TPTP Problem File: SWW473

TPTP Problem File: SWW473_30.p

%------------------------------------------------------------------------------
% File     : SWW473_30 : TPTP v9.0.0. Released v8.2.0.
% Domain   : Software Verification
% Problem  : Hoare's Logic with Procedures line 383, 1000 axioms selected
% Version  : SWW473_3 with the conjecture removed
% English  :

% Refs     : [BN10]  Boehme & Nipkow (2010), Sledgehammer: Judgement Day
%          : [Bla11] Blanchette (2011), Email to Geoff Sutcliffe
% Source   : [TPTP]
% Names    : 

% Status   : Satisfiable
% Rating   : 1.00 v8.2.0
% Syntax   : Number of formulae    : 1810 ( 331 unt; 502 typ;   0 def)
%            Number of atoms       : 3258 ( 905 equ)
%            Maximal formula atoms :    9 (   2 avg)
%            Number of connectives : 2267 ( 317   ~;  77   |; 226   &)
%                                         ( 253 <=>;1394  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   14 (   5 avg)
%            Maximal term depth    :   11 (   2 avg)
%            Number of types       :  140 ( 139 usr)
%            Number of type conns  :  465 ( 261   >; 204   *;   0   +;   0  <<)
%            Number of predicates  :    2 (   1 usr;   0 prp; 1-2 aty)
%            Number of functors    :  362 ( 362 usr; 102 con; 0-2 aty)
%            Number of variables   : 3142 (3033   !; 109   ?;3142   :)
% SPC      : TF0_SAT_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2011-08-09 17:47:48
%------------------------------------------------------------------------------
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tff(ty_ty_tc__fun_Itc__fun_Itc__fun_It__a_Mtc__HOL__Obool_J_Mtc__HOL__Obool_J_Mtc__f_015,type,
    fun_fu418465139l_bool: $tType ).

tff(ty_ty_tc__fun_Itc__fun_Itc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_Mtc__HOL__Obo,type,
    fun_fu399576434l_bool: $tType ).

tff(ty_ty_tc__fun_Itc__fun_Itc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_Mtc__HOL__Obo_016,type,
    fun_fu1730389579ol_nat: $tType ).

tff(ty_ty_tc__fun_Itc__fun_Itc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_Mtc__HOL__Obo_017,type,
    fun_fu2087345469l_bool: $tType ).

tff(ty_ty_tc__fun_Itc__fun_Itc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_Mtc__HOL__Obo_018,type,
    fun_fu1911931399l_bool: $tType ).

tff(ty_ty_tc__fun_Itc__fun_Itc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_Mtc__HOL__Obool,type,
    fun_fu1540142484l_bool: $tType ).

tff(ty_ty_tc__fun_Itc__fun_Itc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_Mtc__HOL__Obool_019,type,
    fun_fu236049833ol_nat: $tType ).

tff(ty_ty_tc__fun_Itc__fun_Itc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_Mtc__HOL__Obool_020,type,
    fun_fu1070826941l_bool: $tType ).

tff(ty_ty_tc__fun_Itc__fun_Itc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_Mtc__HOL__Obool_021,type,
    fun_fu209342027l_bool: $tType ).

tff(ty_ty_tc__fun_Itc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_Mtc__HOL__Obool,type,
    fun_fu814369080l_bool: $tType ).

tff(ty_ty_tc__fun_Itc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_Mtc__HOL__Obool_022,type,
    fun_fu411113733ol_nat: $tType ).

tff(ty_ty_tc__fun_Itc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_Mtc__HOL__Obool_023,type,
    fun_fu616551101l_bool: $tType ).

tff(ty_ty_tc__fun_Itc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_Mtc__HOL__Obool_024,type,
    fun_fu140186515l_bool: $tType ).

tff(ty_ty_tc__fun_Itc__fun_Itc__fun_Itc__fun_It__a_Mtc__HOL__Obool_J_Mtc__HOL__Obool,type,
    fun_fu255076663l_bool: $tType ).

tff(ty_ty_tc__fun_Itc__fun_Itc__fun_Itc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_Mtc_,type,
    fun_fu1438281908l_bool: $tType ).

tff(ty_ty_tc__fun_Itc__fun_Itc__fun_Itc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_Mtc__H,type,
    fun_fu717296342l_bool: $tType ).

tff(ty_ty_tc__fun_Itc__fun_Itc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_Mtc__H,type,
    fun_fu61768826l_bool: $tType ).

%----Explicit typings (363)
tff(sy_c_COMBB_000t__a_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__Com__Opname,type,
    cOMBB_1897541054_pname: ( fun_a_fun_a_bool * fun_pname_a ) > fun_pname_fun_a_bool ).

tff(sy_c_COMBB_000t__a_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__Int__Oint,type,
    cOMBB_375279840ol_int: ( fun_a_fun_a_bool * fun_int_a ) > fun_int_fun_a_bool ).

tff(sy_c_COMBB_000tc__HOL__Obool_000tc__HOL__Obool_000t__a,type,
    cOMBB_bool_bool_a: ( fun_bool_bool * fun_a_bool ) > fun_a_bool ).

tff(sy_c_COMBB_000tc__HOL__Obool_000tc__HOL__Obool_000tc__Com__Opname,type,
    cOMBB_647938656_pname: ( fun_bool_bool * fun_pname_bool ) > fun_pname_bool ).

tff(sy_c_COMBB_000tc__HOL__Obool_000tc__HOL__Obool_000tc__Int__Oint,type,
    cOMBB_bool_bool_int: ( fun_bool_bool * fun_int_bool ) > fun_int_bool ).

tff(sy_c_COMBB_000tc__HOL__Obool_000tc__HOL__Obool_000tc__Nat__Onat,type,
    cOMBB_bool_bool_nat: ( fun_bool_bool * fun_nat_bool ) > fun_nat_bool ).

tff(sy_c_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Obool_J_000t__,type,
    cOMBB_1972296269bool_a: ( fun_bo1549164019l_bool * fun_a_bool ) > fun_a_fun_bool_bool ).

tff(sy_c_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Obool_J_000tc_,type,
    cOMBB_675860798_pname: ( fun_bo1549164019l_bool * fun_pname_bool ) > fun_pn250273176l_bool ).

tff(sy_c_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Obool_J_000tc__025,type,
    cOMBB_1652995168ol_int: ( fun_bo1549164019l_bool * fun_int_bool ) > fun_in531499254l_bool ).

tff(sy_c_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Obool_J_000tc__026,type,
    cOMBB_1015721476ol_nat: ( fun_bo1549164019l_bool * fun_nat_bool ) > fun_na1469252690l_bool ).

tff(sy_c_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Obool_J_000tc__027,type,
    cOMBB_338059395a_bool: ( fun_bo1549164019l_bool * fun_fun_a_bool_bool ) > fun_fu554186387l_bool ).

tff(sy_c_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Obool_J_000tc__028,type,
    cOMBB_2095475776e_bool: ( fun_bo1549164019l_bool * fun_fu1430349052l_bool ) > fun_fu31783638l_bool ).

tff(sy_c_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Obool_J_000tc__029,type,
    cOMBB_1974511970t_bool: ( fun_bo1549164019l_bool * fun_fu819804446l_bool ) > fun_fu989146164l_bool ).

tff(sy_c_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Obool_J_000tc__030,type,
    cOMBB_444170502t_bool: ( fun_bo1549164019l_bool * fun_fu425979586l_bool ) > fun_fu1016514960l_bool ).

tff(sy_c_COMBB_000tc__Int__Oint_000tc__HOL__Obool_000tc__Nat__Onat,type,
    cOMBB_int_bool_nat: ( fun_int_bool * fun_nat_int ) > fun_nat_bool ).

tff(sy_c_COMBB_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__N,type,
    cOMBB_1628441366ol_nat: ( fun_int_fun_int_bool * fun_nat_int ) > fun_nat_fun_int_bool ).

tff(sy_c_COMBB_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__N,type,
    cOMBB_800536526ol_nat: ( fun_nat_fun_nat_bool * fun_nat_nat ) > fun_nat_fun_nat_bool ).

tff(sy_c_COMBB_000tc__Nat__Onat_000tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J,type,
    cOMBB_1146692694ol_nat: ( fun_na1436237685l_bool * fun_nat_nat ) > fun_na1436237685l_bool ).

tff(sy_c_COMBB_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__HOL__Obool_000t__a,type,
    cOMBB_1273333599bool_a: ( fun_fun_a_bool_bool * fun_a_fun_a_bool ) > fun_a_bool ).

tff(sy_c_COMBB_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__fun_Itc__fun_It__a_Mtc__HOL,type,
    cOMBB_313729068ol_int: ( fun_fu1471507361l_bool * fun_int_fun_a_bool ) > fun_in1106722302l_bool ).

tff(sy_c_COMBB_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__fun_Itc__fun_It__a_Mtc__HOL_031,type,
    cOMBB_1823939024ol_nat: ( fun_fu1471507361l_bool * fun_nat_fun_a_bool ) > fun_na1632405922l_bool ).

tff(sy_c_COMBB_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_000tc__fun_Itc__fun_Itc,type,
    cOMBB_607112096ol_int: ( fun_fu802393907l_bool * fun_in2065004473e_bool ) > fun_in711834963l_bool ).

tff(sy_c_COMBB_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_000tc__fun_Itc__fun_Itc_032,type,
    cOMBB_2117322052ol_nat: ( fun_fu802393907l_bool * fun_na936072029e_bool ) > fun_na2122364079l_bool ).

tff(sy_c_COMBB_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__HOL__Obool_000tc__,type,
    cOMBB_1418110531ol_int: ( fun_fu819804446l_bool * fun_int_fun_int_bool ) > fun_int_bool ).

tff(sy_c_COMBB_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__fun_Itc__fun_Itc__,type,
    cOMBB_1761098400ol_int: ( fun_fu1151772339l_bool * fun_int_fun_int_bool ) > fun_in938817141l_bool ).

tff(sy_c_COMBB_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__fun_Itc__fun_Itc___033,type,
    cOMBB_1123824708ol_nat: ( fun_fu1151772339l_bool * fun_nat_fun_int_bool ) > fun_na1583160273l_bool ).

tff(sy_c_COMBB_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__HOL__Obool_000tc__,type,
    cOMBB_955900739ol_nat: ( fun_fu425979586l_bool * fun_nat_fun_nat_bool ) > fun_nat_bool ).

tff(sy_c_COMBB_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__fun_Itc__fun_Itc__,type,
    cOMBB_516767136ol_int: ( fun_fu1217155507l_bool * fun_int_fun_nat_bool ) > fun_in791894553l_bool ).

tff(sy_c_COMBB_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__fun_Itc__fun_Itc___034,type,
    cOMBB_2026977092ol_nat: ( fun_fu1217155507l_bool * fun_nat_fun_nat_bool ) > fun_na1436237685l_bool ).

tff(sy_c_COMBC_000t__a_000t__a_000tc__HOL__Obool,type,
    cOMBC_a_a_bool: fun_a_fun_a_bool > fun_a_fun_a_bool ).

tff(sy_c_COMBC_000t__a_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__HOL__Obool,type,
    cOMBC_1355376034l_bool: fun_a_1255737515l_bool > fun_fu1731003005a_bool ).

tff(sy_c_COMBC_000t__a_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__fun_It__a_Mtc__HOL_,type,
    cOMBC_394484100a_bool: fun_a_1752562919a_bool > fun_fu868269973a_bool ).

tff(sy_c_COMBC_000tc__Com__Opname_000t__a_000tc__HOL__Obool,type,
    cOMBC_pname_a_bool: fun_pname_fun_a_bool > fun_a_fun_pname_bool ).

tff(sy_c_COMBC_000tc__Com__Opname_000tc__Com__Opname_000tc__HOL__Obool,type,
    cOMBC_1149511130e_bool: fun_pn800050071e_bool > fun_pn800050071e_bool ).

tff(sy_c_COMBC_000tc__Com__Opname_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_000t,type,
    cOMBC_1058051404l_bool: fun_pn422929397l_bool > fun_fu410713561e_bool ).

tff(sy_c_COMBC_000tc__Int__Oint_000t__a_000tc__HOL__Obool,type,
    cOMBC_int_a_bool: fun_int_fun_a_bool > fun_a_fun_int_bool ).

tff(sy_c_COMBC_000tc__Int__Oint_000tc__Int__Oint_000tc__HOL__Obool,type,
    cOMBC_int_int_bool: fun_int_fun_int_bool > fun_int_fun_int_bool ).

tff(sy_c_COMBC_000tc__Int__Oint_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__HOL__Obool,type,
    cOMBC_1753430397l_bool: fun_in1106722302l_bool > fun_fu859530750t_bool ).

