_int @ R ) )
     => ~ ! [X2: int,Xs: list_int] :
            ( ( Xs2
              = ( cons_int @ X2 @ Xs ) )
           => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y ) @ R )
             => ~ ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys ) @ ( listrel_int_int @ R ) ) ) ) ) ).

% listrel_Cons2
thf(fact_879_listrel__Cons1,axiom,
    ! [Y: int,Ys: list_int,Xs2: list_int,R: set_Pr958786334691620121nt_int] :
      ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( cons_int @ Y @ Ys ) @ Xs2 ) @ ( listrel_int_int @ R ) )
     => ~ ! [Y3: int,Ys5: list_int] :
            ( ( Xs2
              = ( cons_int @ Y3 @ Ys5 ) )
           => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y @ Y3 ) @ R )
             => ~ ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Ys @ Ys5 ) @ ( listrel_int_int @ R ) ) ) ) ) ).

% listrel_Cons1
thf(fact_880_listrel_OCons,axiom,
    ! [X: int,Y: int,R: set_Pr958786334691620121nt_int,Xs2: list_int,Ys: list_int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R )
     => ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs2 @ Ys ) @ ( listrel_int_int @ R ) )
       => ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( cons_int @ X @ Xs2 ) @ ( cons_int @ Y @ Ys ) ) @ ( listrel_int_int @ R ) ) ) ) ).

% listrel.Cons
thf(fact_881_lenlex__length,axiom,
    ! [Ms: list_P5707943133018811711nt_int,Ns: list_P5707943133018811711nt_int,R: set_Pr2560585780119916871nt_int] :
      ( ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ Ms @ Ns ) @ ( lenlex6370358691973319492nt_int @ R ) )
     => ( ord_less_eq_nat @ ( size_s5157815400016825771nt_int @ Ms ) @ ( size_s5157815400016825771nt_int @ Ns ) ) ) ).

% lenlex_length
thf(fact_882_partition__P,axiom,
    ! [P: product_prod_int_int > $o,Xs2: list_P5707943133018811711nt_int,Yes: list_P5707943133018811711nt_int,No: list_P5707943133018811711nt_int] :
      ( ( ( partit6659055674896272056nt_int @ P @ Xs2 )
        = ( produc1932183703851549015nt_int @ Yes @ No ) )
     => ( ! [X4: product_prod_int_int] :
            ( ( member5262025264175285858nt_int @ X4 @ ( set_Pr2470121279949933262nt_int @ Yes ) )
           => ( P @ X4 ) )
        & ! [X4: product_prod_int_int] :
            ( ( member5262025264175285858nt_int @ X4 @ ( set_Pr2470121279949933262nt_int @ No ) )
           => ~ ( P @ X4 ) ) ) ) ).

% partition_P
thf(fact_883_measures__less,axiom,
    ! [F: int > nat,X: int,Y: int,Fs: list_int_nat] :
      ( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
     => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( measures_int @ ( cons_int_nat @ F @ Fs ) ) ) ) ).

% measures_less
thf(fact_884_nth__take__lemma,axiom,
    ! [K: nat,Xs2: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int] :
      ( ( ord_less_eq_nat @ K @ ( size_s5157815400016825771nt_int @ Xs2 ) )
     => ( ( ord_less_eq_nat @ K @ ( size_s5157815400016825771nt_int @ Ys ) )
       => ( ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ K )
             => ( ( nth_Pr4439495888332055232nt_int @ Xs2 @ I3 )
                = ( nth_Pr4439495888332055232nt_int @ Ys @ I3 ) ) )
         => ( ( take_P8218740963776755879nt_int @ K @ Xs2 )
            = ( take_P8218740963776755879nt_int @ K @ Ys ) ) ) ) ) ).

% nth_take_lemma
thf(fact_885_in__set__enumerate__eq,axiom,
    ! [P2: produc6945250483304103390nt_int,N: nat,Xs2: list_P5707943133018811711nt_int] :
      ( ( member1920110580185997301nt_int @ P2 @ ( set_Pr6407797292369765513nt_int @ ( enumer5233495138702836510nt_int @ N @ Xs2 ) ) )
      = ( ( ord_less_eq_nat @ N @ ( produc4803745454372187940nt_int @ P2 ) )
        & ( ord_less_nat @ ( produc4803745454372187940nt_int @ P2 ) @ ( plus_plus_nat @ ( size_s5157815400016825771nt_int @ Xs2 ) @ N ) )
        & ( ( nth_Pr4439495888332055232nt_int @ Xs2 @ ( minus_minus_nat @ ( produc4803745454372187940nt_int @ P2 ) @ N ) )
          = ( produc3248548876605496418nt_int @ P2 ) ) ) ) ).

% in_set_enumerate_eq
thf(fact_886_nth__image,axiom,
    ! [L: nat,Xs2: list_P5707943133018811711nt_int] :
      ( ( ord_less_eq_nat @ L @ ( size_s5157815400016825771nt_int @ Xs2 ) )
     => ( ( image_2667626500211843362nt_int @ ( nth_Pr4439495888332055232nt_int @ Xs2 ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ L ) )
        = ( set_Pr2470121279949933262nt_int @ ( take_P8218740963776755879nt_int @ L @ Xs2 ) ) ) ) ).

% nth_image
thf(fact_887_insert__subset,axiom,
    ! [X: product_prod_int_int,A: set_Pr958786334691620121nt_int,B2: set_Pr958786334691620121nt_int] :
      ( ( ord_le2843351958646193337nt_int @ ( insert5033312907999012233nt_int @ X @ A ) @ B2 )
      = ( ( member5262025264175285858nt_int @ X @ B2 )
        & ( ord_le2843351958646193337nt_int @ A @ B2 ) ) ) ).

% insert_subset
thf(fact_888_insert__subset,axiom,
    ! [X: int,A: set_int,B2: set_int] :
      ( ( ord_less_eq_set_int @ ( insert_int2 @ X @ A ) @ B2 )
      = ( ( member_int2 @ X @ B2 )
        & ( ord_less_eq_set_int @ A @ B2 ) ) ) ).

% insert_subset
thf(fact_889_Un__subset__iff,axiom,
    ! [A: set_nat,B2: set_nat,C2: set_nat] :
      ( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B2 ) @ C2 )
      = ( ( ord_less_eq_set_nat @ A @ C2 )
        & ( ord_less_eq_set_nat @ B2 @ C2 ) ) ) ).

% Un_subset_iff
thf(fact_890_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A2: nat] :
      ( ( minus_minus_nat @ A2 @ A2 )
      = zero_zero_nat ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_891_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A2: int] :
      ( ( minus_minus_int @ A2 @ A2 )
      = zero_zero_int ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_892_diff__zero,axiom,
    ! [A2: nat] :
      ( ( minus_minus_nat @ A2 @ zero_zero_nat )
      = A2 ) ).

% diff_zero
thf(fact_893_diff__zero,axiom,
    ! [A2: int] :
      ( ( minus_minus_int @ A2 @ zero_zero_int )
      = A2 ) ).

% diff_zero
thf(fact_894_zero__diff,axiom,
    ! [A2: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ A2 )
      = zero_zero_nat ) ).

% zero_diff
thf(fact_895_diff__0__right,axiom,
    ! [A2: int] :
      ( ( minus_minus_int @ A2 @ zero_zero_int )
      = A2 ) ).

% diff_0_right
thf(fact_896_diff__self,axiom,
    ! [A2: int] :
      ( ( minus_minus_int @ A2 @ A2 )
      = zero_zero_int ) ).

% diff_self
thf(fact_897_add__diff__cancel__right_H,axiom,
    ! [A2: nat,B: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ B ) @ B )
      = A2 ) ).

% add_diff_cancel_right'
thf(fact_898_add__diff__cancel__right_H,axiom,
    ! [A2: int,B: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A2 @ B ) @ B )
      = A2 ) ).

% add_diff_cancel_right'
thf(fact_899_add__diff__cancel__right,axiom,
    ! [A2: nat,C: nat,B: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) )
      = ( minus_minus_nat @ A2 @ B ) ) ).

% add_diff_cancel_right
thf(fact_900_add__diff__cancel__right,axiom,
    ! [A2: int,C: int,B: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B @ C ) )
      = ( minus_minus_int @ A2 @ B ) ) ).

% add_diff_cancel_right
thf(fact_901_add__diff__cancel__left_H,axiom,
    ! [A2: nat,B: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ B ) @ A2 )
      = B ) ).

% add_diff_cancel_left'
thf(fact_902_add__diff__cancel__left_H,axiom,
    ! [A2: int,B: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A2 @ B ) @ A2 )
      = B ) ).

% add_diff_cancel_left'
thf(fact_903_add__diff__cancel__left,axiom,
    ! [C: nat,A2: nat,B: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) )
      = ( minus_minus_nat @ A2 @ B ) ) ).

% add_diff_cancel_left
thf(fact_904_add__diff__cancel__left,axiom,
    ! [C: int,A2: int,B: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B ) )
      = ( minus_minus_int @ A2 @ B ) ) ).

% add_diff_cancel_left
thf(fact_905_diff__add__cancel,axiom,
    ! [A2: int,B: int] :
      ( ( plus_plus_int @ ( minus_minus_int @ A2 @ B ) @ B )
      = A2 ) ).

% diff_add_cancel
thf(fact_906_add__diff__cancel,axiom,
    ! [A2: int,B: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A2 @ B ) @ B )
      = A2 ) ).

% add_diff_cancel
thf(fact_907_diff__0__eq__0,axiom,
    ! [N: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ N )
      = zero_zero_nat ) ).

% diff_0_eq_0
thf(fact_908_diff__self__eq__0,axiom,
    ! [M2: nat] :
      ( ( minus_minus_nat @ M2 @ M2 )
      = zero_zero_nat ) ).

% diff_self_eq_0
thf(fact_909_diff__diff__cancel,axiom,
    ! [I: nat,N: nat] :
      ( ( ord_less_eq_nat @ I @ N )
     => ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
        = I ) ) ).

% diff_diff_cancel
thf(fact_910_diff__diff__left,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
      = ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).

% diff_diff_left
thf(fact_911_diff__ge__0__iff__ge,axiom,
    ! [A2: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A2 @ B ) )
      = ( ord_less_eq_int @ B @ A2 ) ) ).

% diff_ge_0_iff_ge
thf(fact_912_diff__gt__0__iff__gt,axiom,
    ! [A2: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A2 @ B ) )
      = ( ord_less_int @ B @ A2 ) ) ).

% diff_gt_0_iff_gt
thf(fact_913_le__add__diff__inverse,axiom,
    ! [B: nat,A2: nat] :
      ( ( ord_less_eq_nat @ B @ A2 )
     => ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A2 @ B ) )
        = A2 ) ) ).

% le_add_diff_inverse
thf(fact_914_le__add__diff__inverse,axiom,
    ! [B: int,A2: int] :
      ( ( ord_less_eq_int @ B @ A2 )
     => ( ( plus_plus_int @ B @ ( minus_minus_int @ A2 @ B ) )
        = A2 ) ) ).

% le_add_diff_inverse
thf(fact_915_le__add__diff__inverse2,axiom,
    ! [B: nat,A2: nat] :
      ( ( ord_less_eq_nat @ B @ A2 )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ A2 @ B ) @ B )
        = A2 ) ) ).

% le_add_diff_inverse2
thf(fact_916_le__add__diff__inverse2,axiom,
    ! [B: int,A2: int] :
      ( ( ord_less_eq_int @ B @ A2 )
     => ( ( plus_plus_int @ ( minus_minus_int @ A2 @ B ) @ B )
        = A2 ) ) ).

% le_add_diff_inverse2
thf(fact_917_diff__add__zero,axiom,
    ! [A2: nat,B: nat] :
      ( ( minus_minus_nat @ A2 @ ( plus_plus_nat @ A2 @ B ) )
      = zero_zero_nat ) ).

% diff_add_zero
thf(fact_918_surj__diff,axiom,
    ! [A2: int] :
      ( ( image_int_int @ ( minus_minus_int @ A2 ) @ top_top_set_int )
      = top_top_set_int ) ).

% surj_diff
thf(fact_919_zero__less__diff,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M2 ) )
      = ( ord_less_nat @ M2 @ N ) ) ).

% zero_less_diff
thf(fact_920_diff__is__0__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( minus_minus_nat @ M2 @ N )
        = zero_zero_nat )
      = ( ord_less_eq_nat @ M2 @ N ) ) ).

% diff_is_0_eq
thf(fact_921_diff__is__0__eq_H,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( minus_minus_nat @ M2 @ N )
        = zero_zero_nat ) ) ).

% diff_is_0_eq'
thf(fact_922_Nat_Odiff__diff__right,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).

% Nat.diff_diff_right
thf(fact_923_Nat_Oadd__diff__assoc2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
        = ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).

% Nat.add_diff_assoc2
thf(fact_924_Nat_Oadd__diff__assoc,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).

% Nat.add_diff_assoc
thf(fact_925_subset__Un__eq,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A5: set_nat,B7: set_nat] :
          ( ( sup_sup_set_nat @ A5 @ B7 )
          = B7 ) ) ) ).

% subset_Un_eq
thf(fact_926_subset__UnE,axiom,
    ! [C2: set_nat,A: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ C2 @ ( sup_sup_set_nat @ A @ B2 ) )
     => ~ ! [A7: set_nat] :
            ( ( ord_less_eq_set_nat @ A7 @ A )
           => ! [B8: set_nat] :
                ( ( ord_less_eq_set_nat @ B8 @ B2 )
               => ( C2
                 != ( sup_sup_set_nat @ A7 @ B8 ) ) ) ) ) ).

% subset_UnE
thf(fact_927_Un__absorb2,axiom,
    ! [B2: set_nat,A: set_nat] :
      ( ( ord_less_eq_set_nat @ B2 @ A )
     => ( ( sup_sup_set_nat @ A @ B2 )
        = A ) ) ).

% Un_absorb2
thf(fact_928_Un__absorb1,axiom,
    ! [A: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B2 )
     => ( ( sup_sup_set_nat @ A @ B2 )
        = B2 ) ) ).

% Un_absorb1
thf(fact_929_Un__upper2,axiom,
    ! [B2: set_nat,A: set_nat] : ( ord_less_eq_set_nat @ B2 @ ( sup_sup_set_nat @ A @ B2 ) ) ).

% Un_upper2
thf(fact_930_Un__upper1,axiom,
    ! [A: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ A @ ( sup_sup_set_nat @ A @ B2 ) ) ).

% Un_upper1
thf(fact_931_Un__least,axiom,
    ! [A: set_nat,C2: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ C2 )
     => ( ( ord_less_eq_set_nat @ B2 @ C2 )
       => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B2 ) @ C2 ) ) ) ).

% Un_least
thf(fact_932_Un__mono,axiom,
    ! [A: set_nat,C2: set_nat,B2: set_nat,D: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ C2 )
     => ( ( ord_less_eq_set_nat @ B2 @ D )
       => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B2 ) @ ( sup_sup_set_nat @ C2 @ D ) ) ) ) ).

% Un_mono
thf(fact_933_subset__insert,axiom,
    ! [X: product_prod_int_int,A: set_Pr958786334691620121nt_int,B2: set_Pr958786334691620121nt_int] :
      ( ~ ( member5262025264175285858nt_int @ X @ A )
     => ( ( ord_le2843351958646193337nt_int @ A @ ( insert5033312907999012233nt_int @ X @ B2 ) )
        = ( ord_le2843351958646193337nt_int @ A @ B2 ) ) ) ).

% subset_insert
thf(fact_934_subset__insert,axiom,
    ! [X: int,A: set_int,B2: set_int] :
      ( ~ ( member_int2 @ X @ A )
     => ( ( ord_less_eq_set_int @ A @ ( insert_int2 @ X @ B2 ) )
        = ( ord_less_eq_set_int @ A @ B2 ) ) ) ).

% subset_insert
thf(fact_935_subset__code_I1_J,axiom,
    ! [Xs2: list_int,B2: set_int] :
      ( ( ord_less_eq_set_int @ ( set_int2 @ Xs2 ) @ B2 )
      = ( ! [X3: int] :
            ( ( member_int2 @ X3 @ ( set_int2 @ Xs2 ) )
           => ( member_int2 @ X3 @ B2 ) ) ) ) ).

% subset_code(1)
thf(fact_936_subset__code_I1_J,axiom,
    ! [Xs2: list_P5707943133018811711nt_int,B2: set_Pr958786334691620121nt_int] :
      ( ( ord_le2843351958646193337nt_int @ ( set_Pr2470121279949933262nt_int @ Xs2 ) @ B2 )
      = ( ! [X3: product_prod_int_int] :
            ( ( member5262025264175285858nt_int @ X3 @ ( set_Pr2470121279949933262nt_int @ Xs2 ) )
           => ( member5262025264175285858nt_int @ X3 @ B2 ) ) ) ) ).

% subset_code(1)
thf(fact_937_less__eq__int__code_I1_J,axiom,
    ord_less_eq_int @ zero_zero_int @ zero_zero_int ).

% less_eq_int_code(1)
thf(fact_938_diff__le__mono2,axiom,
    ! [M2: nat,N: nat,L: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M2 ) ) ) ).

% diff_le_mono2
thf(fact_939_le__diff__iff_H,axiom,
    ! [A2: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ A2 @ C )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A2 ) @ ( minus_minus_nat @ C @ B ) )
          = ( ord_less_eq_nat @ B @ A2 ) ) ) ) ).

% le_diff_iff'
thf(fact_940_diff__le__self,axiom,
    ! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ N ) @ M2 ) ).

% diff_le_self
thf(fact_941_diff__le__mono,axiom,
    ! [M2: nat,N: nat,L: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).

% diff_le_mono
thf(fact_942_Nat_Odiff__diff__eq,axiom,
    ! [K: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M2 )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( minus_minus_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
          = ( minus_minus_nat @ M2 @ N ) ) ) ) ).

% Nat.diff_diff_eq
thf(fact_943_le__diff__iff,axiom,
    ! [K: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M2 )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
          = ( ord_less_eq_nat @ M2 @ N ) ) ) ) ).

% le_diff_iff
thf(fact_944_eq__diff__iff,axiom,
    ! [K: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M2 )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( ( minus_minus_nat @ M2 @ K )
            = ( minus_minus_nat @ N @ K ) )
          = ( M2 = N ) ) ) ) ).

% eq_diff_iff
thf(fact_945_diff__eq__diff__less__eq,axiom,
    ! [A2: int,B: int,C: int,D2: int] :
      ( ( ( minus_minus_int @ A2 @ B )
        = ( minus_minus_int @ C @ D2 ) )
     => ( ( ord_less_eq_int @ A2 @ B )
        = ( ord_less_eq_int @ C @ D2 ) ) ) ).

% diff_eq_diff_less_eq
thf(fact_946_diff__right__mono,axiom,
    ! [A2: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A2 @ B )
     => ( ord_less_eq_int @ ( minus_minus_int @ A2 @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).

% diff_right_mono
thf(fact_947_diff__left__mono,axiom,
    ! [B: int,A2: int,C: int] :
      ( ( ord_less_eq_int @ B @ A2 )
     => ( ord_less_eq_int @ ( minus_minus_int @ C @ A2 ) @ ( minus_minus_int @ C @ B ) ) ) ).

% diff_left_mono
thf(fact_948_diff__mono,axiom,
    ! [A2: int,B: int,D2: int,C: int] :
      ( ( ord_less_eq_int @ A2 @ B )
     => ( ( ord_less_eq_int @ D2 @ C )
       => ( ord_less_eq_int @ ( minus_minus_int @ A2 @ C ) @ ( minus_minus_int @ B @ D2 ) ) ) ) ).

% diff_mono
thf(fact_949_image__mono,axiom,
    ! [A: set_Pr958786334691620121nt_int,B2: set_Pr958786334691620121nt_int,F: product_prod_int_int > int] :
      ( ( ord_le2843351958646193337nt_int @ A @ B2 )
     => ( ord_less_eq_set_int @ ( image_5042161079198086560nt_int @ F @ A ) @ ( image_5042161079198086560nt_int @ F @ B2 ) ) ) ).

% image_mono
thf(fact_950_image__subsetI,axiom,
    ! [A: set_Pr958786334691620121nt_int,F: product_prod_int_int > product_prod_int_int,B2: set_Pr958786334691620121nt_int] :
      ( ! [X2: product_prod_int_int] :
          ( ( member5262025264175285858nt_int @ X2 @ A )
         => ( member5262025264175285858nt_int @ ( F @ X2 ) @ B2 ) )
     => ( ord_le2843351958646193337nt_int @ ( image_2653370878348428101nt_int @ F @ A ) @ B2 ) ) ).

% image_subsetI
thf(fact_951_image__subsetI,axiom,
    ! [A: set_Pr958786334691620121nt_int,F: product_prod_int_int > int,B2: set_int] :
      ( ! [X2: product_prod_int_int] :
          ( ( member5262025264175285858nt_int @ X2 @ A )
         => ( member_int2 @ ( F @ X2 ) @ B2 ) )
     => ( ord_less_eq_set_int @ ( image_5042161079198086560nt_int @ F @ A ) @ B2 ) ) ).

% image_subsetI
thf(fact_952_image__subsetI,axiom,
    ! [A: set_int,F: int > product_prod_int_int,B2: set_Pr958786334691620121nt_int] :
      ( ! [X2: int] :
          ( ( member_int2 @ X2 @ A )
         => ( member5262025264175285858nt_int @ ( F @ X2 ) @ B2 ) )
     => ( ord_le2843351958646193337nt_int @ ( image_5705468584675977158nt_int @ F @ A ) @ B2 ) ) ).

% image_subsetI
thf(fact_953_image__subsetI,axiom,
    ! [A: set_int,F: int > int,B2: set_int] :
      ( ! [X2: int] :
          ( ( member_int2 @ X2 @ A )
         => ( member_int2 @ ( F @ X2 ) @ B2 ) )
     => ( ord_less_eq_set_int @ ( image_int_int @ F @ A ) @ B2 ) ) ).

% image_subsetI
thf(fact_954_subset__imageE,axiom,
    ! [B2: set_int,F: product_prod_int_int > int,A: set_Pr958786334691620121nt_int] :
      ( ( ord_less_eq_set_int @ B2 @ ( image_5042161079198086560nt_int @ F @ A ) )
     => ~ ! [C6: set_Pr958786334691620121nt_int] :
            ( ( ord_le2843351958646193337nt_int @ C6 @ A )
           => ( B2
             != ( image_5042161079198086560nt_int @ F @ C6 ) ) ) ) ).

% subset_imageE
thf(fact_955_image__subset__iff,axiom,
    ! [F: product_prod_int_int > int,A: set_Pr958786334691620121nt_int,B2: set_int] :
      ( ( ord_less_eq_set_int @ ( image_5042161079198086560nt_int @ F @ A ) @ B2 )
      = ( ! [X3: product_prod_int_int] :
            ( ( member5262025264175285858nt_int @ X3 @ A )
           => ( member_int2 @ ( F @ X3 ) @ B2 ) ) ) ) ).

% image_subset_iff
thf(fact_956_subset__image__iff,axiom,
    ! [B2: set_int,F: product_prod_int_int > int,A: set_Pr958786334691620121nt_int] :
      ( ( ord_less_eq_set_int @ B2 @ ( image_5042161079198086560nt_int @ F @ A ) )
      = ( ? [AA: set_Pr958786334691620121nt_int] :
            ( ( ord_le2843351958646193337nt_int @ AA @ A )
            & ( B2
              = ( image_5042161079198086560nt_int @ F @ AA ) ) ) ) ) ).

% subset_image_iff
thf(fact_957_subrelI,axiom,
    ! [R: set_Pr958786334691620121nt_int,S3: set_Pr958786334691620121nt_int] :
      ( ! [X2: int,Y3: int] :
          ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y3 ) @ R )
         => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y3 ) @ S3 ) )
     => ( ord_le2843351958646193337nt_int @ R @ S3 ) ) ).

% subrelI
thf(fact_958_min__diff,axiom,
    ! [M2: nat,I: nat,N: nat] :
      ( ( ord_min_nat @ ( minus_minus_nat @ M2 @ I ) @ ( minus_minus_nat @ N @ I ) )
      = ( minus_minus_nat @ ( ord_min_nat @ M2 @ N ) @ I ) ) ).

% min_diff
thf(fact_959_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y2: int,Z2: int] : ( Y2 = Z2 ) )
    = ( ^ [A3: int,B3: int] :
          ( ( minus_minus_int @ A3 @ B3 )
          = zero_zero_int ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_960_diff__diff__eq,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ A2 @ B ) @ C )
      = ( minus_minus_nat @ A2 @ ( plus_plus_nat @ B @ C ) ) ) ).

% diff_diff_eq
thf(fact_961_diff__diff__eq,axiom,
    ! [A2: int,B: int,C: int] :
      ( ( minus_minus_int @ ( minus_minus_int @ A2 @ B ) @ C )
      = ( minus_minus_int @ A2 @ ( plus_plus_int @ B @ C ) ) ) ).

% diff_diff_eq
thf(fact_962_add__implies__diff,axiom,
    ! [C: nat,B: nat,A2: nat] :
      ( ( ( plus_plus_nat @ C @ B )
        = A2 )
     => ( C
        = ( minus_minus_nat @ A2 @ B ) ) ) ).

% add_implies_diff
thf(fact_963_add__implies__diff,axiom,
    ! [C: int,B: int,A2: int] :
      ( ( ( plus_plus_int @ C @ B )
        = A2 )
     => ( C
        = ( minus_minus_int @ A2 @ B ) ) ) ).

% add_implies_diff
thf(fact_964_diff__add__eq__diff__diff__swap,axiom,
    ! [A2: int,B: int,C: int] :
      ( ( minus_minus_int @ A2 @ ( plus_plus_int @ B @ C ) )
      = ( minus_minus_int @ ( minus_minus_int @ A2 @ C ) @ B ) ) ).

% diff_add_eq_diff_diff_swap
thf(fact_965_diff__add__eq,axiom,
    ! [A2: int,B: int,C: int] :
      ( ( plus_plus_int @ ( minus_minus_int @ A2 @ B ) @ C )
      = ( minus_minus_int @ ( plus_plus_int @ A2 @ C ) @ B ) ) ).

% diff_add_eq
thf(fact_966_diff__diff__eq2,axiom,
    ! [A2: int,B: int,C: int] :
      ( ( minus_minus_int @ A2 @ ( minus_minus_int @ B @ C ) )
      = ( minus_minus_int @ ( plus_plus_int @ A2 @ C ) @ B ) ) ).

% diff_diff_eq2
thf(fact_967_add__diff__eq,axiom,
    ! [A2: int,B: int,C: int] :
      ( ( plus_plus_int @ A2 @ ( minus_minus_int @ B @ C ) )
      = ( minus_minus_int @ ( plus_plus_int @ A2 @ B ) @ C ) ) ).

% add_diff_eq
thf(fact_968_eq__diff__eq,axiom,
    ! [A2: int,C: int,B: int] :
      ( ( A2
        = ( minus_minus_int @ C @ B ) )
      = ( ( plus_plus_int @ A2 @ B )
        = C ) ) ).

% eq_diff_eq
thf(fact_969_diff__eq__eq,axiom,
    ! [A2: int,B: int,C: int] :
      ( ( ( minus_minus_int @ A2 @ B )
        = C )
      = ( A2
        = ( plus_plus_int @ C @ B ) ) ) ).

% diff_eq_eq
thf(fact_970_group__cancel_Osub1,axiom,
    ! [A: int,K: int,A2: int,B: int] :
      ( ( A
        = ( plus_plus_int @ K @ A2 ) )
     => ( ( minus_minus_int @ A @ B )
        = ( plus_plus_int @ K @ ( minus_minus_int @ A2 @ B ) ) ) ) ).

% group_cancel.sub1
thf(fact_971_minus__nat_Odiff__0,axiom,
    ! [M2: nat] :
      ( ( minus_minus_nat @ M2 @ zero_zero_nat )
      = M2 ) ).

% minus_nat.diff_0
thf(fact_972_diffs0__imp__equal,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( minus_minus_nat @ M2 @ N )
        = zero_zero_nat )
     => ( ( ( minus_minus_nat @ N @ M2 )
          = zero_zero_nat )
       => ( M2 = N ) ) ) ).