tff(sy_c_COMBC_000tc__Int__Oint_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_000tc_,type,
    cOMBC_316919022l_bool: fun_in711834963l_bool > fun_fu1159356667t_bool ).

tff(sy_c_COMBC_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__H,type,
    cOMBC_94739984l_bool: fun_in938817141l_bool > fun_fu554825757t_bool ).

tff(sy_c_COMBC_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__f,type,
    cOMBC_1683390479t_bool: fun_in719570932t_bool > fun_fu380257692t_bool ).

tff(sy_c_COMBC_000tc__Int__Oint_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__H,type,
    cOMBC_1848398772l_bool: fun_in791894553l_bool > fun_fu1080509377t_bool ).

tff(sy_c_COMBC_000tc__Nat__Onat_000tc__Int__Oint_000tc__HOL__Obool,type,
    cOMBC_nat_int_bool: fun_nat_fun_int_bool > fun_int_fun_nat_bool ).

tff(sy_c_COMBC_000tc__Nat__Onat_000tc__Nat__Onat_000tc__HOL__Obool,type,
    cOMBC_nat_nat_bool: fun_nat_fun_nat_bool > fun_nat_fun_nat_bool ).

tff(sy_c_COMBC_000tc__Nat__Onat_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__HOL__Obool,type,
    cOMBC_1928494297l_bool: fun_na1632405922l_bool > fun_fu600484770t_bool ).

tff(sy_c_COMBC_000tc__Nat__Onat_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_000tc_,type,
    cOMBC_386238098l_bool: fun_na2122364079l_bool > fun_fu900310687t_bool ).

tff(sy_c_COMBC_000tc__Nat__Onat_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__H,type,
    cOMBC_620423604l_bool: fun_na1583160273l_bool > fun_fu295779777t_bool ).

tff(sy_c_COMBC_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__H,type,
    cOMBC_226598744l_bool: fun_na1436237685l_bool > fun_fu821463397t_bool ).

tff(sy_c_COMBC_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__f,type,
    cOMBC_178881787t_bool: fun_na35521688t_bool > fun_fu429110920t_bool ).

tff(sy_c_COMBC_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__fun_It__a_Mtc__HOL__Obool_J,type,
    cOMBC_1732670874l_bool: fun_fu1471507361l_bool > fun_fu1471507361l_bool ).

tff(sy_c_COMBC_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_000tc__fun_Itc__Com__Op,type,
    cOMBC_1284144636l_bool: fun_fu802393907l_bool > fun_fu802393907l_bool ).

tff(sy_c_COMBC_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__fun_Itc__Int__Oint,type,
    cOMBC_605892544l_bool: fun_fu1151772339l_bool > fun_fu1151772339l_bool ).

tff(sy_c_COMBC_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__fun_Itc__Nat__Onat,type,
    cOMBC_1693257480l_bool: fun_fu1217155507l_bool > fun_fu1217155507l_bool ).

tff(sy_c_COMBC_000tc__fun_Itc__fun_It__a_Mtc__HOL__Obool_J_Mtc__HOL__Obool_J_000tc__,type,
    cOMBC_331553030l_bool: fun_fu418465139l_bool > fun_fu418465139l_bool ).

tff(sy_c_COMBC_000tc__fun_Itc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_Mtc__HOL__Oboo,type,
    cOMBC_7971162l_bool: fun_fu1911931399l_bool > fun_fu1911931399l_bool ).

tff(sy_c_COMBC_000tc__fun_Itc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_Mtc__HOL__Obool_,type,
    cOMBC_348868698l_bool: fun_fu209342027l_bool > fun_fu209342027l_bool ).

tff(sy_c_COMBC_000tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_Mtc__HOL__Obool_,type,
    cOMBC_595898202l_bool: fun_fu140186515l_bool > fun_fu140186515l_bool ).

tff(sy_c_COMBI_000tc__Nat__Onat,type,
    cOMBI_nat: fun_nat_nat ).

tff(sy_c_COMBK_000t__a_000tc__Com__Opname,type,
    cOMBK_a_pname: x_a > fun_pname_a ).

tff(sy_c_COMBK_000tc__HOL__Obool_000t__a,type,
    cOMBK_bool_a: bool > fun_a_bool ).

tff(sy_c_COMBK_000tc__HOL__Obool_000tc__Int__Oint,type,
    cOMBK_bool_int: bool > fun_int_bool ).

tff(sy_c_COMBK_000tc__HOL__Obool_000tc__Nat__Onat,type,
    cOMBK_bool_nat: bool > fun_nat_bool ).

tff(sy_c_COMBK_000tc__Int__Oint_000tc__Nat__Onat,type,
    cOMBK_int_nat: int > fun_nat_int ).

tff(sy_c_COMBS_000t__a_000tc__HOL__Obool_000tc__HOL__Obool,type,
    cOMBS_a_bool_bool: ( fun_a_fun_bool_bool * fun_a_bool ) > fun_a_bool ).

tff(sy_c_COMBS_000tc__Com__Opname_000tc__HOL__Obool_000tc__HOL__Obool,type,
    cOMBS_568398431l_bool: ( fun_pn250273176l_bool * fun_pname_bool ) > fun_pname_bool ).

tff(sy_c_COMBS_000tc__Int__Oint_000tc__HOL__Obool_000tc__HOL__Obool,type,
    cOMBS_int_bool_bool: ( fun_in531499254l_bool * fun_int_bool ) > fun_int_bool ).

tff(sy_c_COMBS_000tc__Nat__Onat_000tc__HOL__Obool_000tc__HOL__Obool,type,
    cOMBS_nat_bool_bool: ( fun_na1469252690l_bool * fun_nat_bool ) > fun_nat_bool ).

tff(sy_c_COMBS_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__HOL__Obool_000tc__HOL__Oboo,type,
    cOMBS_1035972772l_bool: ( fun_fu554186387l_bool * fun_fun_a_bool_bool ) > fun_fun_a_bool_bool ).

tff(sy_c_COMBS_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_000tc__HOL__Obool_000tc,type,
    cOMBS_350070575l_bool: ( fun_fu31783638l_bool * fun_fu1430349052l_bool ) > fun_fu1430349052l_bool ).

tff(sy_c_COMBS_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__HOL__Obool_000tc__,type,
    cOMBS_1912792529l_bool: ( fun_fu989146164l_bool * fun_fu819804446l_bool ) > fun_fu819804446l_bool ).

tff(sy_c_COMBS_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__HOL__Obool_000tc__,type,
    cOMBS_1187019125l_bool: ( fun_fu1016514960l_bool * fun_fu425979586l_bool ) > fun_fu425979586l_bool ).

tff(sy_c_Com_Obody,type,
    body: pname > option_com ).

tff(sy_c_Finite__Set_Ocard_000t__a,type,
    finite_card_a: fun_fun_a_bool_nat ).

tff(sy_c_Finite__Set_Ocard_000tc__Com__Opname,type,
    finite_card_pname: fun_fu1668467777ol_nat ).

tff(sy_c_Finite__Set_Ocard_000tc__Int__Oint,type,
    finite_card_int: fun_fun_int_bool_nat ).

tff(sy_c_Finite__Set_Ocard_000tc__Nat__Onat,type,
    finite_card_nat: fun_fun_nat_bool_nat ).

tff(sy_c_Finite__Set_Ocard_000tc__fun_It__a_Mtc__HOL__Obool_J,type,
    finite1306199131a_bool: fun_fu2020802748ol_nat ).

tff(sy_c_Finite__Set_Ocard_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J,type,
    finite1340463720e_bool: fun_fu1730389579ol_nat ).

tff(sy_c_Finite__Set_Ocard_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J,type,
    finite1876863882t_bool: fun_fu236049833ol_nat ).

tff(sy_c_Finite__Set_Ocard_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J,type,
    finite346522414t_bool: fun_fu411113733ol_nat ).

tff(sy_c_Finite__Set_Ofinite_000t__a,type,
    finite_finite_a: fun_fun_a_bool_bool ).

tff(sy_c_Finite__Set_Ofinite_000tc__Com__Opname,type,
    finite_finite_pname: fun_fu1430349052l_bool ).

tff(sy_c_Finite__Set_Ofinite_000tc__Int__Oint,type,
    finite_finite_int: fun_fu819804446l_bool ).

tff(sy_c_Finite__Set_Ofinite_000tc__Nat__Onat,type,
    finite_finite_nat: fun_fu425979586l_bool ).

tff(sy_c_Finite__Set_Ofinite_000tc__fun_It__a_Mtc__HOL__Obool_J,type,
    finite347923420a_bool: fun_fu885608257l_bool ).

tff(sy_c_Finite__Set_Ofinite_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J,type,
    finite595471783e_bool: fun_fu399576434l_bool ).

tff(sy_c_Finite__Set_Ofinite_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J,type,
    finite1395289673t_bool: fun_fu1540142484l_bool ).

tff(sy_c_Finite__Set_Ofinite_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J,type,
    finite2012431853t_bool: fun_fu814369080l_bool ).

tff(sy_c_Finite__Set_Ofinite_000tc__fun_Itc__fun_It__a_Mtc__HOL__Obool_J_Mtc__HOL__O,type,
    finite1381704300l_bool: fun_fu255076663l_bool ).

tff(sy_c_Finite__Set_Ofinite_000tc__fun_Itc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_,type,
    finite786885583l_bool: fun_fu1438281908l_bool ).

tff(sy_c_Finite__Set_Ofinite_000tc__fun_Itc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_Mt,type,
    finite2095298929l_bool: fun_fu717296342l_bool ).

tff(sy_c_Finite__Set_Ofinite_000tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_Mt,type,
    finite1701474069l_bool: fun_fu61768826l_bool ).

tff(sy_c_Finite__Set_Ofolding__one_000t__a,type,
    finite_folding_one_a: ( fun_a_fun_a_a * fun_fun_a_bool_a ) > bool ).

tff(sy_c_Finite__Set_Ofolding__one_000tc__Com__Opname,type,
    finite1282449217_pname: ( fun_pn1208287926_pname * fun_fu1499449723_pname ) > bool ).

tff(sy_c_Finite__Set_Ofolding__one_000tc__Int__Oint,type,
    finite1626084323ne_int: ( fun_int_fun_int_int * fun_fun_int_bool_int ) > bool ).

tff(sy_c_Finite__Set_Ofolding__one_000tc__Nat__Onat,type,
    finite988810631ne_nat: ( fun_nat_fun_nat_nat * fun_fun_nat_bool_nat ) > bool ).

tff(sy_c_Finite__Set_Ofolding__one__idem_000t__a,type,
    finite1819937229idem_a: ( fun_a_fun_a_a * fun_fun_a_bool_a ) > bool ).

tff(sy_c_Finite__Set_Ofolding__one__idem_000tc__Com__Opname,type,
    finite89670078_pname: ( fun_pn1208287926_pname * fun_fu1499449723_pname ) > bool ).

tff(sy_c_Finite__Set_Ofolding__one__idem_000tc__Int__Oint,type,
    finite1432773856em_int: ( fun_int_fun_int_int * fun_fun_int_bool_int ) > bool ).

tff(sy_c_Finite__Set_Ofolding__one__idem_000tc__Nat__Onat,type,
    finite795500164em_nat: ( fun_nat_fun_nat_nat * fun_fun_nat_bool_nat ) > bool ).

tff(sy_c_Groups_Oabs__class_Oabs_000tc__Int__Oint,type,
    abs_abs_int: int > int ).

tff(sy_c_Groups_Ominus__class_Ominus_000tc__Int__Oint,type,
    minus_minus_int: fun_int_fun_int_int ).

tff(sy_c_Groups_Ominus__class_Ominus_000tc__Nat__Onat,type,
    minus_minus_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Groups_Ominus__class_Ominus_000tc__fun_It__a_Mtc__HOL__Obool_J,type,
    minus_1762468890a_bool: fun_fu1663643505a_bool ).

tff(sy_c_Groups_Ominus__class_Ominus_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J,type,
    minus_1015773161e_bool: fun_fu2101097872e_bool ).

tff(sy_c_Groups_Ominus__class_Ominus_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J,type,
    minus_1449998731t_bool: fun_fu1075512626t_bool ).

tff(sy_c_Groups_Ominus__class_Ominus_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J,type,
    minus_2067140911t_bool: fun_fu915923926t_bool ).

tff(sy_c_Groups_Oone__class_Oone_000tc__Int__Oint,type,
    one_one_int: int ).

tff(sy_c_Groups_Oone__class_Oone_000tc__Nat__Onat,type,
    one_one_nat: nat ).