% diffs0_imp_equal
thf(fact_973_diff__add__inverse2,axiom,
    ! [M2: nat,N: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N ) @ N )
      = M2 ) ).

% diff_add_inverse2
thf(fact_974_diff__add__inverse,axiom,
    ! [N: nat,M2: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ N @ M2 ) @ N )
      = M2 ) ).

% diff_add_inverse
thf(fact_975_diff__cancel2,axiom,
    ! [M2: nat,K: nat,N: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ K ) @ ( plus_plus_nat @ N @ K ) )
      = ( minus_minus_nat @ M2 @ N ) ) ).

% diff_cancel2
thf(fact_976_Nat_Odiff__cancel,axiom,
    ! [K: nat,M2: nat,N: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N ) )
      = ( minus_minus_nat @ M2 @ N ) ) ).

% Nat.diff_cancel
thf(fact_977_min__diff__distrib__left,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( minus_minus_int @ ( ord_min_int @ X @ Y ) @ Z )
      = ( ord_min_int @ ( minus_minus_int @ X @ Z ) @ ( minus_minus_int @ Y @ Z ) ) ) ).

% min_diff_distrib_left
thf(fact_978_diff__eq__diff__eq,axiom,
    ! [A2: int,B: int,C: int,D2: int] :
      ( ( ( minus_minus_int @ A2 @ B )
        = ( minus_minus_int @ C @ D2 ) )
     => ( ( A2 = B )
        = ( C = D2 ) ) ) ).

% diff_eq_diff_eq
thf(fact_979_diff__right__commute,axiom,
    ! [A2: nat,C: nat,B: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ A2 @ C ) @ B )
      = ( minus_minus_nat @ ( minus_minus_nat @ A2 @ B ) @ C ) ) ).

% diff_right_commute
thf(fact_980_diff__right__commute,axiom,
    ! [A2: int,C: int,B: int] :
      ( ( minus_minus_int @ ( minus_minus_int @ A2 @ C ) @ B )
      = ( minus_minus_int @ ( minus_minus_int @ A2 @ B ) @ C ) ) ).

% diff_right_commute
thf(fact_981_diff__commute,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
      = ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).

% diff_commute
thf(fact_982_diff__mult__distrib,axiom,
    ! [M2: nat,N: nat,K: nat] :
      ( ( times_times_nat @ ( minus_minus_nat @ M2 @ N ) @ K )
      = ( minus_minus_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N @ K ) ) ) ).

% diff_mult_distrib
thf(fact_983_diff__mult__distrib2,axiom,
    ! [K: nat,M2: nat,N: nat] :
      ( ( times_times_nat @ K @ ( minus_minus_nat @ M2 @ N ) )
      = ( minus_minus_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) ) ) ).

% diff_mult_distrib2
thf(fact_984_diff__less__mono2,axiom,
    ! [M2: nat,N: nat,L: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( ( ord_less_nat @ M2 @ L )
       => ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M2 ) ) ) ) ).

% diff_less_mono2
thf(fact_985_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_986_diff__strict__mono,axiom,
    ! [A2: int,B: int,D2: int,C: int] :
      ( ( ord_less_int @ A2 @ B )
     => ( ( ord_less_int @ D2 @ C )
       => ( ord_less_int @ ( minus_minus_int @ A2 @ C ) @ ( minus_minus_int @ B @ D2 ) ) ) ) ).

% diff_strict_mono
thf(fact_987_diff__eq__diff__less,axiom,
    ! [A2: int,B: int,C: int,D2: int] :
      ( ( ( minus_minus_int @ A2 @ B )
        = ( minus_minus_int @ C @ D2 ) )
     => ( ( ord_less_int @ A2 @ B )
        = ( ord_less_int @ C @ D2 ) ) ) ).

% diff_eq_diff_less
thf(fact_988_diff__strict__left__mono,axiom,
    ! [B: int,A2: int,C: int] :
      ( ( ord_less_int @ B @ A2 )
     => ( ord_less_int @ ( minus_minus_int @ C @ A2 ) @ ( minus_minus_int @ C @ B ) ) ) ).

% diff_strict_left_mono
thf(fact_989_diff__strict__right__mono,axiom,
    ! [A2: int,B: int,C: int] :
      ( ( ord_less_int @ A2 @ B )
     => ( ord_less_int @ ( minus_minus_int @ A2 @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).

% diff_strict_right_mono
thf(fact_990_le__iff__diff__le__0,axiom,
    ( ord_less_eq_int
    = ( ^ [A3: int,B3: int] : ( ord_less_eq_int @ ( minus_minus_int @ A3 @ B3 ) @ zero_zero_int ) ) ) ).

% le_iff_diff_le_0
thf(fact_991_less__iff__diff__less__0,axiom,
    ( ord_less_int
    = ( ^ [A3: int,B3: int] : ( ord_less_int @ ( minus_minus_int @ A3 @ B3 ) @ zero_zero_int ) ) ) ).

% less_iff_diff_less_0
thf(fact_992_add__le__imp__le__diff,axiom,
    ! [I: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
     => ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N @ K ) ) ) ).

% add_le_imp_le_diff
thf(fact_993_add__le__imp__le__diff,axiom,
    ! [I: int,K: int,N: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
     => ( ord_less_eq_int @ I @ ( minus_minus_int @ N @ K ) ) ) ).

% add_le_imp_le_diff
thf(fact_994_add__le__add__imp__diff__le,axiom,
    ! [I: nat,K: nat,N: nat,J: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
     => ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
       => ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
         => ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
           => ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K ) @ J ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_995_add__le__add__imp__diff__le,axiom,
    ! [I: int,K: int,N: int,J: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
     => ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
       => ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
         => ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
           => ( ord_less_eq_int @ ( minus_minus_int @ N @ K ) @ J ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_996_diff__le__eq,axiom,
    ! [A2: int,B: int,C: int] :
      ( ( ord_less_eq_int @ ( minus_minus_int @ A2 @ B ) @ C )
      = ( ord_less_eq_int @ A2 @ ( plus_plus_int @ C @ B ) ) ) ).

% diff_le_eq
thf(fact_997_le__diff__eq,axiom,
    ! [A2: int,C: int,B: int] :
      ( ( ord_less_eq_int @ A2 @ ( minus_minus_int @ C @ B ) )
      = ( ord_less_eq_int @ ( plus_plus_int @ A2 @ B ) @ C ) ) ).

% le_diff_eq
thf(fact_998_diff__add,axiom,
    ! [A2: nat,B: nat] :
      ( ( ord_less_eq_nat @ A2 @ B )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A2 ) @ A2 )
        = B ) ) ).

% diff_add
thf(fact_999_le__add__diff,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B )
     => ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A2 ) ) ) ).

% le_add_diff
thf(fact_1000_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B )
     => ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A2 ) )
        = ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ B ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_1001_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B )
     => ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A2 ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A2 ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_1002_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A2 )
        = ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A2 ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_1003_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A2 ) @ C )
        = ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A2 ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_1004_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A2 )
        = ( plus_plus_nat @ ( minus_minus_nat @ B @ A2 ) @ C ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_1005_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B )
     => ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A2 ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ C @ A2 ) @ B ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_1006_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
    ! [A2: nat,B: nat] :
      ( ( ord_less_eq_nat @ A2 @ B )
     => ( ( plus_plus_nat @ A2 @ ( minus_minus_nat @ B @ A2 ) )
        = B ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_1007_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B )
     => ( ( ord_less_eq_nat @ A2 @ B )
       => ( ( ( minus_minus_nat @ B @ A2 )
            = C )
          = ( B
            = ( plus_plus_nat @ C @ A2 ) ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_1008_diff__less__eq,axiom,
    ! [A2: int,B: int,C: int] :
      ( ( ord_less_int @ ( minus_minus_int @ A2 @ B ) @ C )
      = ( ord_less_int @ A2 @ ( plus_plus_int @ C @ B ) ) ) ).

% diff_less_eq
thf(fact_1009_less__diff__eq,axiom,
    ! [A2: int,C: int,B: int] :
      ( ( ord_less_int @ A2 @ ( minus_minus_int @ C @ B ) )
      = ( ord_less_int @ ( plus_plus_int @ A2 @ B ) @ C ) ) ).

% less_diff_eq
thf(fact_1010_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A2: nat,B: nat] :
      ( ~ ( ord_less_nat @ A2 @ B )
     => ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A2 @ B ) )
        = A2 ) ) ).

% linordered_semidom_class.add_diff_inverse
thf(fact_1011_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A2: int,B: int] :
      ( ~ ( ord_less_int @ A2 @ B )
     => ( ( plus_plus_int @ B @ ( minus_minus_int @ A2 @ B ) )
        = A2 ) ) ).

% linordered_semidom_class.add_diff_inverse
thf(fact_1012_square__diff__square__factored,axiom,
    ! [X: int,Y: int] :
      ( ( minus_minus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) )
      = ( times_times_int @ ( plus_plus_int @ X @ Y ) @ ( minus_minus_int @ X @ Y ) ) ) ).

% square_diff_square_factored
thf(fact_1013_eq__add__iff2,axiom,
    ! [A2: int,E: int,C: int,B: int,D2: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ A2 @ E ) @ C )
        = ( plus_plus_int @ ( times_times_int @ B @ E ) @ D2 ) )
      = ( C
        = ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A2 ) @ E ) @ D2 ) ) ) ).

% eq_add_iff2
thf(fact_1014_eq__add__iff1,axiom,
    ! [A2: int,E: int,C: int,B: int,D2: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ A2 @ E ) @ C )
        = ( plus_plus_int @ ( times_times_int @ B @ E ) @ D2 ) )
      = ( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A2 @ B ) @ E ) @ C )
        = D2 ) ) ).

% eq_add_iff1
thf(fact_1015_ex__nat__less__eq,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ? [M4: nat] :
            ( ( ord_less_nat @ M4 @ N )
            & ( P @ M4 ) ) )
      = ( ? [X3: nat] :
            ( ( member_nat2 @ X3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
            & ( P @ X3 ) ) ) ) ).

% ex_nat_less_eq
thf(fact_1016_all__nat__less__eq,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ! [M4: nat] :
            ( ( ord_less_nat @ M4 @ N )
           => ( P @ M4 ) ) )
      = ( ! [X3: nat] :
            ( ( member_nat2 @ X3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
           => ( P @ X3 ) ) ) ) ).

% all_nat_less_eq
thf(fact_1017_diff__less,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_nat @ zero_zero_nat @ M2 )
       => ( ord_less_nat @ ( minus_minus_nat @ M2 @ N ) @ M2 ) ) ) ).

% diff_less
thf(fact_1018_diff__add__0,axiom,
    ! [N: nat,M2: nat] :
      ( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M2 ) )
      = zero_zero_nat ) ).

% diff_add_0
thf(fact_1019_diff__less__mono,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A2 @ B )
     => ( ( ord_less_eq_nat @ C @ A2 )
       => ( ord_less_nat @ ( minus_minus_nat @ A2 @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).

% diff_less_mono
thf(fact_1020_less__diff__iff,axiom,
    ! [K: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M2 )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( ord_less_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
          = ( ord_less_nat @ M2 @ N ) ) ) ) ).

% less_diff_iff
thf(fact_1021_add__diff__inverse__nat,axiom,
    ! [M2: nat,N: nat] :
      ( ~ ( ord_less_nat @ M2 @ N )
     => ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M2 @ N ) )
        = M2 ) ) ).

% add_diff_inverse_nat
thf(fact_1022_less__diff__conv,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
      = ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).

% less_diff_conv
thf(fact_1023_Nat_Ole__imp__diff__is__add,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ( minus_minus_nat @ J @ I )
          = K )
        = ( J
          = ( plus_plus_nat @ K @ I ) ) ) ) ).

% Nat.le_imp_diff_is_add
thf(fact_1024_Nat_Odiff__add__assoc2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
        = ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).

% Nat.diff_add_assoc2
thf(fact_1025_Nat_Odiff__add__assoc,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
        = ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).

% Nat.diff_add_assoc
thf(fact_1026_Nat_Ole__diff__conv2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
        = ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).

% Nat.le_diff_conv2
thf(fact_1027_le__diff__conv,axiom,
    ! [J: nat,K: nat,I: nat] :
      ( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
      = ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).

% le_diff_conv
thf(fact_1028_range__subsetD,axiom,
    ! [F: product_prod_int_int > int,B2: set_int,I: product_prod_int_int] :
      ( ( ord_less_eq_set_int @ ( image_5042161079198086560nt_int @ F @ top_to4366644338036079209nt_int ) @ B2 )
     => ( member_int2 @ ( F @ I ) @ B2 ) ) ).

% range_subsetD
thf(fact_1029_set__subset__Cons,axiom,
    ! [Xs2: list_P5707943133018811711nt_int,X: product_prod_int_int] : ( ord_le2843351958646193337nt_int @ ( set_Pr2470121279949933262nt_int @ Xs2 ) @ ( set_Pr2470121279949933262nt_int @ ( cons_P3334398858971670639nt_int @ X @ Xs2 ) ) ) ).

% set_subset_Cons
thf(fact_1030_set__subset__Cons,axiom,
    ! [Xs2: list_int,X: int] : ( ord_less_eq_set_int @ ( set_int2 @ Xs2 ) @ ( set_int2 @ ( cons_int @ X @ Xs2 ) ) ) ).

% set_subset_Cons
thf(fact_1031_set__take__subset,axiom,
    ! [N: nat,Xs2: list_P5707943133018811711nt_int] : ( ord_le2843351958646193337nt_int @ ( set_Pr2470121279949933262nt_int @ ( take_P8218740963776755879nt_int @ N @ Xs2 ) ) @ ( set_Pr2470121279949933262nt_int @ Xs2 ) ) ).

% set_take_subset
thf(fact_1032_le__add__iff1,axiom,
    ! [A2: int,E: int,C: int,B: int,D2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A2 @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D2 ) )
      = ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A2 @ B ) @ E ) @ C ) @ D2 ) ) ).

% le_add_iff1
thf(fact_1033_le__add__iff2,axiom,
    ! [A2: int,E: int,C: int,B: int,D2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A2 @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D2 ) )
      = ( ord_less_eq_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A2 ) @ E ) @ D2 ) ) ) ).

% le_add_iff2
thf(fact_1034_less__add__iff2,axiom,
    ! [A2: int,E: int,C: int,B: int,D2: int] :
      ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A2 @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D2 ) )
      = ( ord_less_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A2 ) @ E ) @ D2 ) ) ) ).

% less_add_iff2
thf(fact_1035_less__add__iff1,axiom,
    ! [A2: int,E: int,C: int,B: int,D2: int] :
      ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A2 @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D2 ) )
      = ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A2 @ B ) @ E ) @ C ) @ D2 ) ) ).

% less_add_iff1
thf(fact_1036_nat__diff__split,axiom,
    ! [P: nat > $o,A2: nat,B: nat] :
      ( ( P @ ( minus_minus_nat @ A2 @ B ) )
      = ( ( ( ord_less_nat @ A2 @ B )
         => ( P @ zero_zero_nat ) )
        & ! [D3: nat] :
            ( ( A2
              = ( plus_plus_nat @ B @ D3 ) )
           => ( P @ D3 ) ) ) ) ).

% nat_diff_split
thf(fact_1037_nat__diff__split__asm,axiom,
    ! [P: nat > $o,A2: nat,B: nat] :
      ( ( P @ ( minus_minus_nat @ A2 @ B ) )
      = ( ~ ( ( ( ord_less_nat @ A2 @ B )
              & ~ ( P @ zero_zero_nat ) )
            | ? [D3: nat] :
                ( ( A2
                  = ( plus_plus_nat @ B @ D3 ) )
                & ~ ( P @ D3 ) ) ) ) ) ).

% nat_diff_split_asm
thf(fact_1038_less__diff__conv2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I )
        = ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ) ).

% less_diff_conv2
thf(fact_1039_nat__eq__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M2 )
          = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M2 )
          = N ) ) ) ).

% nat_eq_add_iff1
thf(fact_1040_nat__eq__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M2 )
          = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( M2
          = ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).

% nat_eq_add_iff2
thf(fact_1041_nat__le__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M2 ) @ N ) ) ) ).

% nat_le_add_iff1
thf(fact_1042_nat__le__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( ord_less_eq_nat @ M2 @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).

% nat_le_add_iff2
thf(fact_1043_nat__diff__add__eq1,axiom,
    ! [J: nat,I: nat,U: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M2 ) @ N ) ) ) ).

% nat_diff_add_eq1
thf(fact_1044_nat__diff__add__eq2,axiom,
    ! [I: nat,J: nat,U: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( minus_minus_nat @ M2 @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).

% nat_diff_add_eq2
thf(fact_1045_set__take__subset__set__take,axiom,
    ! [M2: nat,N: nat,Xs2: list_P5707943133018811711nt_int] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ord_le2843351958646193337nt_int @ ( set_Pr2470121279949933262nt_int @ ( take_P8218740963776755879nt_int @ M2 @ Xs2 ) ) @ ( set_Pr2470121279949933262nt_int @ ( take_P8218740963776755879nt_int @ N @ Xs2 ) ) ) ) ).

% set_take_subset_set_take
thf(fact_1046_subset__subseqs,axiom,
    ! [X10: set_Pr958786334691620121nt_int,Xs2: list_P5707943133018811711nt_int] :
      ( ( ord_le2843351958646193337nt_int @ X10 @ ( set_Pr2470121279949933262nt_int @ Xs2 ) )
     => ( member2340774599025711042nt_int @ X10 @ ( image_689400715899363487nt_int @ set_Pr2470121279949933262nt_int @ ( set_li2659200638379878868nt_int @ ( subseq1357044202310323342nt_int @ Xs2 ) ) ) ) ) ).

% subset_subseqs
thf(fact_1047_nat__less__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( ord_less_nat @ M2 @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).

% nat_less_add_iff2
thf(fact_1048_nat__less__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M2 ) @ N ) ) ) ).

% nat_less_add_iff1
thf(fact_1049_Collect__split__mono__strong,axiom,
    ! [X10: set_int,A: set_Pr958786334691620121nt_int,Y10: set_int,P: int > int > $o,Q2: int > int > $o] :
      ( ( X10
        = ( image_5042161079198086560nt_int @ product_fst_int_int @ A ) )
     => ( ( Y10
          = ( image_5042161079198086560nt_int @ product_snd_int_int @ A ) )
       => ( ! [X2: int] :
              ( ( member_int2 @ X2 @ X10 )
             => ! [Xa: int] :
                  ( ( member_int2 @ Xa @ Y10 )
                 => ( ( P @ X2 @ Xa )
                   => ( Q2 @ X2 @ Xa ) ) ) )
         => ( ( ord_le2843351958646193337nt_int @ A @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ P ) ) )
           => ( ord_le2843351958646193337nt_int @ A @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ Q2 ) ) ) ) ) ) ) ).

% Collect_split_mono_strong
thf(fact_1050_image__add__atLeastLessThan,axiom,
    ! [K: int,I: int,J: int] :
      ( ( image_int_int @ ( plus_plus_int @ K ) @ ( set_or4662586982721622107an_int @ I @ J ) )
      = ( set_or4662586982721622107an_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ K ) ) ) ).

% image_add_atLeastLessThan
thf(fact_1051_image__add__atLeastLessThan,axiom,
    ! [K: nat,I: nat,J: nat] :
      ( ( image_nat_nat @ ( plus_plus_nat @ K ) @ ( set_or4665077453230672383an_nat @ I @ J ) )
      = ( set_or4665077453230672383an_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).

% image_add_atLeastLessThan
thf(fact_1052_incr__mult__lemma,axiom,
    ! [D2: int,P: int > $o,K: int] :
      ( ( ord_less_int @ zero_zero_int @ D2 )
     => ( ! [X2: int] :
            ( ( P @ X2 )
           => ( P @ ( plus_plus_int @ X2 @ D2 ) ) )
       => ( ( ord_less_eq_int @ zero_zero_int @ K )
         => ! [X4: int] :
              ( ( P @ X4 )
             => ( P @ ( plus_plus_int @ X4 @ ( times_times_int @ K @ D2 ) ) ) ) ) ) ) ).

% incr_mult_lemma
thf(fact_1053_atLeastLessThan__add__Un,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( set_or4665077453230672383an_nat @ I @ ( plus_plus_nat @ J @ K ) )
        = ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ I @ J ) @ ( set_or4665077453230672383an_nat @ J @ ( plus_plus_nat @ J @ K ) ) ) ) ) ).

% atLeastLessThan_add_Un
thf(fact_1054_subsetI,axiom,
    ! [A: set_Pr958786334691620121nt_int,B2: set_Pr958786334691620121nt_int] :
      ( ! [X2: product_prod_int_int] :
          ( ( member5262025264175285858nt_int @ X2 @ A )
         => ( member5262025264175285858nt_int @ X2 @ B2 ) )
     => ( ord_le2843351958646193337nt_int @ A @ B2 ) ) ).

% subsetI
thf(fact_1055_subsetI,axiom,
    ! [A: set_int,B2: set_int] :
      ( ! [X2: int] :
          ( ( member_int2 @ X2 @ A )
         => ( member_int2 @ X2 @ B2 ) )
     => ( ord_less_eq_set_int @ A @ B2 ) ) ).

% subsetI
thf(fact_1056_insert__Diff1,axiom,
    ! [X: product_prod_int_int,B2: set_Pr958786334691620121nt_int,A: set_Pr958786334691620121nt_int] :
      ( ( member5262025264175285858nt_int @ X @ B2 )
     => ( ( minus_1052850069191792384nt_int @ ( insert5033312907999012233nt_int @ X @ A ) @ B2 )
        = ( minus_1052850069191792384nt_int @ A @ B2 ) ) ) ).

% insert_Diff1
thf(fact_1057_insert__Diff1,axiom,
    ! [X: int,B2: set_int,A: set_int] :
      ( ( member_int2 @ X @ B2 )
     => ( ( minus_minus_set_int @ ( insert_int2 @ X @ A ) @ B2 )
        = ( minus_minus_set_int @ A @ B2 ) ) ) ).

% insert_Diff1
thf(fact_1058_Diff__insert0,axiom,
    ! [X: product_prod_int_int,A: set_Pr958786334691620121nt_int,B2: set_Pr958786334691620121nt_int] :
      ( ~ ( member5262025264175285858nt_int @ X @ A )
     => ( ( minus_1052850069191792384nt_int @ A @ ( insert5033312907999012233nt_int @ X @ B2 ) )
        = ( minus_1052850069191792384nt_int @ A @ B2 ) ) ) ).

% Diff_insert0
thf(fact_1059_Diff__insert0,axiom,
    ! [X: int,A: set_int,B2: set_int] :
      ( ~ ( member_int2 @ X @ A )
     => ( ( minus_minus_set_int @ A @ ( insert_int2 @ X @ B2 ) )
        = ( minus_minus_set_int @ A @ B2 ) ) ) ).

% Diff_insert0
thf(fact_1060_Un__Diff__cancel2,axiom,
    ! [B2: set_nat,A: set_nat] :
      ( ( sup_sup_set_nat @ ( minus_minus_set_nat @ B2 @ A ) @ A )
      = ( sup_sup_set_nat @ B2 @ A ) ) ).

% Un_Diff_cancel2
thf(fact_1061_Un__Diff__cancel,axiom,
    ! [A: set_nat,B2: set_nat] :
      ( ( sup_sup_set_nat @ A @ ( minus_minus_set_nat @ B2 @ A ) )
      = ( sup_sup_set_nat @ A @ B2 ) ) ).

% Un_Diff_cancel
thf(fact_1062_Diff__subset__conv,axiom,
    ! [A: set_nat,B2: set_nat,C2: set_nat] :
      ( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ B2 ) @ C2 )
      = ( ord_less_eq_set_nat @ A @ ( sup_sup_set_nat @ B2 @ C2 ) ) ) ).

% Diff_subset_conv
thf(fact_1063_Diff__partition,axiom,
    ! [A: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B2 )
     => ( ( sup_sup_set_nat @ A @ ( minus_minus_set_nat @ B2 @ A ) )
        = B2 ) ) ).

% Diff_partition
thf(fact_1064_subset__Diff__insert,axiom,
    ! [A: set_Pr958786334691620121nt_int,B2: set_Pr958786334691620121nt_int,X: product_prod_int_int,C2: set_Pr958786334691620121nt_int] :
      ( ( ord_le2843351958646193337nt_int @ A @ ( minus_1052850069191792384nt_int @ B2 @ ( insert5033312907999012233nt_int @ X @ C2 ) ) )
      = ( ( ord_le2843351958646193337nt_int @ A @ ( minus_1052850069191792384nt_int @ B2 @ C2 ) )
        & ~ ( member5262025264175285858nt_int @ X @ A ) ) ) ).

% subset_Diff_insert
thf(fact_1065_subset__Diff__insert,axiom,
    ! [A: set_int,B2: set_int,X: int,C2: set_int] :
      ( ( ord_less_eq_set_int @ A @ ( minus_minus_set_int @ B2 @ ( insert_int2 @ X @ C2 ) ) )
      = ( ( ord_less_eq_set_int @ A @ ( minus_minus_set_int @ B2 @ C2 ) )
        & ~ ( member_int2 @ X @ A ) ) ) ).

% subset_Diff_insert
thf(fact_1066_image__diff__subset,axiom,
    ! [F: product_prod_int_int > int,A: set_Pr958786334691620121nt_int,B2: set_Pr958786334691620121nt_int] : ( ord_less_eq_set_int @ ( minus_minus_set_int @ ( image_5042161079198086560nt_int @ F @ A ) @ ( image_5042161079198086560nt_int @ F @ B2 ) ) @ ( image_5042161079198086560nt_int @ F @ ( minus_1052850069191792384nt_int @ A @ B2 ) ) ) ).

% image_diff_subset
thf(fact_1067_subset__iff,axiom,
    ( ord_le2843351958646193337nt_int
    = ( ^ [A5: set_Pr958786334691620121nt_int,B7: set_Pr958786334691620121nt_int] :
        ! [T: product_prod_int_int] :
          ( ( member5262025264175285858nt_int @ T @ A5 )
         => ( member5262025264175285858nt_int @ T @ B7 ) ) ) ) ).

% subset_iff
thf(fact_1068_subset__iff,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A5: set_int,B7: set_int] :
        ! [T: int] :
          ( ( member_int2 @ T @ A5 )
         => ( member_int2 @ T @ B7 ) ) ) ) ).

% subset_iff
thf(fact_1069_subset__eq,axiom,
    ( ord_le2843351958646193337nt_int
    = ( ^ [A5: set_Pr958786334691620121nt_int,B7: set_Pr958786334691620121nt_int] :
        ! [X3: product_prod_int_int] :
          ( ( member5262025264175285858nt_int @ X3 @ A5 )
         => ( member5262025264175285858nt_int @ X3 @ B7 ) ) ) ) ).

% subset_eq
thf(fact_1070_subset__eq,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A5: set_int,B7: set_int] :
        ! [X3: int] :
          ( ( member_int2 @ X3 @ A5 )
         => ( member_int2 @ X3 @ B7 ) ) ) ) ).

% subset_eq
thf(fact_1071_subsetD,axiom,
    ! [A: set_Pr958786334691620121nt_int,B2: set_Pr958786334691620121nt_int,C: product_prod_int_int] :
      ( ( ord_le2843351958646193337nt_int @ A @ B2 )
     => ( ( member5262025264175285858nt_int @ C @ A )
       => ( member5262025264175285858nt_int @ C @ B2 ) ) ) ).

% subsetD
thf(fact_1072_subsetD,axiom,
    ! [A: set_int,B2: set_int,C: int] :
      ( ( ord_less_eq_set_int @ A @ B2 )
     => ( ( member_int2 @ C @ A )
       => ( member_int2 @ C @ B2 ) ) ) ).

% subsetD
thf(fact_1073_in__mono,axiom,
    ! [A: set_Pr958786334691620121nt_int,B2: set_Pr958786334691620121nt_int,X: product_prod_int_int] :
      ( ( ord_le2843351958646193337nt_int @ A @ B2 )
     => ( ( member5262025264175285858nt_int @ X @ A )
       => ( member5262025264175285858nt_int @ X @ B2 ) ) ) ).