tff(sy_c_Groups_Oplus__class_Oplus_000tc__Int__Oint,type,
    plus_plus_int: fun_int_fun_int_int ).

tff(sy_c_Groups_Oplus__class_Oplus_000tc__Nat__Onat,type,
    plus_plus_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Groups_Otimes__class_Otimes_000tc__Int__Oint,type,
    times_times_int: fun_int_fun_int_int ).

tff(sy_c_Groups_Otimes__class_Otimes_000tc__Nat__Onat,type,
    times_times_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Groups_Ozero__class_Ozero_000tc__Int__Oint,type,
    zero_zero_int: int ).

tff(sy_c_Groups_Ozero__class_Ozero_000tc__Nat__Onat,type,
    zero_zero_nat: nat ).

tff(sy_c_HOL_OThe_000t__a,type,
    the_a: fun_fun_a_bool_a ).

tff(sy_c_HOL_OThe_000tc__Int__Oint,type,
    the_int: fun_fun_int_bool_int ).

tff(sy_c_HOL_OThe_000tc__Nat__Onat,type,
    the_nat: fun_fun_nat_bool_nat ).

tff(sy_c_If_000tc__Nat__Onat,type,
    if_nat: bool > fun_nat_fun_nat_nat ).

tff(sy_c_Int_OBit1,type,
    bit1: int > int ).

tff(sy_c_Int_OPls,type,
    pls: int ).

tff(sy_c_Int_Onumber__class_Onumber__of_000tc__Int__Oint,type,
    number_number_of_int: int > int ).

tff(sy_c_Int_Onumber__class_Onumber__of_000tc__Nat__Onat,type,
    number_number_of_nat: int > nat ).

tff(sy_c_Int_Osucc,type,
    succ: int > int ).

tff(sy_c_Nat_OSuc,type,
    suc: fun_nat_nat ).

tff(sy_c_Nat_Onat_Onat__case_000tc__HOL__Obool,type,
    nat_case_bool: ( bool * fun_nat_bool ) > fun_nat_bool ).

tff(sy_c_Nat_Onat_Onat__case_000tc__Nat__Onat,type,
    nat_case_nat: ( nat * fun_nat_nat ) > fun_nat_nat ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_000tc__Int__Oint,type,
    semiri1621563631at_int: fun_nat_int ).

tff(sy_c_Nat__Numeral_Oneg,type,
    nat_neg: fun_int_bool ).

tff(sy_c_Nat__Transfer_Otsub,type,
    nat_tsub: fun_int_fun_int_int ).

tff(sy_c_Option_Othe_000tc__Com__Ocom,type,
    the_com: option_com > com ).

tff(sy_c_Orderings_Obot__class_Obot_000tc__HOL__Obool,type,
    bot_bot_bool: bool ).

tff(sy_c_Orderings_Obot__class_Obot_000tc__Nat__Onat,type,
    bot_bot_nat: nat ).

tff(sy_c_Orderings_Obot__class_Obot_000tc__fun_It__a_Mtc__HOL__Obool_J,type,
    bot_bot_fun_a_bool: fun_a_bool ).

tff(sy_c_Orderings_Obot__class_Obot_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J,type,
    bot_bo844097828e_bool: fun_pname_bool ).

tff(sy_c_Orderings_Obot__class_Obot_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J,type,
    bot_bot_fun_int_bool: fun_int_bool ).

tff(sy_c_Orderings_Obot__class_Obot_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J,type,
    bot_bot_fun_nat_bool: fun_nat_bool ).

tff(sy_c_Orderings_Oord__class_Oless_000tc__Int__Oint,type,
    ord_less_int: fun_int_fun_int_bool ).

tff(sy_c_Orderings_Oord__class_Oless_000tc__Nat__Onat,type,
    ord_less_nat: fun_nat_fun_nat_bool ).

tff(sy_c_Orderings_Oord__class_Oless_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J,type,
    ord_le1912455174t_bool: fun_fu1151772339l_bool ).

tff(sy_c_Orderings_Oord__class_Oless_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J,type,
    ord_le382113706t_bool: fun_fu1217155507l_bool ).

tff(sy_c_Orderings_Oord__class_Oless__eq_000tc__HOL__Obool,type,
    ord_less_eq_bool: fun_bo1549164019l_bool ).

tff(sy_c_Orderings_Oord__class_Oless__eq_000tc__Int__Oint,type,
    ord_less_eq_int: fun_int_fun_int_bool ).

tff(sy_c_Orderings_Oord__class_Oless__eq_000tc__Nat__Onat,type,
    ord_less_eq_nat: fun_nat_fun_nat_bool ).

tff(sy_c_Orderings_Oord__class_Oless__eq_000tc__fun_It__a_Mtc__HOL__Obool_J,type,
    ord_le1311769555a_bool: fun_fu1471507361l_bool ).

tff(sy_c_Orderings_Oord__class_Oless__eq_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool,type,
    ord_le313189616e_bool: fun_fu802393907l_bool ).

tff(sy_c_Orderings_Oord__class_Oless__eq_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J,type,
    ord_le951220754t_bool: fun_fu1151772339l_bool ).

tff(sy_c_Orderings_Oord__class_Oless__eq_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J,type,
    ord_le1568362934t_bool: fun_fu1217155507l_bool ).

tff(sy_c_Orderings_Oord__class_Oless__eq_000tc__fun_Itc__fun_It__a_Mtc__HOL__Obool_J,type,
    ord_le1375614389l_bool: fun_fu418465139l_bool ).

tff(sy_c_Orderings_Oord__class_Oless__eq_000tc__fun_Itc__fun_Itc__Com__Opname_Mtc__H,type,
    ord_le675606854l_bool: fun_fu1911931399l_bool ).

tff(sy_c_Orderings_Oord__class_Oless__eq_000tc__fun_Itc__fun_Itc__Int__Oint_Mtc__HOL,type,
    ord_le1848167016l_bool: fun_fu209342027l_bool ).

tff(sy_c_Orderings_Oord__class_Oless__eq_000tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HOL,type,
    ord_le1454342156l_bool: fun_fu140186515l_bool ).

tff(sy_c_Set_OCollect_000t__a,type,
    collect_a: fun_a_bool > fun_a_bool ).

tff(sy_c_Set_OCollect_000tc__Com__Opname,type,
    collect_pname: fun_pname_bool > fun_pname_bool ).

tff(sy_c_Set_OCollect_000tc__Int__Oint,type,
    collect_int: fun_int_bool > fun_int_bool ).

tff(sy_c_Set_OCollect_000tc__Nat__Onat,type,
    collect_nat: fun_nat_bool > fun_nat_bool ).

tff(sy_c_Set_OCollect_000tc__fun_It__a_Mtc__HOL__Obool_J,type,
    collect_fun_a_bool: fun_fun_a_bool_bool > fun_fun_a_bool_bool ).

tff(sy_c_Set_OCollect_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J,type,
    collec1974731493e_bool: fun_fu1430349052l_bool > fun_fu1430349052l_bool ).

tff(sy_c_Set_OCollect_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J,type,
    collect_fun_int_bool: fun_fu819804446l_bool > fun_fu819804446l_bool ).

tff(sy_c_Set_OCollect_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J,type,
    collect_fun_nat_bool: fun_fu425979586l_bool > fun_fu425979586l_bool ).

tff(sy_c_Set_OCollect_000tc__fun_Itc__fun_It__a_Mtc__HOL__Obool_J_Mtc__HOL__Obool_J,type,
    collec707592106l_bool: fun_fu885608257l_bool > fun_fu885608257l_bool ).

tff(sy_c_Set_OCollect_000tc__fun_Itc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_Mtc__HO,type,
    collec1613912337l_bool: fun_fu399576434l_bool > fun_fu399576434l_bool ).

tff(sy_c_Set_OCollect_000tc__fun_Itc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_Mtc__HOL_,type,
    collec1409689523l_bool: fun_fu1540142484l_bool > fun_fu1540142484l_bool ).

tff(sy_c_Set_OCollect_000tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_Mtc__HOL_,type,
    collec1015864663l_bool: fun_fu814369080l_bool > fun_fu814369080l_bool ).

tff(sy_c_Set_Oimage_000t__a_000t__a,type,
    image_a_a: ( fun_a_a * fun_a_bool ) > fun_a_bool ).

tff(sy_c_Set_Oimage_000t__a_000tc__Com__Opname,type,
    image_a_pname: ( fun_a_pname * fun_a_bool ) > fun_pname_bool ).

tff(sy_c_Set_Oimage_000t__a_000tc__Int__Oint,type,
    image_a_int: ( fun_a_int * fun_a_bool ) > fun_int_bool ).

tff(sy_c_Set_Oimage_000t__a_000tc__Nat__Onat,type,
    image_a_nat: ( fun_a_nat * fun_a_bool ) > fun_nat_bool ).

tff(sy_c_Set_Oimage_000t__a_000tc__fun_It__a_Mtc__HOL__Obool_J,type,
    image_a_fun_a_bool: ( fun_a_fun_a_bool * fun_a_bool ) > fun_fun_a_bool_bool ).

tff(sy_c_Set_Oimage_000t__a_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J,type,
    image_819518260e_bool: ( fun_a_fun_pname_bool * fun_a_bool ) > fun_fu1430349052l_bool ).

tff(sy_c_Set_Oimage_000t__a_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J,type,
    image_a_fun_int_bool: ( fun_a_fun_int_bool * fun_a_bool ) > fun_fu819804446l_bool ).

tff(sy_c_Set_Oimage_000t__a_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J,type,
    image_a_fun_nat_bool: ( fun_a_fun_nat_bool * fun_a_bool ) > fun_fu425979586l_bool ).

tff(sy_c_Set_Oimage_000tc__Com__Opname_000t__a,type,
    image_pname_a: ( fun_pname_a * fun_pname_bool ) > fun_a_bool ).

tff(sy_c_Set_Oimage_000tc__Com__Opname_000tc__Com__Opname,type,
    image_pname_pname: ( fun_pname_pname * fun_pname_bool ) > fun_pname_bool ).

tff(sy_c_Set_Oimage_000tc__Com__Opname_000tc__Int__Oint,type,
    image_pname_int: ( fun_pname_int * fun_pname_bool ) > fun_int_bool ).

tff(sy_c_Set_Oimage_000tc__Com__Opname_000tc__Nat__Onat,type,
    image_pname_nat: ( fun_pname_nat * fun_pname_bool ) > fun_nat_bool ).

tff(sy_c_Set_Oimage_000tc__Com__Opname_000tc__fun_It__a_Mtc__HOL__Obool_J,type,
    image_112932426a_bool: ( fun_pname_fun_a_bool * fun_pname_bool ) > fun_fun_a_bool_bool ).

tff(sy_c_Set_Oimage_000tc__Com__Opname_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J,type,
    image_47868345e_bool: ( fun_pn800050071e_bool * fun_pname_bool ) > fun_fu1430349052l_bool ).

tff(sy_c_Set_Oimage_000tc__Com__Opname_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J,type,
    image_1512837979t_bool: ( fun_pn665169337t_bool * fun_pname_bool ) > fun_fu819804446l_bool ).

tff(sy_c_Set_Oimage_000tc__Com__Opname_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J,type,
    image_2129980159t_bool: ( fun_pn406123357t_bool * fun_pname_bool ) > fun_fu425979586l_bool ).

tff(sy_c_Set_Oimage_000tc__Int__Oint_000t__a,type,
    image_int_a: ( fun_int_a * fun_int_bool ) > fun_a_bool ).

tff(sy_c_Set_Oimage_000tc__Int__Oint_000tc__Com__Opname,type,
    image_int_pname: ( fun_int_pname * fun_int_bool ) > fun_pname_bool ).

tff(sy_c_Set_Oimage_000tc__Int__Oint_000tc__fun_It__a_Mtc__HOL__Obool_J,type,
    image_int_fun_a_bool: ( fun_int_fun_a_bool * fun_int_bool ) > fun_fun_a_bool_bool ).

tff(sy_c_Set_Oimage_000tc__Int__Oint_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J,type,
    image_637364955e_bool: ( fun_in2065004473e_bool * fun_int_bool ) > fun_fu1430349052l_bool ).

tff(sy_c_Set_Oimage_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J,type,
    image_275383677t_bool: ( fun_int_fun_int_bool * fun_int_bool ) > fun_fu819804446l_bool ).

tff(sy_c_Set_Oimage_000tc__Int__Oint_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J,type,
    image_892525857t_bool: ( fun_int_fun_nat_bool * fun_int_bool ) > fun_fu425979586l_bool ).