% in_mono
thf(fact_1074_in__mono,axiom,
    ! [A: set_int,B2: set_int,X: int] :
      ( ( ord_less_eq_set_int @ A @ B2 )
     => ( ( member_int2 @ X @ A )
       => ( member_int2 @ X @ B2 ) ) ) ).

% in_mono
thf(fact_1075_verit__la__generic,axiom,
    ! [A2: int,X: int] :
      ( ( ord_less_eq_int @ A2 @ X )
      | ( A2 = X )
      | ( ord_less_eq_int @ X @ A2 ) ) ).

% verit_la_generic
thf(fact_1076_minus__int__code_I1_J,axiom,
    ! [K: int] :
      ( ( minus_minus_int @ K @ zero_zero_int )
      = K ) ).

% minus_int_code(1)
thf(fact_1077_insert__Diff__if,axiom,
    ! [X: product_prod_int_int,B2: set_Pr958786334691620121nt_int,A: set_Pr958786334691620121nt_int] :
      ( ( ( member5262025264175285858nt_int @ X @ B2 )
       => ( ( minus_1052850069191792384nt_int @ ( insert5033312907999012233nt_int @ X @ A ) @ B2 )
          = ( minus_1052850069191792384nt_int @ A @ B2 ) ) )
      & ( ~ ( member5262025264175285858nt_int @ X @ B2 )
       => ( ( minus_1052850069191792384nt_int @ ( insert5033312907999012233nt_int @ X @ A ) @ B2 )
          = ( insert5033312907999012233nt_int @ X @ ( minus_1052850069191792384nt_int @ A @ B2 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_1078_insert__Diff__if,axiom,
    ! [X: int,B2: set_int,A: set_int] :
      ( ( ( member_int2 @ X @ B2 )
       => ( ( minus_minus_set_int @ ( insert_int2 @ X @ A ) @ B2 )
          = ( minus_minus_set_int @ A @ B2 ) ) )
      & ( ~ ( member_int2 @ X @ B2 )
       => ( ( minus_minus_set_int @ ( insert_int2 @ X @ A ) @ B2 )
          = ( insert_int2 @ X @ ( minus_minus_set_int @ A @ B2 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_1079_int__distrib_I3_J,axiom,
    ! [Z12: int,Z23: int,W: int] :
      ( ( times_times_int @ ( minus_minus_int @ Z12 @ Z23 ) @ W )
      = ( minus_minus_int @ ( times_times_int @ Z12 @ W ) @ ( times_times_int @ Z23 @ W ) ) ) ).

% int_distrib(3)
thf(fact_1080_int__distrib_I4_J,axiom,
    ! [W: int,Z12: int,Z23: int] :
      ( ( times_times_int @ W @ ( minus_minus_int @ Z12 @ Z23 ) )
      = ( minus_minus_int @ ( times_times_int @ W @ Z12 ) @ ( times_times_int @ W @ Z23 ) ) ) ).

% int_distrib(4)
thf(fact_1081_Un__Diff,axiom,
    ! [A: set_nat,B2: set_nat,C2: set_nat] :
      ( ( minus_minus_set_nat @ ( sup_sup_set_nat @ A @ B2 ) @ C2 )
      = ( sup_sup_set_nat @ ( minus_minus_set_nat @ A @ C2 ) @ ( minus_minus_set_nat @ B2 @ C2 ) ) ) ).

% Un_Diff
thf(fact_1082_translation__diff,axiom,
    ! [A2: int,S3: set_int,T2: set_int] :
      ( ( image_int_int @ ( plus_plus_int @ A2 ) @ ( minus_minus_set_int @ S3 @ T2 ) )
      = ( minus_minus_set_int @ ( image_int_int @ ( plus_plus_int @ A2 ) @ S3 ) @ ( image_int_int @ ( plus_plus_int @ A2 ) @ T2 ) ) ) ).

% translation_diff
thf(fact_1083_plusinfinity,axiom,
    ! [D2: int,P5: int > $o,P: int > $o] :
      ( ( ord_less_int @ zero_zero_int @ D2 )
     => ( ! [X2: int,K2: int] :
            ( ( P5 @ X2 )
            = ( P5 @ ( minus_minus_int @ X2 @ ( times_times_int @ K2 @ D2 ) ) ) )
       => ( ? [Z5: int] :
            ! [X2: int] :
              ( ( ord_less_int @ Z5 @ X2 )
             => ( ( P @ X2 )
                = ( P5 @ X2 ) ) )
         => ( ? [X_1: int] : ( P5 @ X_1 )
           => ? [X_12: int] : ( P @ X_12 ) ) ) ) ) ).

% plusinfinity
thf(fact_1084_minusinfinity,axiom,
    ! [D2: int,P1: int > $o,P: int > $o] :
      ( ( ord_less_int @ zero_zero_int @ D2 )
     => ( ! [X2: int,K2: int] :
            ( ( P1 @ X2 )
            = ( P1 @ ( minus_minus_int @ X2 @ ( times_times_int @ K2 @ D2 ) ) ) )
       => ( ? [Z5: int] :
            ! [X2: int] :
              ( ( ord_less_int @ X2 @ Z5 )
             => ( ( P @ X2 )
                = ( P1 @ X2 ) ) )
         => ( ? [X_1: int] : ( P1 @ X_1 )
           => ? [X_12: int] : ( P @ X_12 ) ) ) ) ) ).

% minusinfinity
thf(fact_1085_decr__mult__lemma,axiom,
    ! [D2: int,P: int > $o,K: int] :
      ( ( ord_less_int @ zero_zero_int @ D2 )
     => ( ! [X2: int] :
            ( ( P @ X2 )
           => ( P @ ( minus_minus_int @ X2 @ D2 ) ) )
       => ( ( ord_less_eq_int @ zero_zero_int @ K )
         => ! [X4: int] :
              ( ( P @ X4 )
             => ( P @ ( minus_minus_int @ X4 @ ( times_times_int @ K @ D2 ) ) ) ) ) ) ) ).

% decr_mult_lemma
thf(fact_1086_conj__le__cong,axiom,
    ! [X: int,X11: int,P: $o,P5: $o] :
      ( ( X = X11 )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ X11 )
         => ( P = P5 ) )
       => ( ( ( ord_less_eq_int @ zero_zero_int @ X )
            & P )
          = ( ( ord_less_eq_int @ zero_zero_int @ X11 )
            & P5 ) ) ) ) ).

% conj_le_cong
thf(fact_1087_imp__le__cong,axiom,
    ! [X: int,X11: int,P: $o,P5: $o] :
      ( ( X = X11 )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ X11 )
         => ( P = P5 ) )
       => ( ( ( ord_less_eq_int @ zero_zero_int @ X )
           => P )
          = ( ( ord_less_eq_int @ zero_zero_int @ X11 )
           => P5 ) ) ) ) ).

% imp_le_cong
thf(fact_1088_all__subset__image,axiom,
    ! [F: product_prod_int_int > int,A: set_Pr958786334691620121nt_int,P: set_int > $o] :
      ( ( ! [B7: set_int] :
            ( ( ord_less_eq_set_int @ B7 @ ( image_5042161079198086560nt_int @ F @ A ) )
           => ( P @ B7 ) ) )
      = ( ! [B7: set_Pr958786334691620121nt_int] :
            ( ( ord_le2843351958646193337nt_int @ B7 @ A )
           => ( P @ ( image_5042161079198086560nt_int @ F @ B7 ) ) ) ) ) ).

% all_subset_image
thf(fact_1089_nth__Cons__pos,axiom,
    ! [N: nat,X: int,Xs2: list_int] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( nth_int @ ( cons_int @ X @ Xs2 ) @ N )
        = ( nth_int @ Xs2 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).

% nth_Cons_pos
thf(fact_1090_DiffI,axiom,
    ! [C: product_prod_int_int,A: set_Pr958786334691620121nt_int,B2: set_Pr958786334691620121nt_int] :
      ( ( member5262025264175285858nt_int @ C @ A )
     => ( ~ ( member5262025264175285858nt_int @ C @ B2 )
       => ( member5262025264175285858nt_int @ C @ ( minus_1052850069191792384nt_int @ A @ B2 ) ) ) ) ).

% DiffI
thf(fact_1091_DiffI,axiom,
    ! [C: int,A: set_int,B2: set_int] :
      ( ( member_int2 @ C @ A )
     => ( ~ ( member_int2 @ C @ B2 )
       => ( member_int2 @ C @ ( minus_minus_set_int @ A @ B2 ) ) ) ) ).

% DiffI
thf(fact_1092_Diff__iff,axiom,
    ! [C: product_prod_int_int,A: set_Pr958786334691620121nt_int,B2: set_Pr958786334691620121nt_int] :
      ( ( member5262025264175285858nt_int @ C @ ( minus_1052850069191792384nt_int @ A @ B2 ) )
      = ( ( member5262025264175285858nt_int @ C @ A )
        & ~ ( member5262025264175285858nt_int @ C @ B2 ) ) ) ).

% Diff_iff
thf(fact_1093_Diff__iff,axiom,
    ! [C: int,A: set_int,B2: set_int] :
      ( ( member_int2 @ C @ ( minus_minus_set_int @ A @ B2 ) )
      = ( ( member_int2 @ C @ A )
        & ~ ( member_int2 @ C @ B2 ) ) ) ).

% Diff_iff
thf(fact_1094_mult__1,axiom,
    ! [A2: nat] :
      ( ( times_times_nat @ one_one_nat @ A2 )
      = A2 ) ).

% mult_1
thf(fact_1095_mult__1,axiom,
    ! [A2: int] :
      ( ( times_times_int @ one_one_int @ A2 )
      = A2 ) ).

% mult_1
thf(fact_1096_mult_Oright__neutral,axiom,
    ! [A2: nat] :
      ( ( times_times_nat @ A2 @ one_one_nat )
      = A2 ) ).

% mult.right_neutral
thf(fact_1097_mult_Oright__neutral,axiom,
    ! [A2: int] :
      ( ( times_times_int @ A2 @ one_one_int )
      = A2 ) ).

% mult.right_neutral
thf(fact_1098_nat__mult__eq__1__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( times_times_nat @ M2 @ N )
        = one_one_nat )
      = ( ( M2 = one_one_nat )
        & ( N = one_one_nat ) ) ) ).

% nat_mult_eq_1_iff
thf(fact_1099_nat__1__eq__mult__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( one_one_nat
        = ( times_times_nat @ M2 @ N ) )
      = ( ( M2 = one_one_nat )
        & ( N = one_one_nat ) ) ) ).

% nat_1_eq_mult_iff
thf(fact_1100_mult__cancel__right2,axiom,
    ! [A2: int,C: int] :
      ( ( ( times_times_int @ A2 @ C )
        = C )
      = ( ( C = zero_zero_int )
        | ( A2 = one_one_int ) ) ) ).

% mult_cancel_right2
thf(fact_1101_mult__cancel__right1,axiom,
    ! [C: int,B: int] :
      ( ( C
        = ( times_times_int @ B @ C ) )
      = ( ( C = zero_zero_int )
        | ( B = one_one_int ) ) ) ).

% mult_cancel_right1
thf(fact_1102_mult__cancel__left2,axiom,
    ! [C: int,A2: int] :
      ( ( ( times_times_int @ C @ A2 )
        = C )
      = ( ( C = zero_zero_int )
        | ( A2 = one_one_int ) ) ) ).

% mult_cancel_left2
thf(fact_1103_mult__cancel__left1,axiom,
    ! [C: int,B: int] :
      ( ( C
        = ( times_times_int @ C @ B ) )
      = ( ( C = zero_zero_int )
        | ( B = one_one_int ) ) ) ).

% mult_cancel_left1
thf(fact_1104_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_int @ one_one_int @ one_one_int )
    = zero_zero_int ) ).

% diff_numeral_special(9)
thf(fact_1105_min__0__1_I2_J,axiom,
    ( ( ord_min_int @ one_one_int @ zero_zero_int )
    = zero_zero_int ) ).

% min_0_1(2)
thf(fact_1106_min__0__1_I2_J,axiom,
    ( ( ord_min_nat @ one_one_nat @ zero_zero_nat )
    = zero_zero_nat ) ).

% min_0_1(2)
thf(fact_1107_min__0__1_I1_J,axiom,
    ( ( ord_min_int @ zero_zero_int @ one_one_int )
    = zero_zero_int ) ).

% min_0_1(1)
thf(fact_1108_min__0__1_I1_J,axiom,
    ( ( ord_min_nat @ zero_zero_nat @ one_one_nat )
    = zero_zero_nat ) ).

% min_0_1(1)
thf(fact_1109_less__one,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ N @ one_one_nat )
      = ( N = zero_zero_nat ) ) ).

% less_one
thf(fact_1110_le__numeral__extra_I4_J,axiom,
    ord_less_eq_nat @ one_one_nat @ one_one_nat ).

% le_numeral_extra(4)
thf(fact_1111_le__numeral__extra_I4_J,axiom,
    ord_less_eq_int @ one_one_int @ one_one_int ).

% le_numeral_extra(4)
thf(fact_1112_DiffE,axiom,
    ! [C: product_prod_int_int,A: set_Pr958786334691620121nt_int,B2: set_Pr958786334691620121nt_int] :
      ( ( member5262025264175285858nt_int @ C @ ( minus_1052850069191792384nt_int @ A @ B2 ) )
     => ~ ( ( member5262025264175285858nt_int @ C @ A )
         => ( member5262025264175285858nt_int @ C @ B2 ) ) ) ).

% DiffE
thf(fact_1113_DiffE,axiom,
    ! [C: int,A: set_int,B2: set_int] :
      ( ( member_int2 @ C @ ( minus_minus_set_int @ A @ B2 ) )
     => ~ ( ( member_int2 @ C @ A )
         => ( member_int2 @ C @ B2 ) ) ) ).

% DiffE
thf(fact_1114_DiffD1,axiom,
    ! [C: product_prod_int_int,A: set_Pr958786334691620121nt_int,B2: set_Pr958786334691620121nt_int] :
      ( ( member5262025264175285858nt_int @ C @ ( minus_1052850069191792384nt_int @ A @ B2 ) )
     => ( member5262025264175285858nt_int @ C @ A ) ) ).

% DiffD1
thf(fact_1115_DiffD1,axiom,
    ! [C: int,A: set_int,B2: set_int] :
      ( ( member_int2 @ C @ ( minus_minus_set_int @ A @ B2 ) )
     => ( member_int2 @ C @ A ) ) ).

% DiffD1
thf(fact_1116_DiffD2,axiom,
    ! [C: product_prod_int_int,A: set_Pr958786334691620121nt_int,B2: set_Pr958786334691620121nt_int] :
      ( ( member5262025264175285858nt_int @ C @ ( minus_1052850069191792384nt_int @ A @ B2 ) )
     => ~ ( member5262025264175285858nt_int @ C @ B2 ) ) ).

% DiffD2
thf(fact_1117_DiffD2,axiom,
    ! [C: int,A: set_int,B2: set_int] :
      ( ( member_int2 @ C @ ( minus_minus_set_int @ A @ B2 ) )
     => ~ ( member_int2 @ C @ B2 ) ) ).

% DiffD2
thf(fact_1118_psubset__imp__ex__mem,axiom,
    ! [A: set_Pr958786334691620121nt_int,B2: set_Pr958786334691620121nt_int] :
      ( ( ord_le7563427860532173253nt_int @ A @ B2 )
     => ? [B4: product_prod_int_int] : ( member5262025264175285858nt_int @ B4 @ ( minus_1052850069191792384nt_int @ B2 @ A ) ) ) ).

% psubset_imp_ex_mem
thf(fact_1119_psubset__imp__ex__mem,axiom,
    ! [A: set_int,B2: set_int] :
      ( ( ord_less_set_int @ A @ B2 )
     => ? [B4: int] : ( member_int2 @ B4 @ ( minus_minus_set_int @ B2 @ A ) ) ) ).

% psubset_imp_ex_mem
thf(fact_1120_not__one__le__zero,axiom,
    ~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).

% not_one_le_zero
thf(fact_1121_not__one__le__zero,axiom,
    ~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).

% not_one_le_zero
thf(fact_1122_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).

% linordered_nonzero_semiring_class.zero_le_one
thf(fact_1123_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    ord_less_eq_int @ zero_zero_int @ one_one_int ).

% linordered_nonzero_semiring_class.zero_le_one
thf(fact_1124_zero__less__one__class_Ozero__le__one,axiom,
    ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).

% zero_less_one_class.zero_le_one
thf(fact_1125_zero__less__one__class_Ozero__le__one,axiom,
    ord_less_eq_int @ zero_zero_int @ one_one_int ).

% zero_less_one_class.zero_le_one
thf(fact_1126_zero__less__one,axiom,
    ord_less_nat @ zero_zero_nat @ one_one_nat ).

% zero_less_one
thf(fact_1127_zero__less__one,axiom,
    ord_less_int @ zero_zero_int @ one_one_int ).

% zero_less_one
thf(fact_1128_not__one__less__zero,axiom,
    ~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).

% not_one_less_zero
thf(fact_1129_not__one__less__zero,axiom,
    ~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).

% not_one_less_zero
thf(fact_1130_less__numeral__extra_I1_J,axiom,
    ord_less_nat @ zero_zero_nat @ one_one_nat ).

% less_numeral_extra(1)
thf(fact_1131_less__numeral__extra_I1_J,axiom,
    ord_less_int @ zero_zero_int @ one_one_int ).

% less_numeral_extra(1)
thf(fact_1132_less__add__one,axiom,
    ! [A2: nat] : ( ord_less_nat @ A2 @ ( plus_plus_nat @ A2 @ one_one_nat ) ) ).

% less_add_one
thf(fact_1133_less__add__one,axiom,
    ! [A2: int] : ( ord_less_int @ A2 @ ( plus_plus_int @ A2 @ one_one_int ) ) ).

% less_add_one
thf(fact_1134_add__mono1,axiom,
    ! [A2: nat,B: nat] :
      ( ( ord_less_nat @ A2 @ B )
     => ( ord_less_nat @ ( plus_plus_nat @ A2 @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).

% add_mono1
thf(fact_1135_add__mono1,axiom,
    ! [A2: int,B: int] :
      ( ( ord_less_int @ A2 @ B )
     => ( ord_less_int @ ( plus_plus_int @ A2 @ one_one_int ) @ ( plus_plus_int @ B @ one_one_int ) ) ) ).

% add_mono1
thf(fact_1136_mult__eq__self__implies__10,axiom,
    ! [M2: nat,N: nat] :
      ( ( M2
        = ( times_times_nat @ M2 @ N ) )
     => ( ( N = one_one_nat )
        | ( M2 = zero_zero_nat ) ) ) ).

% mult_eq_self_implies_10
thf(fact_1137_zero__neq__one,axiom,
    zero_zero_nat != one_one_nat ).

% zero_neq_one
thf(fact_1138_zero__neq__one,axiom,
    zero_zero_int != one_one_int ).

% zero_neq_one
thf(fact_1139_one__natural_Orsp,axiom,
    one_one_nat = one_one_nat ).

% one_natural.rsp
thf(fact_1140_one__reorient,axiom,
    ! [X: nat] :
      ( ( one_one_nat = X )
      = ( X = one_one_nat ) ) ).

% one_reorient
thf(fact_1141_one__reorient,axiom,
    ! [X: int] :
      ( ( one_one_int = X )
      = ( X = one_one_int ) ) ).

% one_reorient
thf(fact_1142_nat__mult__1__right,axiom,
    ! [N: nat] :
      ( ( times_times_nat @ N @ one_one_nat )
      = N ) ).

% nat_mult_1_right
thf(fact_1143_nat__mult__1,axiom,
    ! [N: nat] :
      ( ( times_times_nat @ one_one_nat @ N )
      = N ) ).

% nat_mult_1
thf(fact_1144_mult_Ocomm__neutral,axiom,
    ! [A2: nat] :
      ( ( times_times_nat @ A2 @ one_one_nat )
      = A2 ) ).

% mult.comm_neutral
thf(fact_1145_mult_Ocomm__neutral,axiom,
    ! [A2: int] :
      ( ( times_times_int @ A2 @ one_one_int )
      = A2 ) ).

% mult.comm_neutral
thf(fact_1146_comm__monoid__mult__class_Omult__1,axiom,
    ! [A2: nat] :
      ( ( times_times_nat @ one_one_nat @ A2 )
      = A2 ) ).

% comm_monoid_mult_class.mult_1
thf(fact_1147_comm__monoid__mult__class_Omult__1,axiom,
    ! [A2: int] :
      ( ( times_times_int @ one_one_int @ A2 )
      = A2 ) ).

% comm_monoid_mult_class.mult_1
thf(fact_1148_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).

% less_numeral_extra(4)
thf(fact_1149_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_int @ one_one_int @ one_one_int ) ).

% less_numeral_extra(4)
thf(fact_1150_psubsetD,axiom,
    ! [A: set_Pr958786334691620121nt_int,B2: set_Pr958786334691620121nt_int,C: product_prod_int_int] :
      ( ( ord_le7563427860532173253nt_int @ A @ B2 )
     => ( ( member5262025264175285858nt_int @ C @ A )
       => ( member5262025264175285858nt_int @ C @ B2 ) ) ) ).

% psubsetD
thf(fact_1151_psubsetD,axiom,
    ! [A: set_int,B2: set_int,C: int] :
      ( ( ord_less_set_int @ A @ B2 )
     => ( ( member_int2 @ C @ A )
       => ( member_int2 @ C @ B2 ) ) ) ).

% psubsetD
thf(fact_1152_mult__left__le__one__le,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ( ord_less_eq_int @ Y @ one_one_int )
         => ( ord_less_eq_int @ ( times_times_int @ Y @ X ) @ X ) ) ) ) ).

% mult_left_le_one_le
thf(fact_1153_mult__right__le__one__le,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ( ord_less_eq_int @ Y @ one_one_int )
         => ( ord_less_eq_int @ ( times_times_int @ X @ Y ) @ X ) ) ) ) ).

% mult_right_le_one_le
thf(fact_1154_mult__le__one,axiom,
    ! [A2: nat,B: nat] :
      ( ( ord_less_eq_nat @ A2 @ one_one_nat )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ( ord_less_eq_nat @ B @ one_one_nat )
         => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B ) @ one_one_nat ) ) ) ) ).

% mult_le_one
thf(fact_1155_mult__le__one,axiom,
    ! [A2: int,B: int] :
      ( ( ord_less_eq_int @ A2 @ one_one_int )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ( ord_less_eq_int @ B @ one_one_int )
         => ( ord_less_eq_int @ ( times_times_int @ A2 @ B ) @ one_one_int ) ) ) ) ).

% mult_le_one
thf(fact_1156_mult__left__le,axiom,
    ! [C: nat,A2: nat] :
      ( ( ord_less_eq_nat @ C @ one_one_nat )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
       => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ A2 ) ) ) ).

% mult_left_le
thf(fact_1157_mult__left__le,axiom,
    ! [C: int,A2: int] :
      ( ( ord_less_eq_int @ C @ one_one_int )
     => ( ( ord_less_eq_int @ zero_zero_int @ A2 )
       => ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ A2 ) ) ) ).

% mult_left_le
thf(fact_1158_zero__less__two,axiom,
    ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).

% zero_less_two
thf(fact_1159_zero__less__two,axiom,
    ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).

% zero_less_two
thf(fact_1160_square__diff__one__factored,axiom,
    ! [X: int] :
      ( ( minus_minus_int @ ( times_times_int @ X @ X ) @ one_one_int )
      = ( times_times_int @ ( plus_plus_int @ X @ one_one_int ) @ ( minus_minus_int @ X @ one_one_int ) ) ) ).

% square_diff_one_factored
thf(fact_1161_list__len__g__1__split,axiom,
    ! [Xs2: list_int] :
      ( ( ord_less_nat @ one_one_nat @ ( size_size_list_int @ Xs2 ) )
     => ? [X_12: int,X_2: int,Xs4: list_int] :
          ( Xs2
          = ( cons_int @ X_12 @ ( cons_int @ X_2 @ Xs4 ) ) ) ) ).

% list_len_g_1_split
thf(fact_1162_list__len__g__1__split,axiom,
    ! [Xs2: list_P5707943133018811711nt_int] :
      ( ( ord_less_nat @ one_one_nat @ ( size_s5157815400016825771nt_int @ Xs2 ) )
     => ? [X_12: product_prod_int_int,X_2: product_prod_int_int,Xs4: list_P5707943133018811711nt_int] :
          ( Xs2
          = ( cons_P3334398858971670639nt_int @ X_12 @ ( cons_P3334398858971670639nt_int @ X_2 @ Xs4 ) ) ) ) ).

% list_len_g_1_split
thf(fact_1163_count__list_Osimps_I2_J,axiom,
    ! [X: int,Y: int,Xs2: list_int] :
      ( ( ( X = Y )
       => ( ( count_list_int @ ( cons_int @ X @ Xs2 ) @ Y )
          = ( plus_plus_nat @ ( count_list_int @ Xs2 @ Y ) @ one_one_nat ) ) )
      & ( ( X != Y )
       => ( ( count_list_int @ ( cons_int @ X @ Xs2 ) @ Y )
          = ( count_list_int @ Xs2 @ Y ) ) ) ) ).

% count_list.simps(2)
thf(fact_1164_mult__le__cancel__left1,axiom,
    ! [C: int,B: int] :
      ( ( ord_less_eq_int @ C @ ( times_times_int @ C @ B ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ one_one_int @ B ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).

% mult_le_cancel_left1
thf(fact_1165_mult__le__cancel__left2,axiom,
    ! [C: int,A2: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ C )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ A2 @ one_one_int ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ one_one_int @ A2 ) ) ) ) ).

% mult_le_cancel_left2
thf(fact_1166_mult__le__cancel__right1,axiom,
    ! [C: int,B: int] :
      ( ( ord_less_eq_int @ C @ ( times_times_int @ B @ C ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ one_one_int @ B ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).

% mult_le_cancel_right1
thf(fact_1167_mult__le__cancel__right2,axiom,
    ! [A2: int,C: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ C )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ A2 @ one_one_int ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ one_one_int @ A2 ) ) ) ) ).

% mult_le_cancel_right2
thf(fact_1168_mult__less__cancel__left1,axiom,
    ! [C: int,B: int] :
      ( ( ord_less_int @ C @ ( times_times_int @ C @ B ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ one_one_int @ B ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ B @ one_one_int ) ) ) ) ).

% mult_less_cancel_left1
thf(fact_1169_mult__less__cancel__left2,axiom,
    ! [C: int,A2: int] :
      ( ( ord_less_int @ ( times_times_int @ C @ A2 ) @ C )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ A2 @ one_one_int ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ one_one_int @ A2 ) ) ) ) ).

% mult_less_cancel_left2
thf(fact_1170_mult__less__cancel__right1,axiom,
    ! [C: int,B: int] :
      ( ( ord_less_int @ C @ ( times_times_int @ B @ C ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ one_one_int @ B ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ B @ one_one_int ) ) ) ) ).

% mult_less_cancel_right1
thf(fact_1171_mult__less__cancel__right2,axiom,
    ! [A2: int,C: int] :
      ( ( ord_less_int @ ( times_times_int @ A2 @ C ) @ C )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ A2 @ one_one_int ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ one_one_int @ A2 ) ) ) ) ).

% mult_less_cancel_right2
thf(fact_1172_convex__bound__le,axiom,
    ! [X: int,A2: int,Y: int,U: int,V: int] :
      ( ( ord_less_eq_int @ X @ A2 )
     => ( ( ord_less_eq_int @ Y @ A2 )
       => ( ( ord_less_eq_int @ zero_zero_int @ U )
         => ( ( ord_less_eq_int @ zero_zero_int @ V )
           => ( ( ( plus_plus_int @ U @ V )
                = one_one_int )
             => ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ U @ X ) @ ( times_times_int @ V @ Y ) ) @ A2 ) ) ) ) ) ) ).

% convex_bound_le
thf(fact_1173_mult__eq__if,axiom,
    ( times_times_nat
    = ( ^ [M4: nat,N3: nat] : ( if_nat @ ( M4 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N3 @ ( times_times_nat @ ( minus_minus_nat @ M4 @ one_one_nat ) @ N3 ) ) ) ) ) ).