tff(sy_c_Set_Oimage_000tc__Nat__Onat_000t__a,type,
    image_nat_a: ( fun_nat_a * fun_nat_bool ) > fun_a_bool ).

tff(sy_c_Set_Oimage_000tc__Nat__Onat_000tc__Com__Opname,type,
    image_nat_pname: ( fun_nat_pname * fun_nat_bool ) > fun_pname_bool ).

tff(sy_c_Set_Oimage_000tc__Nat__Onat_000tc__Int__Oint,type,
    image_nat_int: ( fun_nat_int * fun_nat_bool ) > fun_int_bool ).

tff(sy_c_Set_Oimage_000tc__Nat__Onat_000tc__fun_It__a_Mtc__HOL__Obool_J,type,
    image_nat_fun_a_bool: ( fun_nat_fun_a_bool * fun_nat_bool ) > fun_fun_a_bool_bool ).

tff(sy_c_Set_Oimage_000tc__Nat__Onat_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J,type,
    image_1655916159e_bool: ( fun_na936072029e_bool * fun_nat_bool ) > fun_fu1430349052l_bool ).

tff(sy_c_Set_Oimage_000tc__Nat__Onat_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J,type,
    image_1556378401t_bool: ( fun_nat_fun_int_bool * fun_nat_bool ) > fun_fu819804446l_bool ).

tff(sy_c_Set_Oimage_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J,type,
    image_26036933t_bool: ( fun_nat_fun_nat_bool * fun_nat_bool ) > fun_fu425979586l_bool ).

tff(sy_c_Set_Oimage_000tc__fun_It__a_Mtc__HOL__Obool_J_000t__a,type,
    image_fun_a_bool_a: ( fun_fun_a_bool_a * fun_fun_a_bool_bool ) > fun_a_bool ).

tff(sy_c_Set_Oimage_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__Com__Opname,type,
    image_1854862208_pname: ( fun_fun_a_bool_pname * fun_fun_a_bool_bool ) > fun_pname_bool ).

tff(sy_c_Set_Oimage_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__Int__Oint,type,
    image_fun_a_bool_int: ( fun_fun_a_bool_int * fun_fun_a_bool_bool ) > fun_int_bool ).

tff(sy_c_Set_Oimage_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__Nat__Onat,type,
    image_fun_a_bool_nat: ( fun_fun_a_bool_nat * fun_fun_a_bool_bool ) > fun_nat_bool ).

tff(sy_c_Set_Oimage_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_000t__a,type,
    image_876012084bool_a: ( fun_fun_pname_bool_a * fun_fu1430349052l_bool ) > fun_a_bool ).

tff(sy_c_Set_Oimage_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_000tc__Com__Opname,type,
    image_1283814551_pname: ( fun_fu1499449723_pname * fun_fu1430349052l_bool ) > fun_pname_bool ).

tff(sy_c_Set_Oimage_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_000tc__Int__Oint,type,
    image_41399353ol_int: ( fun_fu1830380701ol_int * fun_fu1430349052l_bool ) > fun_int_bool ).

tff(sy_c_Set_Oimage_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_000tc__Nat__Onat,type,
    image_1551609309ol_nat: ( fun_fu1668467777ol_nat * fun_fu1430349052l_bool ) > fun_nat_bool ).

tff(sy_c_Set_Oimage_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000t__a,type,
    image_fun_int_bool_a: ( fun_fun_int_bool_a * fun_fu819804446l_bool ) > fun_a_bool ).

tff(sy_c_Set_Oimage_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__Com__Opname,type,
    image_62782965_pname: ( fun_fu953875929_pname * fun_fu819804446l_bool ) > fun_pname_bool ).

tff(sy_c_Set_Oimage_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__Int__Oint,type,
    image_2069007511ol_int: ( fun_fun_int_bool_int * fun_fu819804446l_bool ) > fun_int_bool ).

tff(sy_c_Set_Oimage_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__Nat__Onat,type,
    image_1431733819ol_nat: ( fun_fun_int_bool_nat * fun_fu819804446l_bool ) > fun_nat_bool ).

tff(sy_c_Set_Oimage_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000t__a,type,
    image_fun_nat_bool_a: ( fun_fun_nat_bool_a * fun_fu425979586l_bool ) > fun_a_bool ).

tff(sy_c_Set_Oimage_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__Com__Opname,type,
    image_1921560913_pname: ( fun_fu665170229_pname * fun_fu425979586l_bool ) > fun_pname_bool ).

tff(sy_c_Set_Oimage_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__Int__Oint,type,
    image_1133522419ol_int: ( fun_fun_nat_bool_int * fun_fu425979586l_bool ) > fun_int_bool ).

tff(sy_c_Set_Oimage_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__Nat__Onat,type,
    image_496248727ol_nat: ( fun_fun_nat_bool_nat * fun_fu425979586l_bool ) > fun_nat_bool ).

tff(sy_c_Set_Oinsert_000t__a,type,
    insert_a: fun_a_1752562919a_bool ).

tff(sy_c_Set_Oinsert_000tc__Com__Opname,type,
    insert_pname: fun_pn479984722e_bool ).

tff(sy_c_Set_Oinsert_000tc__Int__Oint,type,
    insert_int: fun_in719570932t_bool ).

tff(sy_c_Set_Oinsert_000tc__Nat__Onat,type,
    insert_nat: fun_na35521688t_bool ).

tff(sy_c_Set_Oinsert_000tc__fun_It__a_Mtc__HOL__Obool_J,type,
    insert_fun_a_bool: fun_fu1434634199l_bool ).

tff(sy_c_Set_Oinsert_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J,type,
    insert1325755072e_bool: fun_fu1232732404l_bool ).

tff(sy_c_Set_Oinsert_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J,type,
    insert_fun_int_bool: fun_fu587680722l_bool ).

tff(sy_c_Set_Oinsert_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J,type,
    insert_fun_nat_bool: fun_fu1516990510l_bool ).

tff(sy_c_Set_Othe__elem_000t__a,type,
    the_elem_a: fun_fun_a_bool_a ).

tff(sy_c_Set_Othe__elem_000tc__Int__Oint,type,
    the_elem_int: fun_fun_int_bool_int ).

tff(sy_c_Set_Othe__elem_000tc__Nat__Onat,type,
    the_elem_nat: fun_fun_nat_bool_nat ).

tff(sy_c_fFalse,type,
    fFalse: bool ).

tff(sy_c_fNot,type,
    fNot: fun_bool_bool ).

tff(sy_c_fTrue,type,
    fTrue: bool ).

tff(sy_c_fconj,type,
    fconj: fun_bo1549164019l_bool ).

tff(sy_c_fdisj,type,
    fdisj: fun_bo1549164019l_bool ).

tff(sy_c_fequal_000t__a,type,
    fequal_a: fun_a_fun_a_bool ).

tff(sy_c_fequal_000tc__Com__Opname,type,
    fequal_pname: fun_pn800050071e_bool ).

tff(sy_c_fequal_000tc__Int__Oint,type,
    fequal_int: fun_int_fun_int_bool ).

tff(sy_c_fequal_000tc__Nat__Onat,type,
    fequal_nat: fun_nat_fun_nat_bool ).

tff(sy_c_fequal_000tc__fun_It__a_Mtc__HOL__Obool_J,type,
    fequal_fun_a_bool: fun_fu1471507361l_bool ).

tff(sy_c_fequal_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J,type,
    fequal533582459e_bool: fun_fu802393907l_bool ).

tff(sy_c_fequal_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J,type,
    fequal_fun_int_bool: fun_fu1151772339l_bool ).

tff(sy_c_fequal_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J,type,
    fequal_fun_nat_bool: fun_fu1217155507l_bool ).

tff(sy_c_fimplies,type,
    fimplies: fun_bo1549164019l_bool ).

tff(sy_c_hAPP_000t__a_000t__a,type,
    hAPP_a_a: ( fun_a_a * x_a ) > x_a ).

tff(sy_c_hAPP_000t__a_000tc__HOL__Obool,type,
    hAPP_a_bool: ( fun_a_bool * x_a ) > bool ).

tff(sy_c_hAPP_000t__a_000tc__fun_It__a_Mt__a_J,type,
    hAPP_a_fun_a_a: ( fun_a_fun_a_a * x_a ) > fun_a_a ).

tff(sy_c_hAPP_000t__a_000tc__fun_It__a_Mtc__HOL__Obool_J,type,
    hAPP_a_fun_a_bool: ( fun_a_fun_a_bool * x_a ) > fun_a_bool ).

tff(sy_c_hAPP_000t__a_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J,type,
    hAPP_a93125764e_bool: ( fun_a_fun_pname_bool * x_a ) > fun_pname_bool ).

tff(sy_c_hAPP_000t__a_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Obool_J,type,
    hAPP_a_fun_bool_bool: ( fun_a_fun_bool_bool * x_a ) > fun_bool_bool ).

tff(sy_c_hAPP_000t__a_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J,type,
    hAPP_a_fun_int_bool: ( fun_a_fun_int_bool * x_a ) > fun_int_bool ).

tff(sy_c_hAPP_000t__a_000tc__fun_Itc__fun_It__a_Mtc__HOL__Obool_J_Mtc__HOL__Obool_J,type,
    hAPP_a85458249l_bool: ( fun_a_1255737515l_bool * x_a ) > fun_fun_a_bool_bool ).

tff(sy_c_hAPP_000t__a_000tc__fun_Itc__fun_It__a_Mtc__HOL__Obool_J_Mtc__fun_It__a_Mtc,type,
    hAPP_a1206381875a_bool: ( fun_a_1752562919a_bool * x_a ) > fun_fu1731003005a_bool ).

tff(sy_c_hAPP_000tc__Com__Opname_000t__a,type,
    hAPP_pname_a: ( fun_pname_a * pname ) > x_a ).

tff(sy_c_hAPP_000tc__Com__Opname_000tc__Com__Opname,type,
    hAPP_pname_pname: ( fun_pname_pname * pname ) > pname ).

tff(sy_c_hAPP_000tc__Com__Opname_000tc__HOL__Obool,type,
    hAPP_pname_bool: ( fun_pname_bool * pname ) > bool ).

tff(sy_c_hAPP_000tc__Com__Opname_000tc__fun_It__a_Mtc__HOL__Obool_J,type,
    hAPP_p1534023578a_bool: ( fun_pname_fun_a_bool * pname ) > fun_a_bool ).

tff(sy_c_hAPP_000tc__Com__Opname_000tc__fun_Itc__Com__Opname_Mtc__Com__Opname_J,type,
    hAPP_p1170154830_pname: ( fun_pn1208287926_pname * pname ) > fun_pname_pname ).

tff(sy_c_hAPP_000tc__Com__Opname_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J,type,
    hAPP_p61793385e_bool: ( fun_pn800050071e_bool * pname ) > fun_pname_bool ).

tff(sy_c_hAPP_000tc__Com__Opname_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Obool_J,type,
    hAPP_p393069232l_bool: ( fun_pn250273176l_bool * pname ) > fun_bool_bool ).

tff(sy_c_hAPP_000tc__Com__Opname_000tc__fun_Itc__fun_Itc__Com__Opname_Mtc__HOL__Oboo,type,
    hAPP_p338031245l_bool: ( fun_pn422929397l_bool * pname ) > fun_fu1430349052l_bool ).

tff(sy_c_hAPP_000tc__Com__Opname_000tc__fun_Itc__fun_Itc__Com__Opname_Mtc__HOL__Oboo_035,type,
    hAPP_p905327722e_bool: ( fun_pn479984722e_bool * pname ) > fun_fu410713561e_bool ).

tff(sy_c_hAPP_000tc__HOL__Obool_000tc__HOL__Obool,type,
    hAPP_bool_bool: ( fun_bool_bool * bool ) > bool ).

tff(sy_c_hAPP_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Obool_J,type,
    hAPP_b589554111l_bool: ( fun_bo1549164019l_bool * bool ) > fun_bool_bool ).

tff(sy_c_hAPP_000tc__Int__Oint_000t__a,type,
    hAPP_int_a: ( fun_int_a * int ) > x_a ).

tff(sy_c_hAPP_000tc__Int__Oint_000tc__HOL__Obool,type,
    hAPP_int_bool: ( fun_int_bool * int ) > bool ).

tff(sy_c_hAPP_000tc__Int__Oint_000tc__Int__Oint,type,
    hAPP_int_int: ( fun_int_int * int ) > int ).

tff(sy_c_hAPP_000tc__Int__Oint_000tc__fun_It__a_Mtc__HOL__Obool_J,type,
    hAPP_int_fun_a_bool: ( fun_int_fun_a_bool * int ) > fun_a_bool ).

tff(sy_c_hAPP_000tc__Int__Oint_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J,type,
    hAPP_i7355787e_bool: ( fun_in2065004473e_bool * int ) > fun_pname_bool ).

tff(sy_c_hAPP_000tc__Int__Oint_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Obool_J,type,
    hAPP_i68813070l_bool: ( fun_in531499254l_bool * int ) > fun_bool_bool ).

tff(sy_c_hAPP_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J,type,
    hAPP_i1948725293t_bool: ( fun_int_fun_int_bool * int ) > fun_int_bool ).

tff(sy_c_hAPP_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__Int__Oint_J,type,
    hAPP_int_fun_int_int: ( fun_int_fun_int_int * int ) > fun_int_int ).

tff(sy_c_hAPP_000tc__Int__Oint_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J,type,
    hAPP_i418383825t_bool: ( fun_int_fun_nat_bool * int ) > fun_nat_bool ).

tff(sy_c_hAPP_000tc__Int__Oint_000tc__fun_Itc__fun_It__a_Mtc__HOL__Obool_J_Mtc__HOL_,type,
    hAPP_i888906320l_bool: ( fun_in1106722302l_bool * int ) > fun_fun_a_bool_bool ).

tff(sy_c_hAPP_000tc__Int__Oint_000tc__fun_Itc__fun_Itc__Com__Opname_Mtc__HOL__Obool_,type,
    hAPP_i1587699435l_bool: ( fun_in711834963l_bool * int ) > fun_fu1430349052l_bool ).

tff(sy_c_hAPP_000tc__Int__Oint_000tc__fun_Itc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_,type,
    hAPP_i2112223885l_bool: ( fun_in938817141l_bool * int ) > fun_fu819804446l_bool ).

tff(sy_c_hAPP_000tc__Int__Oint_000tc__fun_Itc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J__036,type,
    hAPP_i1529485324t_bool: ( fun_in719570932t_bool * int ) > fun_fu554825757t_bool ).

tff(sy_c_hAPP_000tc__Int__Oint_000tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_,type,
    hAPP_i1718399025l_bool: ( fun_in791894553l_bool * int ) > fun_fu425979586l_bool ).

tff(sy_c_hAPP_000tc__Nat__Onat_000tc__HOL__Obool,type,
    hAPP_nat_bool: ( fun_nat_bool * nat ) > bool ).

tff(sy_c_hAPP_000tc__Nat__Onat_000tc__Int__Oint,type,
    hAPP_nat_int: ( fun_nat_int * nat ) > int ).

tff(sy_c_hAPP_000tc__Nat__Onat_000tc__Nat__Onat,type,
    hAPP_nat_nat: ( fun_nat_nat * nat ) > nat ).

tff(sy_c_hAPP_000tc__Nat__Onat_000tc__fun_It__a_Mtc__HOL__Obool_J,type,
    hAPP_nat_fun_a_bool: ( fun_nat_fun_a_bool * nat ) > fun_a_bool ).

tff(sy_c_hAPP_000tc__Nat__Onat_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J,type,
    hAPP_n1025906991e_bool: ( fun_na936072029e_bool * nat ) > fun_pname_bool ).

tff(sy_c_hAPP_000tc__Nat__Onat_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Obool_J,type,
    hAPP_n1006566506l_bool: ( fun_na1469252690l_bool * nat ) > fun_bool_bool ).

tff(sy_c_hAPP_000tc__Nat__Onat_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J,type,
    hAPP_n1082236369t_bool: ( fun_nat_fun_int_bool * nat ) > fun_int_bool ).

tff(sy_c_hAPP_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J,type,
    hAPP_n1699378549t_bool: ( fun_nat_fun_nat_bool * nat ) > fun_nat_bool ).

tff(sy_c_hAPP_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mtc__Nat__Onat_J,type,
    hAPP_nat_fun_nat_nat: ( fun_nat_fun_nat_nat * nat ) > fun_nat_nat ).

tff(sy_c_hAPP_000tc__Nat__Onat_000tc__fun_Itc__fun_It__a_Mtc__HOL__Obool_J_Mtc__HOL_,type,
    hAPP_n1414589940l_bool: ( fun_na1632405922l_bool * nat ) > fun_fun_a_bool_bool ).

tff(sy_c_hAPP_000tc__Nat__Onat_000tc__fun_Itc__fun_Itc__Com__Opname_Mtc__HOL__Obool_,type,
    hAPP_n850744903l_bool: ( fun_na2122364079l_bool * nat ) > fun_fu1430349052l_bool ).

tff(sy_c_hAPP_000tc__Nat__Onat_000tc__fun_Itc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_,type,
    hAPP_n609083369l_bool: ( fun_na1583160273l_bool * nat ) > fun_fu819804446l_bool ).

tff(sy_c_hAPP_000tc__Nat__Onat_000tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_,type,
    hAPP_n215258509l_bool: ( fun_na1436237685l_bool * nat ) > fun_fu425979586l_bool ).

tff(sy_c_hAPP_000tc__Nat__Onat_000tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J__037,type,
    hAPP_n1512601776t_bool: ( fun_na35521688t_bool * nat ) > fun_fu821463397t_bool ).

tff(sy_c_hAPP_000tc__fun_It__a_Mtc__HOL__Obool_J_000t__a,type,
    hAPP_fun_a_bool_a: ( fun_fun_a_bool_a * fun_a_bool ) > x_a ).

tff(sy_c_hAPP_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__HOL__Obool,type,
    hAPP_fun_a_bool_bool: ( fun_fun_a_bool_bool * fun_a_bool ) > bool ).

tff(sy_c_hAPP_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__Nat__Onat,type,
    hAPP_fun_a_bool_nat: ( fun_fun_a_bool_nat * fun_a_bool ) > nat ).

tff(sy_c_hAPP_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__fun_It__a_Mtc__HOL__Obool_J,type,
    hAPP_f2050579477a_bool: ( fun_fu1731003005a_bool * fun_a_bool ) > fun_a_bool ).

tff(sy_c_hAPP_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__fun_It__a_Mtc__fun_It__a_Mtc,type,
    hAPP_f1038053735a_bool: ( fun_fu868269973a_bool * fun_a_bool ) > fun_a_fun_a_bool ).

tff(sy_c_hAPP_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__fun_Itc__HOL__Obool_Mtc__HOL,type,
    hAPP_f198738859l_bool: ( fun_fu554186387l_bool * fun_a_bool ) > fun_bool_bool ).

tff(sy_c_hAPP_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__fun_Itc__Int__Oint_Mtc__HOL_,type,
    hAPP_f1812805200t_bool: ( fun_fu859530750t_bool * fun_a_bool ) > fun_int_bool ).

tff(sy_c_hAPP_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__fun_Itc__Nat__Onat_Mtc__HOL_,type,
    hAPP_f282463732t_bool: ( fun_fu600484770t_bool * fun_a_bool ) > fun_nat_bool ).

tff(sy_c_hAPP_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__fun_Itc__fun_It__a_Mtc__HOL_,type,
    hAPP_f1631501043l_bool: ( fun_fu1471507361l_bool * fun_a_bool ) > fun_fun_a_bool_bool ).

tff(sy_c_hAPP_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__fun_Itc__fun_It__a_Mtc__HOL__038,type,
    hAPP_f1791771145a_bool: ( fun_fu1663643505a_bool * fun_a_bool ) > fun_fu1731003005a_bool ).

tff(sy_c_hAPP_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__fun_Itc__fun_Itc__fun_It__a_,type,
    hAPP_f285962445l_bool: ( fun_fu386216885l_bool * fun_a_bool ) > fun_fu885608257l_bool ).

tff(sy_c_hAPP_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__fun_Itc__fun_Itc__fun_It__a__039,type,
    hAPP_f1531278575l_bool: ( fun_fu1434634199l_bool * fun_a_bool ) > fun_fu911136611l_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_000tc__Com__Opname,type,
    hAPP_f1297739591_pname: ( fun_fu1499449723_pname * fun_pname_bool ) > pname ).

tff(sy_c_hAPP_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_000tc__HOL__Obool,type,
    hAPP_f1664156314l_bool: ( fun_fu1430349052l_bool * fun_pname_bool ) > bool ).

tff(sy_c_hAPP_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_000tc__Nat__Onat,type,
    hAPP_f921600141ol_nat: ( fun_fu1668467777ol_nat * fun_pname_bool ) > nat ).

tff(sy_c_hAPP_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_000tc__fun_Itc__Com__Opn,type,
    hAPP_f759274231e_bool: ( fun_fu410713561e_bool * fun_pname_bool ) > fun_pname_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_000tc__fun_Itc__HOL__Obo,type,
    hAPP_f1476298914l_bool: ( fun_fu31783638l_bool * fun_pname_bool ) > fun_bool_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_000tc__fun_Itc__Int__Oin,type,
    hAPP_f449020825t_bool: ( fun_fu1159356667t_bool * fun_pname_bool ) > fun_int_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_000tc__fun_Itc__Nat__Ona,type,
    hAPP_f1066163005t_bool: ( fun_fu900310687t_bool * fun_pname_bool ) > fun_nat_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_000tc__fun_Itc__fun_Itc_,type,
    hAPP_f434788991l_bool: ( fun_fu802393907l_bool * fun_pname_bool ) > fun_fu1430349052l_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_000tc__fun_Itc__fun_Itc__040,type,
    hAPP_f1388330588e_bool: ( fun_fu2101097872e_bool * fun_pname_bool ) > fun_fu410713561e_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_000tc__fun_Itc__fun_Itc__041,type,
    hAPP_f556039215l_bool: ( fun_fu931343505l_bool * fun_pname_bool ) > fun_fu399576434l_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_000tc__fun_Itc__fun_Itc__042,type,
    hAPP_f1320879424l_bool: ( fun_fu1232732404l_bool * fun_pname_bool ) > fun_fu2087345469l_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__HOL__Obool,type,
    hAPP_f448129468l_bool: ( fun_fu819804446l_bool * fun_int_bool ) > bool ).

tff(sy_c_hAPP_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__Int__Oint,type,
    hAPP_f1594865479ol_int: ( fun_fun_int_bool_int * fun_int_bool ) > int ).

tff(sy_c_hAPP_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__Nat__Onat,type,
    hAPP_f957591787ol_nat: ( fun_fun_int_bool_nat * fun_int_bool ) > nat ).

tff(sy_c_hAPP_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__fun_Itc__HOL__Obool,type,
    hAPP_f1721100032l_bool: ( fun_fu989146164l_bool * fun_int_bool ) > fun_bool_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__fun_Itc__Int__Oint_,type,
    hAPP_f1805168059t_bool: ( fun_fu554825757t_bool * fun_int_bool ) > fun_int_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__fun_Itc__Int__Oint__043,type,
    hAPP_f1048215610t_bool: ( fun_fu380257692t_bool * fun_int_bool ) > fun_int_fun_int_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__fun_Itc__Nat__Onat_,type,
    hAPP_f274826591t_bool: ( fun_fu295779777t_bool * fun_int_bool ) > fun_nat_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__fun_Itc__fun_Itc__I,type,
    hAPP_f284875647l_bool: ( fun_fu1151772339l_bool * fun_int_bool ) > fun_fu819804446l_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__fun_Itc__fun_Itc__I_044,type,
    hAPP_f1223193598t_bool: ( fun_fu1075512626t_bool * fun_int_bool ) > fun_fu554825757t_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__fun_Itc__fun_Itc__f,type,
    hAPP_f628503027l_bool: ( fun_fu1637550549l_bool * fun_int_bool ) > fun_fu1540142484l_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__fun_Itc__fun_Itc__f_045,type,
    hAPP_f587666462l_bool: ( fun_fu587680722l_bool * fun_int_bool ) > fun_fu1070826941l_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__HOL__Obool,type,
    hAPP_f54304608l_bool: ( fun_fu425979586l_bool * fun_nat_bool ) > bool ).

tff(sy_c_hAPP_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__Nat__Onat,type,
    hAPP_f22106695ol_nat: ( fun_fun_nat_bool_nat * fun_nat_bool ) > nat ).