% mult_eq_if
thf(fact_1174_nth__Cons_H,axiom,
    ! [N: nat,X: int,Xs2: list_int] :
      ( ( ( N = zero_zero_nat )
       => ( ( nth_int @ ( cons_int @ X @ Xs2 ) @ N )
          = X ) )
      & ( ( N != zero_zero_nat )
       => ( ( nth_int @ ( cons_int @ X @ Xs2 ) @ N )
          = ( nth_int @ Xs2 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).

% nth_Cons'
thf(fact_1175_convex__bound__lt,axiom,
    ! [X: int,A2: int,Y: int,U: int,V: int] :
      ( ( ord_less_int @ X @ A2 )
     => ( ( ord_less_int @ Y @ A2 )
       => ( ( ord_less_eq_int @ zero_zero_int @ U )
         => ( ( ord_less_eq_int @ zero_zero_int @ V )
           => ( ( ( plus_plus_int @ U @ V )
                = one_one_int )
             => ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ U @ X ) @ ( times_times_int @ V @ Y ) ) @ A2 ) ) ) ) ) ) ).

% convex_bound_lt
thf(fact_1176_nth__non__equal__first__eq,axiom,
    ! [X: int,Y: int,Xs2: list_int,N: nat] :
      ( ( X != Y )
     => ( ( ( nth_int @ ( cons_int @ X @ Xs2 ) @ N )
          = Y )
        = ( ( ( nth_int @ Xs2 @ ( minus_minus_nat @ N @ one_one_nat ) )
            = Y )
          & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).

% nth_non_equal_first_eq
thf(fact_1177_restrict__map__upds,axiom,
    ! [Xs2: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int,D: set_Pr958786334691620121nt_int,M2: product_prod_int_int > option4624381673175914239nt_int] :
      ( ( ( size_s5157815400016825771nt_int @ Xs2 )
        = ( size_s5157815400016825771nt_int @ Ys ) )
     => ( ( ord_le2843351958646193337nt_int @ ( set_Pr2470121279949933262nt_int @ Xs2 ) @ D )
       => ( ( restri2103409392168779760nt_int @ ( map_up6499465237031507894nt_int @ M2 @ Xs2 @ Ys ) @ D )
          = ( map_up6499465237031507894nt_int @ ( restri2103409392168779760nt_int @ M2 @ ( minus_1052850069191792384nt_int @ D @ ( set_Pr2470121279949933262nt_int @ Xs2 ) ) ) @ Xs2 @ Ys ) ) ) ) ).

% restrict_map_upds
thf(fact_1178_zle__add1__eq__le,axiom,
    ! [W: int,Z: int] :
      ( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
      = ( ord_less_eq_int @ W @ Z ) ) ).

% zle_add1_eq_le
thf(fact_1179_zle__diff1__eq,axiom,
    ! [W: int,Z: int] :
      ( ( ord_less_eq_int @ W @ ( minus_minus_int @ Z @ one_one_int ) )
      = ( ord_less_int @ W @ Z ) ) ).

% zle_diff1_eq
thf(fact_1180_int__ge__induct,axiom,
    ! [K: int,I: int,P: int > $o] :
      ( ( ord_less_eq_int @ K @ I )
     => ( ( P @ K )
       => ( ! [I3: int] :
              ( ( ord_less_eq_int @ K @ I3 )
             => ( ( P @ I3 )
               => ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
         => ( P @ I ) ) ) ) ).

% int_ge_induct
thf(fact_1181_int__le__induct,axiom,
    ! [I: int,K: int,P: int > $o] :
      ( ( ord_less_eq_int @ I @ K )
     => ( ( P @ K )
       => ( ! [I3: int] :
              ( ( ord_less_eq_int @ I3 @ K )
             => ( ( P @ I3 )
               => ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
         => ( P @ I ) ) ) ) ).

% int_le_induct
thf(fact_1182_int__less__induct,axiom,
    ! [I: int,K: int,P: int > $o] :
      ( ( ord_less_int @ I @ K )
     => ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
       => ( ! [I3: int] :
              ( ( ord_less_int @ I3 @ K )
             => ( ( P @ I3 )
               => ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
         => ( P @ I ) ) ) ) ).

% int_less_induct
thf(fact_1183_odd__nonzero,axiom,
    ! [Z: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z )
     != zero_zero_int ) ).

% odd_nonzero
thf(fact_1184_int__gr__induct,axiom,
    ! [K: int,I: int,P: int > $o] :
      ( ( ord_less_int @ K @ I )
     => ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
       => ( ! [I3: int] :
              ( ( ord_less_int @ K @ I3 )
             => ( ( P @ I3 )
               => ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
         => ( P @ I ) ) ) ) ).

% int_gr_induct
thf(fact_1185_zless__add1__eq,axiom,
    ! [W: int,Z: int] :
      ( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
      = ( ( ord_less_int @ W @ Z )
        | ( W = Z ) ) ) ).

% zless_add1_eq
thf(fact_1186_graph__restrictD_I1_J,axiom,
    ! [K: int,V: int,M2: int > option_int,A: set_int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ K @ V ) @ ( graph_int_int @ ( restrict_map_int_int @ M2 @ A ) ) )
     => ( member_int2 @ K @ A ) ) ).

% graph_restrictD(1)
thf(fact_1187_int__one__le__iff__zero__less,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ one_one_int @ Z )
      = ( ord_less_int @ zero_zero_int @ Z ) ) ).

% int_one_le_iff_zero_less
thf(fact_1188_odd__less__0__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z ) @ zero_zero_int )
      = ( ord_less_int @ Z @ zero_zero_int ) ) ).

% odd_less_0_iff
thf(fact_1189_pos__zmult__eq__1__iff,axiom,
    ! [M2: int,N: int] :
      ( ( ord_less_int @ zero_zero_int @ M2 )
     => ( ( ( times_times_int @ M2 @ N )
          = one_one_int )
        = ( ( M2 = one_one_int )
          & ( N = one_one_int ) ) ) ) ).

% pos_zmult_eq_1_iff
thf(fact_1190_zless__imp__add1__zle,axiom,
    ! [W: int,Z: int] :
      ( ( ord_less_int @ W @ Z )
     => ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z ) ) ).

% zless_imp_add1_zle
thf(fact_1191_add1__zle__eq,axiom,
    ! [W: int,Z: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z )
      = ( ord_less_int @ W @ Z ) ) ).

% add1_zle_eq
thf(fact_1192_int__induct,axiom,
    ! [P: int > $o,K: int,I: int] :
      ( ( P @ K )
     => ( ! [I3: int] :
            ( ( ord_less_eq_int @ K @ I3 )
           => ( ( P @ I3 )
             => ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
       => ( ! [I3: int] :
              ( ( ord_less_eq_int @ I3 @ K )
             => ( ( P @ I3 )
               => ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
         => ( P @ I ) ) ) ) ).

% int_induct
thf(fact_1193_graph__restrictD_I2_J,axiom,
    ! [K: int,V: int,M2: int > option_int,A: set_int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ K @ V ) @ ( graph_int_int @ ( restrict_map_int_int @ M2 @ A ) ) )
     => ( ( M2 @ K )
        = ( some_int @ V ) ) ) ).

% graph_restrictD(2)
thf(fact_1194_le__imp__0__less,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z ) ) ) ).

% le_imp_0_less
thf(fact_1195_one__integer_Orsp,axiom,
    one_one_int = one_one_int ).

% one_integer.rsp
thf(fact_1196_upto__aux__rec,axiom,
    ( upto_aux
    = ( ^ [I2: int,J2: int,Js: list_int] : ( if_list_int @ ( ord_less_int @ J2 @ I2 ) @ Js @ ( upto_aux @ I2 @ ( minus_minus_int @ J2 @ one_one_int ) @ ( cons_int @ J2 @ Js ) ) ) ) ) ).

% upto_aux_rec
thf(fact_1197_nat0__intermed__int__val,axiom,
    ! [N: nat,F: nat > int,K: int] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ N )
         => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I3 @ one_one_nat ) ) @ ( F @ I3 ) ) ) @ one_one_int ) )
     => ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
       => ( ( ord_less_eq_int @ K @ ( F @ N ) )
         => ? [I3: nat] :
              ( ( ord_less_eq_nat @ I3 @ N )
              & ( ( F @ I3 )
                = K ) ) ) ) ) ).

% nat0_intermed_int_val
thf(fact_1198_butlast__take,axiom,
    ! [N: nat,Xs2: list_P5707943133018811711nt_int] :
      ( ( ord_less_eq_nat @ N @ ( size_s5157815400016825771nt_int @ Xs2 ) )
     => ( ( butlas2390654596765537291nt_int @ ( take_P8218740963776755879nt_int @ N @ Xs2 ) )
        = ( take_P8218740963776755879nt_int @ ( minus_minus_nat @ N @ one_one_nat ) @ Xs2 ) ) ) ).

% butlast_take
thf(fact_1199_abs__idempotent,axiom,
    ! [A2: int] :
      ( ( abs_abs_int @ ( abs_abs_int @ A2 ) )
      = ( abs_abs_int @ A2 ) ) ).

% abs_idempotent
thf(fact_1200_abs__0,axiom,
    ( ( abs_abs_int @ zero_zero_int )
    = zero_zero_int ) ).

% abs_0
thf(fact_1201_abs__zero,axiom,
    ( ( abs_abs_int @ zero_zero_int )
    = zero_zero_int ) ).

% abs_zero
thf(fact_1202_abs__eq__0,axiom,
    ! [A2: int] :
      ( ( ( abs_abs_int @ A2 )
        = zero_zero_int )
      = ( A2 = zero_zero_int ) ) ).

% abs_eq_0
thf(fact_1203_abs__0__eq,axiom,
    ! [A2: int] :
      ( ( zero_zero_int
        = ( abs_abs_int @ A2 ) )
      = ( A2 = zero_zero_int ) ) ).

% abs_0_eq
thf(fact_1204_abs__add__abs,axiom,
    ! [A2: int,B: int] :
      ( ( abs_abs_int @ ( plus_plus_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ B ) ) )
      = ( plus_plus_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ B ) ) ) ).

% abs_add_abs
thf(fact_1205_abs__of__nonneg,axiom,
    ! [A2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A2 )
     => ( ( abs_abs_int @ A2 )
        = A2 ) ) ).

% abs_of_nonneg
thf(fact_1206_abs__le__self__iff,axiom,
    ! [A2: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ A2 ) @ A2 )
      = ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ).

% abs_le_self_iff
thf(fact_1207_abs__le__zero__iff,axiom,
    ! [A2: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ A2 ) @ zero_zero_int )
      = ( A2 = zero_zero_int ) ) ).

% abs_le_zero_iff
thf(fact_1208_zero__less__abs__iff,axiom,
    ! [A2: int] :
      ( ( ord_less_int @ zero_zero_int @ ( abs_abs_int @ A2 ) )
      = ( A2 != zero_zero_int ) ) ).

% zero_less_abs_iff
thf(fact_1209_zabs__less__one__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_int @ ( abs_abs_int @ Z ) @ one_one_int )
      = ( Z = zero_zero_int ) ) ).

% zabs_less_one_iff
thf(fact_1210_length__butlast,axiom,
    ! [Xs2: list_P5707943133018811711nt_int] :
      ( ( size_s5157815400016825771nt_int @ ( butlas2390654596765537291nt_int @ Xs2 ) )
      = ( minus_minus_nat @ ( size_s5157815400016825771nt_int @ Xs2 ) @ one_one_nat ) ) ).

% length_butlast
thf(fact_1211_abs__mult__pos_H,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( times_times_int @ X @ ( abs_abs_int @ Y ) )
        = ( abs_abs_int @ ( times_times_int @ X @ Y ) ) ) ) ).

% abs_mult_pos'
thf(fact_1212_abs__eq__mult,axiom,
    ! [A2: int,B: int] :
      ( ( ( ( ord_less_eq_int @ zero_zero_int @ A2 )
          | ( ord_less_eq_int @ A2 @ zero_zero_int ) )
        & ( ( ord_less_eq_int @ zero_zero_int @ B )
          | ( ord_less_eq_int @ B @ zero_zero_int ) ) )
     => ( ( abs_abs_int @ ( times_times_int @ A2 @ B ) )
        = ( times_times_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ B ) ) ) ) ).

% abs_eq_mult
thf(fact_1213_abs__mult__pos,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( times_times_int @ ( abs_abs_int @ Y ) @ X )
        = ( abs_abs_int @ ( times_times_int @ Y @ X ) ) ) ) ).

% abs_mult_pos
thf(fact_1214_abs__diff__le__iff,axiom,
    ! [X: int,A2: int,R: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ X @ A2 ) ) @ R )
      = ( ( ord_less_eq_int @ ( minus_minus_int @ A2 @ R ) @ X )
        & ( ord_less_eq_int @ X @ ( plus_plus_int @ A2 @ R ) ) ) ) ).

% abs_diff_le_iff
thf(fact_1215_abs__diff__triangle__ineq,axiom,
    ! [A2: int,B: int,C: int,D2: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( plus_plus_int @ A2 @ B ) @ ( plus_plus_int @ C @ D2 ) ) ) @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ A2 @ C ) ) @ ( abs_abs_int @ ( minus_minus_int @ B @ D2 ) ) ) ) ).

% abs_diff_triangle_ineq
thf(fact_1216_abs__triangle__ineq4,axiom,
    ! [A2: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ A2 @ B ) ) @ ( plus_plus_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ B ) ) ) ).

% abs_triangle_ineq4
thf(fact_1217_abs__diff__less__iff,axiom,
    ! [X: int,A2: int,R: int] :
      ( ( ord_less_int @ ( abs_abs_int @ ( minus_minus_int @ X @ A2 ) ) @ R )
      = ( ( ord_less_int @ ( minus_minus_int @ A2 @ R ) @ X )
        & ( ord_less_int @ X @ ( plus_plus_int @ A2 @ R ) ) ) ) ).

% abs_diff_less_iff
thf(fact_1218_abs__ge__self,axiom,
    ! [A2: int] : ( ord_less_eq_int @ A2 @ ( abs_abs_int @ A2 ) ) ).

% abs_ge_self
thf(fact_1219_abs__le__D1,axiom,
    ! [A2: int,B: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ A2 ) @ B )
     => ( ord_less_eq_int @ A2 @ B ) ) ).