tff(sy_c_hAPP_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__fun_Itc__HOL__Obool,type,
    hAPP_f1748468828l_bool: ( fun_fu1016514960l_bool * fun_nat_bool ) > fun_bool_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__fun_Itc__Int__Oint_,type,
    hAPP_f183368031t_bool: ( fun_fu1080509377t_bool * fun_nat_bool ) > fun_int_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__fun_Itc__Nat__Onat_,type,
    hAPP_f800510211t_bool: ( fun_fu821463397t_bool * fun_nat_bool ) > fun_nat_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__fun_Itc__Nat__Onat__046,type,
    hAPP_f1570313510t_bool: ( fun_fu429110920t_bool * fun_nat_bool ) > fun_nat_fun_nat_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__fun_Itc__fun_Itc__N,type,
    hAPP_f103356543l_bool: ( fun_fu1217155507l_bool * fun_nat_bool ) > fun_fu425979586l_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__fun_Itc__fun_Itc__N_047,type,
    hAPP_f1730770594t_bool: ( fun_fu915923926t_bool * fun_nat_bool ) > fun_fu821463397t_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__fun_Itc__fun_Itc__f,type,
    hAPP_f1951378235l_bool: ( fun_fu1436348701l_bool * fun_nat_bool ) > fun_fu814369080l_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__fun_Itc__fun_Itc__f_048,type,
    hAPP_f633452666l_bool: ( fun_fu1516990510l_bool * fun_nat_bool ) > fun_fu616551101l_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_It__a_Mtc__HOL__Obool_J_Mtc__HOL__Obool_J_000tc__H,type,
    hAPP_f621171935l_bool: ( fun_fu885608257l_bool * fun_fun_a_bool_bool ) > bool ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_It__a_Mtc__HOL__Obool_J_Mtc__HOL__Obool_J_000tc__N,type,
    hAPP_f2009550088ol_nat: ( fun_fu2020802748ol_nat * fun_fun_a_bool_bool ) > nat ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_It__a_Mtc__HOL__Obool_J_Mtc__HOL__Obool_J_000tc__f,type,
    hAPP_f2117159681l_bool: ( fun_fu911136611l_bool * fun_fun_a_bool_bool ) > fun_fun_a_bool_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_It__a_Mtc__HOL__Obool_J_Mtc__HOL__Obool_J_000tc__f_049,type,
    hAPP_f1434722111l_bool: ( fun_fu418465139l_bool * fun_fun_a_bool_bool ) > fun_fu885608257l_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_Mtc__HOL__Obool,type,
    hAPP_f1935102916l_bool: ( fun_fu399576434l_bool * fun_fu1430349052l_bool ) > bool ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_Mtc__HOL__Obool_050,type,
    hAPP_f55526627ol_nat: ( fun_fu1730389579ol_nat * fun_fu1430349052l_bool ) > nat ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_Mtc__HOL__Obool_051,type,
    hAPP_f559147733l_bool: ( fun_fu2087345469l_bool * fun_fu1430349052l_bool ) > fun_fu1430349052l_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_Mtc__HOL__Obool_052,type,
    hAPP_f510955609l_bool: ( fun_fu1911931399l_bool * fun_fu1430349052l_bool ) > fun_fu399576434l_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_Mtc__HOL__Obool_J,type,
    hAPP_f215623910l_bool: ( fun_fu1540142484l_bool * fun_fu819804446l_bool ) > bool ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_Mtc__HOL__Obool_J_053,type,
    hAPP_f521865025ol_nat: ( fun_fu236049833ol_nat * fun_fu819804446l_bool ) > nat ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_Mtc__HOL__Obool_J_054,type,
    hAPP_f1948010709l_bool: ( fun_fu1070826941l_bool * fun_fu819804446l_bool ) > fun_fu819804446l_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_Mtc__HOL__Obool_J_055,type,
    hAPP_f317860125l_bool: ( fun_fu209342027l_bool * fun_fu819804446l_bool ) > fun_fu1540142484l_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_Mtc__HOL__Obool_J,type,
    hAPP_f1637334154l_bool: ( fun_fu814369080l_bool * fun_fu425979586l_bool ) > bool ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_Mtc__HOL__Obool_J_056,type,
    hAPP_f696928925ol_nat: ( fun_fu411113733ol_nat * fun_fu425979586l_bool ) > nat ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_Mtc__HOL__Obool_J_057,type,
    hAPP_f1246832597l_bool: ( fun_fu616551101l_bool * fun_fu425979586l_bool ) > fun_fu425979586l_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_Mtc__HOL__Obool_J_058,type,
    hAPP_f1772781669l_bool: ( fun_fu140186515l_bool * fun_fu425979586l_bool ) > fun_fu814369080l_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__fun_It__a_Mtc__HOL__Obool_J_Mtc__HOL__Obool_J,type,
    hAPP_f292226953l_bool: ( fun_fu255076663l_bool * fun_fu885608257l_bool ) > bool ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_Mtc__H,type,
    hAPP_f389811538l_bool: ( fun_fu1438281908l_bool * fun_fu399576434l_bool ) > bool ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_Mtc__HOL,type,
    hAPP_f1593524852l_bool: ( fun_fu717296342l_bool * fun_fu1540142484l_bool ) > bool ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_Mtc__HOL,type,
    hAPP_f937997336l_bool: ( fun_fu61768826l_bool * fun_fu814369080l_bool ) > bool ).

tff(sy_c_hBOOL,type,
    hBOOL: bool > $o ).

tff(sy_c_member_000t__a,type,
    member_a: fun_a_1255737515l_bool ).

tff(sy_c_member_000tc__Com__Opname,type,
    member_pname: fun_pn422929397l_bool ).

tff(sy_c_member_000tc__Int__Oint,type,
    member_int: fun_in938817141l_bool ).

tff(sy_c_member_000tc__Nat__Onat,type,
    member_nat: fun_na1436237685l_bool ).

tff(sy_c_member_000tc__fun_It__a_Mtc__HOL__Obool_J,type,
    member_fun_a_bool: fun_fu386216885l_bool ).

tff(sy_c_member_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J,type,
    member799430823e_bool: fun_fu931343505l_bool ).

tff(sy_c_member_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J,type,
    member_fun_int_bool: fun_fu1637550549l_bool ).

tff(sy_c_member_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J,type,
    member_fun_nat_bool: fun_fu1436348701l_bool ).

tff(sy_v_G,type,
    g: fun_a_bool ).

tff(sy_v_P,type,
    p: fun_a_bool > fun_fun_a_bool_bool ).

tff(sy_v_U,type,
    u: fun_pname_bool ).

tff(sy_v_mgt,type,
    mgt: com > x_a ).

tff(sy_v_mgt__call,type,
    mgt_call: fun_pname_a ).

tff(sy_v_na,type,
    na: nat ).

tff(sy_v_pn,type,
    pn: pname ).

tff(sy_v_wt,type,
    wt: com > bool ).

%----Relevant facts (1199)
tff(fact_0_assms_I1_J,axiom,
    ! [Ts: fun_a_bool,G: fun_a_bool] :
      ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,Ts),G))
     => hBOOL(hAPP_fun_a_bool_bool(p(G),Ts)) ) ).

tff(fact_1_finite__Collect__subsets,axiom,
    ! [A: fun_fu819804446l_bool] :
      ( hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,A))
     => hBOOL(hAPP_f1593524852l_bool(finite2095298929l_bool,collec1409689523l_bool(hAPP_f317860125l_bool(cOMBC_348868698l_bool(ord_le1848167016l_bool),A)))) ) ).

tff(fact_2_finite__Collect__subsets,axiom,
    ! [A: fun_fu425979586l_bool] :
      ( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,A))
     => hBOOL(hAPP_f937997336l_bool(finite1701474069l_bool,collec1015864663l_bool(hAPP_f1772781669l_bool(cOMBC_595898202l_bool(ord_le1454342156l_bool),A)))) ) ).

tff(fact_3_finite__Collect__subsets,axiom,
    ! [A: fun_fu1430349052l_bool] :
      ( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,A))
     => hBOOL(hAPP_f389811538l_bool(finite786885583l_bool,collec1613912337l_bool(hAPP_f510955609l_bool(cOMBC_7971162l_bool(ord_le675606854l_bool),A)))) ) ).

tff(fact_4_finite__Collect__subsets,axiom,
    ! [A: fun_fun_a_bool_bool] :
      ( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,A))
     => hBOOL(hAPP_f292226953l_bool(finite1381704300l_bool,collec707592106l_bool(hAPP_f1434722111l_bool(cOMBC_331553030l_bool(ord_le1375614389l_bool),A)))) ) ).

tff(fact_5_finite__Collect__subsets,axiom,
    ! [A: fun_a_bool] :
      ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A))
     => hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,collect_fun_a_bool(hAPP_f1631501043l_bool(cOMBC_1732670874l_bool(ord_le1311769555a_bool),A)))) ) ).

tff(fact_6_finite__Collect__subsets,axiom,
    ! [A: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A))
     => hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,collec1974731493e_bool(hAPP_f434788991l_bool(cOMBC_1284144636l_bool(ord_le313189616e_bool),A)))) ) ).

tff(fact_7_finite__Collect__subsets,axiom,
    ! [A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,collect_fun_nat_bool(hAPP_f103356543l_bool(cOMBC_1693257480l_bool(ord_le1568362934t_bool),A)))) ) ).

tff(fact_8_finite__Collect__subsets,axiom,
    ! [A: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
     => hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,collect_fun_int_bool(hAPP_f284875647l_bool(cOMBC_605892544l_bool(ord_le951220754t_bool),A)))) ) ).

tff(fact_9_finite__imageI,axiom,
    ! [H: fun_pn665169337t_bool,F_1: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,F_1))
     => hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,image_1512837979t_bool(H,F_1))) ) ).

tff(fact_10_finite__imageI,axiom,
    ! [H: fun_pn406123357t_bool,F_1: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,F_1))
     => hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,image_2129980159t_bool(H,F_1))) ) ).

tff(fact_11_finite__imageI,axiom,
    ! [H: fun_pn800050071e_bool,F_1: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,F_1))
     => hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,image_47868345e_bool(H,F_1))) ) ).

tff(fact_12_finite__imageI,axiom,
    ! [H: fun_pname_fun_a_bool,F_1: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,F_1))
     => hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,image_112932426a_bool(H,F_1))) ) ).

tff(fact_13_finite__imageI,axiom,
    ! [H: fun_nat_a,F_1: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F_1))
     => hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,image_nat_a(H,F_1))) ) ).

tff(fact_14_finite__imageI,axiom,
    ! [H: fun_nat_fun_int_bool,F_1: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F_1))
     => hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,image_1556378401t_bool(H,F_1))) ) ).

tff(fact_15_finite__imageI,axiom,
    ! [H: fun_nat_fun_nat_bool,F_1: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F_1))
     => hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,image_26036933t_bool(H,F_1))) ) ).

tff(fact_16_finite__imageI,axiom,
    ! [H: fun_na936072029e_bool,F_1: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F_1))
     => hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,image_1655916159e_bool(H,F_1))) ) ).

tff(fact_17_finite__imageI,axiom,
    ! [H: fun_nat_fun_a_bool,F_1: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F_1))
     => hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,image_nat_fun_a_bool(H,F_1))) ) ).

tff(fact_18_finite__imageI,axiom,
    ! [H: fun_int_a,F_1: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,F_1))
     => hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,image_int_a(H,F_1))) ) ).

tff(fact_19_finite__imageI,axiom,
    ! [H: fun_int_fun_int_bool,F_1: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,F_1))
     => hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,image_275383677t_bool(H,F_1))) ) ).

tff(fact_20_finite__imageI,axiom,
    ! [H: fun_int_fun_nat_bool,F_1: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,F_1))
     => hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,image_892525857t_bool(H,F_1))) ) ).

tff(fact_21_finite__imageI,axiom,
    ! [H: fun_in2065004473e_bool,F_1: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,F_1))
     => hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,image_637364955e_bool(H,F_1))) ) ).

tff(fact_22_finite__imageI,axiom,
    ! [H: fun_int_fun_a_bool,F_1: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,F_1))
     => hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,image_int_fun_a_bool(H,F_1))) ) ).

tff(fact_23_finite__imageI,axiom,
    ! [H: fun_a_pname,F_1: fun_a_bool] :
      ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,F_1))
     => hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,image_a_pname(H,F_1))) ) ).

tff(fact_24_finite__imageI,axiom,
    ! [H: fun_fu953875929_pname,F_1: fun_fu819804446l_bool] :
      ( hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,F_1))
     => hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,image_62782965_pname(H,F_1))) ) ).

tff(fact_25_finite__imageI,axiom,
    ! [H: fun_fu665170229_pname,F_1: fun_fu425979586l_bool] :
      ( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,F_1))
     => hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,image_1921560913_pname(H,F_1))) ) ).

tff(fact_26_finite__imageI,axiom,
    ! [H: fun_fu1499449723_pname,F_1: fun_fu1430349052l_bool] :
      ( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,F_1))
     => hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,image_1283814551_pname(H,F_1))) ) ).

tff(fact_27_finite__imageI,axiom,
    ! [H: fun_fun_a_bool_pname,F_1: fun_fun_a_bool_bool] :
      ( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,F_1))
     => hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,image_1854862208_pname(H,F_1))) ) ).

tff(fact_28_finite__imageI,axiom,
    ! [H: fun_a_nat,F_1: fun_a_bool] :
      ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,F_1))
     => hBOOL(hAPP_f54304608l_bool(finite_finite_nat,image_a_nat(H,F_1))) ) ).

tff(fact_29_finite__imageI,axiom,
    ! [H: fun_fun_int_bool_nat,F_1: fun_fu819804446l_bool] :
      ( hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,F_1))
     => hBOOL(hAPP_f54304608l_bool(finite_finite_nat,image_1431733819ol_nat(H,F_1))) ) ).

tff(fact_30_finite__imageI,axiom,
    ! [H: fun_fun_nat_bool_nat,F_1: fun_fu425979586l_bool] :
      ( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,F_1))
     => hBOOL(hAPP_f54304608l_bool(finite_finite_nat,image_496248727ol_nat(H,F_1))) ) ).

tff(fact_31_finite__imageI,axiom,
    ! [H: fun_fu1668467777ol_nat,F_1: fun_fu1430349052l_bool] :
      ( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,F_1))
     => hBOOL(hAPP_f54304608l_bool(finite_finite_nat,image_1551609309ol_nat(H,F_1))) ) ).

tff(fact_32_finite__imageI,axiom,
    ! [H: fun_fun_a_bool_nat,F_1: fun_fun_a_bool_bool] :
      ( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,F_1))
     => hBOOL(hAPP_f54304608l_bool(finite_finite_nat,image_fun_a_bool_nat(H,F_1))) ) ).

tff(fact_33_finite__imageI,axiom,
    ! [H: fun_a_int,F_1: fun_a_bool] :
      ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,F_1))
     => hBOOL(hAPP_f448129468l_bool(finite_finite_int,image_a_int(H,F_1))) ) ).

tff(fact_34_finite__imageI,axiom,
    ! [H: fun_fun_int_bool_int,F_1: fun_fu819804446l_bool] :
      ( hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,F_1))
     => hBOOL(hAPP_f448129468l_bool(finite_finite_int,image_2069007511ol_int(H,F_1))) ) ).

tff(fact_35_finite__imageI,axiom,
    ! [H: fun_fun_nat_bool_int,F_1: fun_fu425979586l_bool] :
      ( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,F_1))
     => hBOOL(hAPP_f448129468l_bool(finite_finite_int,image_1133522419ol_int(H,F_1))) ) ).

tff(fact_36_finite__imageI,axiom,
    ! [H: fun_fu1830380701ol_int,F_1: fun_fu1430349052l_bool] :
      ( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,F_1))
     => hBOOL(hAPP_f448129468l_bool(finite_finite_int,image_41399353ol_int(H,F_1))) ) ).

tff(fact_37_finite__imageI,axiom,
    ! [H: fun_fun_a_bool_int,F_1: fun_fun_a_bool_bool] :
      ( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,F_1))
     => hBOOL(hAPP_f448129468l_bool(finite_finite_int,image_fun_a_bool_int(H,F_1))) ) ).

tff(fact_38_finite__imageI,axiom,
    ! [H: fun_pname_a,F_1: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,F_1))
     => hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,image_pname_a(H,F_1))) ) ).

tff(fact_39_finite__imageI,axiom,
    ! [H: fun_nat_int,F_1: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F_1))
     => hBOOL(hAPP_f448129468l_bool(finite_finite_int,image_nat_int(H,F_1))) ) ).

tff(fact_40_finite_OinsertI,axiom,
    ! [A_2: fun_int_bool,A: fun_fu819804446l_bool] :
      ( hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,A))
     => hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,hAPP_f1948010709l_bool(hAPP_f587666462l_bool(insert_fun_int_bool,A_2),A))) ) ).

tff(fact_41_finite_OinsertI,axiom,
    ! [A_2: fun_nat_bool,A: fun_fu425979586l_bool] :
      ( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,A))
     => hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,hAPP_f1246832597l_bool(hAPP_f633452666l_bool(insert_fun_nat_bool,A_2),A))) ) ).

tff(fact_42_finite_OinsertI,axiom,
    ! [A_2: fun_pname_bool,A: fun_fu1430349052l_bool] :
      ( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,A))
     => hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,hAPP_f559147733l_bool(hAPP_f1320879424l_bool(insert1325755072e_bool,A_2),A))) ) ).

tff(fact_43_finite_OinsertI,axiom,
    ! [A_2: fun_a_bool,A: fun_fun_a_bool_bool] :
      ( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,A))
     => hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,hAPP_f2117159681l_bool(hAPP_f1531278575l_bool(insert_fun_a_bool,A_2),A))) ) ).

tff(fact_44_finite_OinsertI,axiom,
    ! [A_2: pname,A: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A))
     => hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_2),A))) ) ).

tff(fact_45_finite_OinsertI,axiom,
    ! [A_2: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_2),A))) ) ).

tff(fact_46_finite_OinsertI,axiom,
    ! [A_2: int,A: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
     => hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_2),A))) ) ).

tff(fact_47_finite_OinsertI,axiom,
    ! [A_2: x_a,A: fun_a_bool] :
      ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A))
     => hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,A_2),A))) ) ).

tff(fact_48_card__image__le,axiom,
    ! [F: fun_pname_pname,A: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f921600141ol_nat(finite_card_pname,image_pname_pname(F,A))),hAPP_f921600141ol_nat(finite_card_pname,A))) ) ).

tff(fact_49_card__image__le,axiom,
    ! [F: fun_a_a,A: fun_a_bool] :
      ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_fun_a_bool_nat(finite_card_a,image_a_a(F,A))),hAPP_fun_a_bool_nat(finite_card_a,A))) ) ).

tff(fact_50_card__image__le,axiom,
    ! [F: fun_fun_int_bool_a,A: fun_fu819804446l_bool] :
      ( hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,A))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_fun_a_bool_nat(finite_card_a,image_fun_int_bool_a(F,A))),hAPP_f521865025ol_nat(finite1876863882t_bool,A))) ) ).

tff(fact_51_card__image__le,axiom,
    ! [F: fun_fun_nat_bool_a,A: fun_fu425979586l_bool] :
      ( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,A))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_fun_a_bool_nat(finite_card_a,image_fun_nat_bool_a(F,A))),hAPP_f696928925ol_nat(finite346522414t_bool,A))) ) ).

tff(fact_52_card__image__le,axiom,
    ! [F: fun_fun_pname_bool_a,A: fun_fu1430349052l_bool] :
      ( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,A))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_fun_a_bool_nat(finite_card_a,image_876012084bool_a(F,A))),hAPP_f55526627ol_nat(finite1340463720e_bool,A))) ) ).

tff(fact_53_card__image__le,axiom,
    ! [F: fun_fun_a_bool_a,A: fun_fun_a_bool_bool] :
      ( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,A))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_fun_a_bool_nat(finite_card_a,image_fun_a_bool_a(F,A))),hAPP_f2009550088ol_nat(finite1306199131a_bool,A))) ) ).

tff(fact_54_card__image__le,axiom,
    ! [F: fun_pname_nat,A: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f22106695ol_nat(finite_card_nat,image_pname_nat(F,A))),hAPP_f921600141ol_nat(finite_card_pname,A))) ) ).

tff(fact_55_card__image__le,axiom,
    ! [F: fun_a_nat,A: fun_a_bool] :
      ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f22106695ol_nat(finite_card_nat,image_a_nat(F,A))),hAPP_fun_a_bool_nat(finite_card_a,A))) ) ).

tff(fact_56_card__image__le,axiom,
    ! [F: fun_fun_int_bool_nat,A: fun_fu819804446l_bool] :
      ( hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,A))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f22106695ol_nat(finite_card_nat,image_1431733819ol_nat(F,A))),hAPP_f521865025ol_nat(finite1876863882t_bool,A))) ) ).

tff(fact_57_card__image__le,axiom,
    ! [F: fun_fun_nat_bool_nat,A: fun_fu425979586l_bool] :
      ( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,A))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f22106695ol_nat(finite_card_nat,image_496248727ol_nat(F,A))),hAPP_f696928925ol_nat(finite346522414t_bool,A))) ) ).

tff(fact_58_card__image__le,axiom,
    ! [F: fun_fu1668467777ol_nat,A: fun_fu1430349052l_bool] :
      ( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,A))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f22106695ol_nat(finite_card_nat,image_1551609309ol_nat(F,A))),hAPP_f55526627ol_nat(finite1340463720e_bool,A))) ) ).

tff(fact_59_card__image__le,axiom,
    ! [F: fun_fun_a_bool_nat,A: fun_fun_a_bool_bool] :
      ( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,A))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f22106695ol_nat(finite_card_nat,image_fun_a_bool_nat(F,A))),hAPP_f2009550088ol_nat(finite1306199131a_bool,A))) ) ).

tff(fact_60_card__image__le,axiom,
    ! [F: fun_pname_int,A: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f957591787ol_nat(finite_card_int,image_pname_int(F,A))),hAPP_f921600141ol_nat(finite_card_pname,A))) ) ).

tff(fact_61_card__image__le,axiom,
    ! [F: fun_a_int,A: fun_a_bool] :
      ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f957591787ol_nat(finite_card_int,image_a_int(F,A))),hAPP_fun_a_bool_nat(finite_card_a,A))) ) ).

tff(fact_62_card__image__le,axiom,
    ! [F: fun_fun_int_bool_int,A: fun_fu819804446l_bool] :
      ( hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,A))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f957591787ol_nat(finite_card_int,image_2069007511ol_int(F,A))),hAPP_f521865025ol_nat(finite1876863882t_bool,A))) ) ).

tff(fact_63_card__image__le,axiom,
    ! [F: fun_fun_nat_bool_int,A: fun_fu425979586l_bool] :
      ( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,A))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f957591787ol_nat(finite_card_int,image_1133522419ol_int(F,A))),hAPP_f696928925ol_nat(finite346522414t_bool,A))) ) ).

tff(fact_64_card__image__le,axiom,
    ! [F: fun_fu1830380701ol_int,A: fun_fu1430349052l_bool] :
      ( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,A))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f957591787ol_nat(finite_card_int,image_41399353ol_int(F,A))),hAPP_f55526627ol_nat(finite1340463720e_bool,A))) ) ).

tff(fact_65_card__image__le,axiom,
    ! [F: fun_fun_a_bool_int,A: fun_fun_a_bool_bool] :
      ( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,A))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f957591787ol_nat(finite_card_int,image_fun_a_bool_int(F,A))),hAPP_f2009550088ol_nat(finite1306199131a_bool,A))) ) ).

tff(fact_66_card__image__le,axiom,
    ! [F: fun_a_pname,A: fun_a_bool] :
      ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f921600141ol_nat(finite_card_pname,image_a_pname(F,A))),hAPP_fun_a_bool_nat(finite_card_a,A))) ) ).

tff(fact_67_card__image__le,axiom,
    ! [F: fun_nat_pname,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f921600141ol_nat(finite_card_pname,image_nat_pname(F,A))),hAPP_f22106695ol_nat(finite_card_nat,A))) ) ).

tff(fact_68_card__image__le,axiom,
    ! [F: fun_int_pname,A: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f921600141ol_nat(finite_card_pname,image_int_pname(F,A))),hAPP_f957591787ol_nat(finite_card_int,A))) ) ).

tff(fact_69_card__image__le,axiom,
    ! [F: fun_pname_a,A: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_fun_a_bool_nat(finite_card_a,image_pname_a(F,A))),hAPP_f921600141ol_nat(finite_card_pname,A))) ) ).

tff(fact_70_card__image__le,axiom,
    ! [F: fun_nat_int,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f957591787ol_nat(finite_card_int,image_nat_int(F,A))),hAPP_f22106695ol_nat(finite_card_nat,A))) ) ).

tff(fact_71_card__mono,axiom,
    ! [A: fun_fu819804446l_bool,B: fun_fu819804446l_bool] :
      ( hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,B))
     => ( hBOOL(hAPP_f215623910l_bool(hAPP_f317860125l_bool(ord_le1848167016l_bool,A),B))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f521865025ol_nat(finite1876863882t_bool,A)),hAPP_f521865025ol_nat(finite1876863882t_bool,B))) ) ) ).

tff(fact_72_card__mono,axiom,
    ! [A: fun_fu425979586l_bool,B: fun_fu425979586l_bool] :
      ( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,B))
     => ( hBOOL(hAPP_f1637334154l_bool(hAPP_f1772781669l_bool(ord_le1454342156l_bool,A),B))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f696928925ol_nat(finite346522414t_bool,A)),hAPP_f696928925ol_nat(finite346522414t_bool,B))) ) ) ).