% abs_le_D1
thf(fact_1220_abs__ge__zero,axiom,
    ! [A2: int] : ( ord_less_eq_int @ zero_zero_int @ ( abs_abs_int @ A2 ) ) ).

% abs_ge_zero
thf(fact_1221_abs__triangle__ineq,axiom,
    ! [A2: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( plus_plus_int @ A2 @ B ) ) @ ( plus_plus_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ B ) ) ) ).

% abs_triangle_ineq
thf(fact_1222_abs__eq__0__iff,axiom,
    ! [A2: int] :
      ( ( ( abs_abs_int @ A2 )
        = zero_zero_int )
      = ( A2 = zero_zero_int ) ) ).

% abs_eq_0_iff
thf(fact_1223_in__set__butlastD,axiom,
    ! [X: int,Xs2: list_int] :
      ( ( member_int2 @ X @ ( set_int2 @ ( butlast_int @ Xs2 ) ) )
     => ( member_int2 @ X @ ( set_int2 @ Xs2 ) ) ) ).

% in_set_butlastD
thf(fact_1224_in__set__butlastD,axiom,
    ! [X: product_prod_int_int,Xs2: list_P5707943133018811711nt_int] :
      ( ( member5262025264175285858nt_int @ X @ ( set_Pr2470121279949933262nt_int @ ( butlas2390654596765537291nt_int @ Xs2 ) ) )
     => ( member5262025264175285858nt_int @ X @ ( set_Pr2470121279949933262nt_int @ Xs2 ) ) ) ).

% in_set_butlastD
thf(fact_1225_abs__not__less__zero,axiom,
    ! [A2: int] :
      ~ ( ord_less_int @ ( abs_abs_int @ A2 ) @ zero_zero_int ) ).

% abs_not_less_zero
thf(fact_1226_abs__of__pos,axiom,
    ! [A2: int] :
      ( ( ord_less_int @ zero_zero_int @ A2 )
     => ( ( abs_abs_int @ A2 )
        = A2 ) ) ).

% abs_of_pos
thf(fact_1227_abs__minus__commute,axiom,
    ! [A2: int,B: int] :
      ( ( abs_abs_int @ ( minus_minus_int @ A2 @ B ) )
      = ( abs_abs_int @ ( minus_minus_int @ B @ A2 ) ) ) ).

% abs_minus_commute
thf(fact_1228_abs__triangle__ineq2,axiom,
    ! [A2: int,B: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ B ) ) @ ( abs_abs_int @ ( minus_minus_int @ A2 @ B ) ) ) ).

% abs_triangle_ineq2
thf(fact_1229_abs__triangle__ineq3,axiom,
    ! [A2: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ B ) ) ) @ ( abs_abs_int @ ( minus_minus_int @ A2 @ B ) ) ) ).

% abs_triangle_ineq3
thf(fact_1230_abs__triangle__ineq2__sym,axiom,
    ! [A2: int,B: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ B ) ) @ ( abs_abs_int @ ( minus_minus_int @ B @ A2 ) ) ) ).

% abs_triangle_ineq2_sym
thf(fact_1231_abs__zmult__eq__1,axiom,
    ! [M2: int,N: int] :
      ( ( ( abs_abs_int @ ( times_times_int @ M2 @ N ) )
        = one_one_int )
     => ( ( abs_abs_int @ M2 )
        = one_one_int ) ) ).

% abs_zmult_eq_1
thf(fact_1232_abs__add__one__gt__zero,axiom,
    ! [X: int] : ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ ( abs_abs_int @ X ) ) ) ).

% abs_add_one_gt_zero
thf(fact_1233_nth__butlast,axiom,
    ! [N: nat,Xs2: list_P5707943133018811711nt_int] :
      ( ( ord_less_nat @ N @ ( size_s5157815400016825771nt_int @ ( butlas2390654596765537291nt_int @ Xs2 ) ) )
     => ( ( nth_Pr4439495888332055232nt_int @ ( butlas2390654596765537291nt_int @ Xs2 ) @ N )
        = ( nth_Pr4439495888332055232nt_int @ Xs2 @ N ) ) ) ).

% nth_butlast
thf(fact_1234_take__butlast,axiom,
    ! [N: nat,Xs2: list_P5707943133018811711nt_int] :
      ( ( ord_less_nat @ N @ ( size_s5157815400016825771nt_int @ Xs2 ) )
     => ( ( take_P8218740963776755879nt_int @ N @ ( butlas2390654596765537291nt_int @ Xs2 ) )
        = ( take_P8218740963776755879nt_int @ N @ Xs2 ) ) ) ).

% take_butlast
thf(fact_1235_butlast__conv__take,axiom,
    ( butlas2390654596765537291nt_int
    = ( ^ [Xs3: list_P5707943133018811711nt_int] : ( take_P8218740963776755879nt_int @ ( minus_minus_nat @ ( size_s5157815400016825771nt_int @ Xs3 ) @ one_one_nat ) @ Xs3 ) ) ) ).

% butlast_conv_take
thf(fact_1236_decr__lemma,axiom,
    ! [D2: int,X: int,Z: int] :
      ( ( ord_less_int @ zero_zero_int @ D2 )
     => ( ord_less_int @ ( minus_minus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z ) ) @ one_one_int ) @ D2 ) ) @ Z ) ) ).

% decr_lemma
thf(fact_1237_incr__lemma,axiom,
    ! [D2: int,Z: int,X: int] :
      ( ( ord_less_int @ zero_zero_int @ D2 )
     => ( ord_less_int @ Z @ ( plus_plus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z ) ) @ one_one_int ) @ D2 ) ) ) ) ).

% incr_lemma
thf(fact_1238_nat__ivt__aux,axiom,
    ! [N: nat,F: nat > int,K: int] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ N )
         => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) ) ) @ one_one_int ) )
     => ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
       => ( ( ord_less_eq_int @ K @ ( F @ N ) )
         => ? [I3: nat] :
              ( ( ord_less_eq_nat @ I3 @ N )
              & ( ( F @ I3 )
                = K ) ) ) ) ) ).

% nat_ivt_aux
thf(fact_1239_nat__intermed__int__val,axiom,
    ! [M2: nat,N: nat,F: nat > int,K: int] :
      ( ! [I3: nat] :
          ( ( ( ord_less_eq_nat @ M2 @ I3 )
            & ( ord_less_nat @ I3 @ N ) )
         => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) ) ) @ one_one_int ) )
     => ( ( ord_less_eq_nat @ M2 @ N )
       => ( ( ord_less_eq_int @ ( F @ M2 ) @ K )
         => ( ( ord_less_eq_int @ K @ ( F @ N ) )
           => ? [I3: nat] :
                ( ( ord_less_eq_nat @ M2 @ I3 )
                & ( ord_less_eq_nat @ I3 @ N )
                & ( ( F @ I3 )
                  = K ) ) ) ) ) ) ).

% nat_intermed_int_val
thf(fact_1240_old_Onat_Oinject,axiom,
    ! [Nat: nat,Nat2: nat] :
      ( ( ( suc @ Nat )
        = ( suc @ Nat2 ) )
      = ( Nat = Nat2 ) ) ).

% old.nat.inject
thf(fact_1241_nat_Oinject,axiom,
    ! [X22: nat,Y22: nat] :
      ( ( ( suc @ X22 )
        = ( suc @ Y22 ) )
      = ( X22 = Y22 ) ) ).

% nat.inject
thf(fact_1242_Suc__less__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) )
      = ( ord_less_nat @ M2 @ N ) ) ).

% Suc_less_eq
thf(fact_1243_Suc__mono,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) ) ) ).

% Suc_mono
thf(fact_1244_lessI,axiom,
    ! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).

% lessI
thf(fact_1245_Suc__le__mono,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M2 ) )
      = ( ord_less_eq_nat @ N @ M2 ) ) ).

% Suc_le_mono
thf(fact_1246_add__Suc__right,axiom,
    ! [M2: nat,N: nat] :
      ( ( plus_plus_nat @ M2 @ ( suc @ N ) )
      = ( suc @ ( plus_plus_nat @ M2 @ N ) ) ) ).

% add_Suc_right
thf(fact_1247_diff__Suc__Suc,axiom,
    ! [M2: nat,N: nat] :
      ( ( minus_minus_nat @ ( suc @ M2 ) @ ( suc @ N ) )
      = ( minus_minus_nat @ M2 @ N ) ) ).

% diff_Suc_Suc
thf(fact_1248_Suc__diff__diff,axiom,
    ! [M2: nat,N: nat,K: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M2 ) @ N ) @ ( suc @ K ) )
      = ( minus_minus_nat @ ( minus_minus_nat @ M2 @ N ) @ K ) ) ).

% Suc_diff_diff
thf(fact_1249_min__Suc__Suc,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_min_nat @ ( suc @ M2 ) @ ( suc @ N ) )
      = ( suc @ ( ord_min_nat @ M2 @ N ) ) ) ).

% min_Suc_Suc
thf(fact_1250_less__Suc0,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
      = ( N = zero_zero_nat ) ) ).

% less_Suc0
thf(fact_1251_zero__less__Suc,axiom,
    ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).

% zero_less_Suc
thf(fact_1252_mult__eq__1__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( times_times_nat @ M2 @ N )
        = ( suc @ zero_zero_nat ) )
      = ( ( M2
          = ( suc @ zero_zero_nat ) )
        & ( N
          = ( suc @ zero_zero_nat ) ) ) ) ).

% mult_eq_1_iff
thf(fact_1253_one__eq__mult__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( suc @ zero_zero_nat )
        = ( times_times_nat @ M2 @ N ) )
      = ( ( M2
          = ( suc @ zero_zero_nat ) )
        & ( N
          = ( suc @ zero_zero_nat ) ) ) ) ).

% one_eq_mult_iff
thf(fact_1254_mult__Suc__right,axiom,
    ! [M2: nat,N: nat] :
      ( ( times_times_nat @ M2 @ ( suc @ N ) )
      = ( plus_plus_nat @ M2 @ ( times_times_nat @ M2 @ N ) ) ) ).

% mult_Suc_right
thf(fact_1255_nth__Cons__Suc,axiom,
    ! [X: int,Xs2: list_int,N: nat] :
      ( ( nth_int @ ( cons_int @ X @ Xs2 ) @ ( suc @ N ) )
      = ( nth_int @ Xs2 @ N ) ) ).

% nth_Cons_Suc
thf(fact_1256_diff__Suc__1,axiom,
    ! [N: nat] :
      ( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
      = N ) ).

% diff_Suc_1
thf(fact_1257_take__Suc__Cons,axiom,
    ! [N: nat,X: int,Xs2: list_int] :
      ( ( take_int @ ( suc @ N ) @ ( cons_int @ X @ Xs2 ) )
      = ( cons_int @ X @ ( take_int @ N @ Xs2 ) ) ) ).

% take_Suc_Cons
thf(fact_1258_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_1259_one__le__mult__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M2 @ N ) )
      = ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M2 )
        & ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).

% one_le_mult_iff
thf(fact_1260_diff__Suc__diff__eq2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I )
        = ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I ) ) ) ) ).

% diff_Suc_diff_eq2
thf(fact_1261_diff__Suc__diff__eq1,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ ( suc @ J ) ) ) ) ).

% diff_Suc_diff_eq1
thf(fact_1262_enumerate__simps_I2_J,axiom,
    ! [N: nat,X: int,Xs2: list_int] :
      ( ( enumerate_int @ N @ ( cons_int @ X @ Xs2 ) )
      = ( cons_P2335045147070616083at_int @ ( product_Pair_nat_int @ N @ X ) @ ( enumerate_int @ ( suc @ N ) @ Xs2 ) ) ) ).

% enumerate_simps(2)
thf(fact_1263_Suc__diff__1,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
        = N ) ) ).

% Suc_diff_1
thf(fact_1264_one__less__mult,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
     => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M2 )
       => ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M2 @ N ) ) ) ) ).

% one_less_mult
thf(fact_1265_n__less__m__mult__n,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M2 )
       => ( ord_less_nat @ N @ ( times_times_nat @ M2 @ N ) ) ) ) ).

% n_less_m_mult_n
thf(fact_1266_n__less__n__mult__m,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M2 )
       => ( ord_less_nat @ N @ ( times_times_nat @ N @ M2 ) ) ) ) ).

% n_less_n_mult_m
thf(fact_1267_diff__Suc__eq__diff__pred,axiom,
    ! [M2: nat,N: nat] :
      ( ( minus_minus_nat @ M2 @ ( suc @ N ) )
      = ( minus_minus_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N ) ) ).

% diff_Suc_eq_diff_pred
thf(fact_1268_Suc__eq__plus1__left,axiom,
    ( suc
    = ( plus_plus_nat @ one_one_nat ) ) ).

% Suc_eq_plus1_left
thf(fact_1269_plus__1__eq__Suc,axiom,
    ( ( plus_plus_nat @ one_one_nat )
    = suc ) ).

% plus_1_eq_Suc

% Helper facts (7)
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 ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
    ! [X: list_int,Y: list_int] :
      ( ( if_list_int @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
    ! [X: list_int,Y: list_int] :
      ( ( if_list_int @ $true @ X @ Y )
      = X ) ).

thf(help_If_3_1_If_001t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_T,axiom,
    ! [P: $o] :
      ( ( P = $true )
      | ( P = $false ) ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_T,axiom,
    ! [X: list_P5707943133018811711nt_int,Y: list_P5707943133018811711nt_int] :
      ( ( if_lis8883190402267401221nt_int @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_T,axiom,
    ! [X: list_P5707943133018811711nt_int,Y: list_P5707943133018811711nt_int] :
      ( ( if_lis8883190402267401221nt_int @ $true @ X @ Y )
      = X ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    ( ( size_s5157815400016825771nt_int @ ps )
    = ( size_s5157815400016825771nt_int @ ( mirror1_aux @ ( plus_plus_int @ ( lattic8263393255366662781ax_int @ ( image_5042161079198086560nt_int @ product_fst_int_int @ ( set_Pr2470121279949933262nt_int @ ps ) ) ) @ ( lattic8718645017227715691in_int @ ( image_5042161079198086560nt_int @ product_fst_int_int @ ( set_Pr2470121279949933262nt_int @ ps ) ) ) ) @ ps ) ) ) ).

%------------------------------------------------------------------------------