tff(fact_73_card__mono,axiom,
    ! [A: fun_fu1430349052l_bool,B: fun_fu1430349052l_bool] :
      ( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,B))
     => ( hBOOL(hAPP_f1935102916l_bool(hAPP_f510955609l_bool(ord_le675606854l_bool,A),B))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f55526627ol_nat(finite1340463720e_bool,A)),hAPP_f55526627ol_nat(finite1340463720e_bool,B))) ) ) ).

tff(fact_74_card__mono,axiom,
    ! [A: fun_fun_a_bool_bool,B: fun_fun_a_bool_bool] :
      ( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,B))
     => ( hBOOL(hAPP_f621171935l_bool(hAPP_f1434722111l_bool(ord_le1375614389l_bool,A),B))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f2009550088ol_nat(finite1306199131a_bool,A)),hAPP_f2009550088ol_nat(finite1306199131a_bool,B))) ) ) ).

tff(fact_75_card__mono,axiom,
    ! [A: fun_pname_bool,B: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,B))
     => ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A),B))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f921600141ol_nat(finite_card_pname,A)),hAPP_f921600141ol_nat(finite_card_pname,B))) ) ) ).

tff(fact_76_card__mono,axiom,
    ! [A: fun_a_bool,B: fun_a_bool] :
      ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,B))
     => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),B))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_fun_a_bool_nat(finite_card_a,A)),hAPP_fun_a_bool_nat(finite_card_a,B))) ) ) ).

tff(fact_77_card__mono,axiom,
    ! [A: fun_nat_bool,B: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f22106695ol_nat(finite_card_nat,A)),hAPP_f22106695ol_nat(finite_card_nat,B))) ) ) ).

tff(fact_78_card__mono,axiom,
    ! [A: fun_int_bool,B: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,B))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),B))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f957591787ol_nat(finite_card_int,A)),hAPP_f957591787ol_nat(finite_card_int,B))) ) ) ).

tff(fact_79_card__seteq,axiom,
    ! [A: fun_fu819804446l_bool,B: fun_fu819804446l_bool] :
      ( hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,B))
     => ( hBOOL(hAPP_f215623910l_bool(hAPP_f317860125l_bool(ord_le1848167016l_bool,A),B))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f521865025ol_nat(finite1876863882t_bool,B)),hAPP_f521865025ol_nat(finite1876863882t_bool,A)))
         => ( A = B ) ) ) ) ).

tff(fact_80_card__seteq,axiom,
    ! [A: fun_fu425979586l_bool,B: fun_fu425979586l_bool] :
      ( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,B))
     => ( hBOOL(hAPP_f1637334154l_bool(hAPP_f1772781669l_bool(ord_le1454342156l_bool,A),B))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f696928925ol_nat(finite346522414t_bool,B)),hAPP_f696928925ol_nat(finite346522414t_bool,A)))
         => ( A = B ) ) ) ) ).

tff(fact_81_card__seteq,axiom,
    ! [A: fun_fu1430349052l_bool,B: fun_fu1430349052l_bool] :
      ( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,B))
     => ( hBOOL(hAPP_f1935102916l_bool(hAPP_f510955609l_bool(ord_le675606854l_bool,A),B))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f55526627ol_nat(finite1340463720e_bool,B)),hAPP_f55526627ol_nat(finite1340463720e_bool,A)))
         => ( A = B ) ) ) ) ).

tff(fact_82_card__seteq,axiom,
    ! [A: fun_fun_a_bool_bool,B: fun_fun_a_bool_bool] :
      ( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,B))
     => ( hBOOL(hAPP_f621171935l_bool(hAPP_f1434722111l_bool(ord_le1375614389l_bool,A),B))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f2009550088ol_nat(finite1306199131a_bool,B)),hAPP_f2009550088ol_nat(finite1306199131a_bool,A)))
         => ( A = B ) ) ) ) ).

tff(fact_83_card__seteq,axiom,
    ! [A: fun_pname_bool,B: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,B))
     => ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A),B))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f921600141ol_nat(finite_card_pname,B)),hAPP_f921600141ol_nat(finite_card_pname,A)))
         => ( A = B ) ) ) ) ).

tff(fact_84_card__seteq,axiom,
    ! [A: fun_a_bool,B: fun_a_bool] :
      ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,B))
     => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),B))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_fun_a_bool_nat(finite_card_a,B)),hAPP_fun_a_bool_nat(finite_card_a,A)))
         => ( A = B ) ) ) ) ).

tff(fact_85_card__seteq,axiom,
    ! [A: fun_nat_bool,B: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f22106695ol_nat(finite_card_nat,B)),hAPP_f22106695ol_nat(finite_card_nat,A)))
         => ( A = B ) ) ) ) ).

tff(fact_86_card__seteq,axiom,
    ! [A: fun_int_bool,B: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,B))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),B))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f957591787ol_nat(finite_card_int,B)),hAPP_f957591787ol_nat(finite_card_int,A)))
         => ( A = B ) ) ) ) ).

tff(fact_87_card__insert__le,axiom,
    ! [X_1: fun_int_bool,A: fun_fu819804446l_bool] :
      ( hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,A))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f521865025ol_nat(finite1876863882t_bool,A)),hAPP_f521865025ol_nat(finite1876863882t_bool,hAPP_f1948010709l_bool(hAPP_f587666462l_bool(insert_fun_int_bool,X_1),A)))) ) ).

tff(fact_88_card__insert__le,axiom,
    ! [X_1: fun_nat_bool,A: fun_fu425979586l_bool] :
      ( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,A))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f696928925ol_nat(finite346522414t_bool,A)),hAPP_f696928925ol_nat(finite346522414t_bool,hAPP_f1246832597l_bool(hAPP_f633452666l_bool(insert_fun_nat_bool,X_1),A)))) ) ).

tff(fact_89_card__insert__le,axiom,
    ! [X_1: fun_pname_bool,A: fun_fu1430349052l_bool] :
      ( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,A))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f55526627ol_nat(finite1340463720e_bool,A)),hAPP_f55526627ol_nat(finite1340463720e_bool,hAPP_f559147733l_bool(hAPP_f1320879424l_bool(insert1325755072e_bool,X_1),A)))) ) ).

tff(fact_90_card__insert__le,axiom,
    ! [X_1: fun_a_bool,A: fun_fun_a_bool_bool] :
      ( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,A))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f2009550088ol_nat(finite1306199131a_bool,A)),hAPP_f2009550088ol_nat(finite1306199131a_bool,hAPP_f2117159681l_bool(hAPP_f1531278575l_bool(insert_fun_a_bool,X_1),A)))) ) ).

tff(fact_91_card__insert__le,axiom,
    ! [X_1: pname,A: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f921600141ol_nat(finite_card_pname,A)),hAPP_f921600141ol_nat(finite_card_pname,hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_1),A)))) ) ).

tff(fact_92_card__insert__le,axiom,
    ! [X_1: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f22106695ol_nat(finite_card_nat,A)),hAPP_f22106695ol_nat(finite_card_nat,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)))) ) ).

tff(fact_93_card__insert__le,axiom,
    ! [X_1: int,A: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f957591787ol_nat(finite_card_int,A)),hAPP_f957591787ol_nat(finite_card_int,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A)))) ) ).

tff(fact_94_card__insert__le,axiom,
    ! [X_1: x_a,A: fun_a_bool] :
      ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_fun_a_bool_nat(finite_card_a,A)),hAPP_fun_a_bool_nat(finite_card_a,hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_1),A)))) ) ).

tff(fact_95_card__insert__if,axiom,
    ! [X_1: fun_int_bool,A: fun_fu819804446l_bool] :
      ( hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,A))
     => ( ( hBOOL(hAPP_f215623910l_bool(hAPP_f628503027l_bool(member_fun_int_bool,X_1),A))
         => ( hAPP_f521865025ol_nat(finite1876863882t_bool,hAPP_f1948010709l_bool(hAPP_f587666462l_bool(insert_fun_int_bool,X_1),A)) = hAPP_f521865025ol_nat(finite1876863882t_bool,A) ) )
        & ( ~ hBOOL(hAPP_f215623910l_bool(hAPP_f628503027l_bool(member_fun_int_bool,X_1),A))
         => ( hAPP_f521865025ol_nat(finite1876863882t_bool,hAPP_f1948010709l_bool(hAPP_f587666462l_bool(insert_fun_int_bool,X_1),A)) = hAPP_nat_nat(suc,hAPP_f521865025ol_nat(finite1876863882t_bool,A)) ) ) ) ) ).

tff(fact_96_card__insert__if,axiom,
    ! [X_1: fun_nat_bool,A: fun_fu425979586l_bool] :
      ( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,A))
     => ( ( hBOOL(hAPP_f1637334154l_bool(hAPP_f1951378235l_bool(member_fun_nat_bool,X_1),A))
         => ( hAPP_f696928925ol_nat(finite346522414t_bool,hAPP_f1246832597l_bool(hAPP_f633452666l_bool(insert_fun_nat_bool,X_1),A)) = hAPP_f696928925ol_nat(finite346522414t_bool,A) ) )
        & ( ~ hBOOL(hAPP_f1637334154l_bool(hAPP_f1951378235l_bool(member_fun_nat_bool,X_1),A))
         => ( hAPP_f696928925ol_nat(finite346522414t_bool,hAPP_f1246832597l_bool(hAPP_f633452666l_bool(insert_fun_nat_bool,X_1),A)) = hAPP_nat_nat(suc,hAPP_f696928925ol_nat(finite346522414t_bool,A)) ) ) ) ) ).

tff(fact_97_card__insert__if,axiom,
    ! [X_1: fun_pname_bool,A: fun_fu1430349052l_bool] :
      ( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,A))
     => ( ( hBOOL(hAPP_f1935102916l_bool(hAPP_f556039215l_bool(member799430823e_bool,X_1),A))
         => ( hAPP_f55526627ol_nat(finite1340463720e_bool,hAPP_f559147733l_bool(hAPP_f1320879424l_bool(insert1325755072e_bool,X_1),A)) = hAPP_f55526627ol_nat(finite1340463720e_bool,A) ) )
        & ( ~ hBOOL(hAPP_f1935102916l_bool(hAPP_f556039215l_bool(member799430823e_bool,X_1),A))
         => ( hAPP_f55526627ol_nat(finite1340463720e_bool,hAPP_f559147733l_bool(hAPP_f1320879424l_bool(insert1325755072e_bool,X_1),A)) = hAPP_nat_nat(suc,hAPP_f55526627ol_nat(finite1340463720e_bool,A)) ) ) ) ) ).

tff(fact_98_card__insert__if,axiom,
    ! [X_1: fun_a_bool,A: fun_fun_a_bool_bool] :
      ( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,A))
     => ( ( hBOOL(hAPP_f621171935l_bool(hAPP_f285962445l_bool(member_fun_a_bool,X_1),A))
         => ( hAPP_f2009550088ol_nat(finite1306199131a_bool,hAPP_f2117159681l_bool(hAPP_f1531278575l_bool(insert_fun_a_bool,X_1),A)) = hAPP_f2009550088ol_nat(finite1306199131a_bool,A) ) )
        & ( ~ hBOOL(hAPP_f621171935l_bool(hAPP_f285962445l_bool(member_fun_a_bool,X_1),A))
         => ( hAPP_f2009550088ol_nat(finite1306199131a_bool,hAPP_f2117159681l_bool(hAPP_f1531278575l_bool(insert_fun_a_bool,X_1),A)) = hAPP_nat_nat(suc,hAPP_f2009550088ol_nat(finite1306199131a_bool,A)) ) ) ) ) ).

tff(fact_99_card__insert__if,axiom,
    ! [X_1: pname,A: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A))
     => ( ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_1),A))
         => ( hAPP_f921600141ol_nat(finite_card_pname,hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_1),A)) = hAPP_f921600141ol_nat(finite_card_pname,A) ) )
        & ( ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_1),A))
         => ( hAPP_f921600141ol_nat(finite_card_pname,hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_1),A)) = hAPP_nat_nat(suc,hAPP_f921600141ol_nat(finite_card_pname,A)) ) ) ) ) ).

tff(fact_100_card__insert__if,axiom,
    ! [X_1: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => ( ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
         => ( hAPP_f22106695ol_nat(finite_card_nat,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)) = hAPP_f22106695ol_nat(finite_card_nat,A) ) )
        & ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
         => ( hAPP_f22106695ol_nat(finite_card_nat,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)) = hAPP_nat_nat(suc,hAPP_f22106695ol_nat(finite_card_nat,A)) ) ) ) ) ).

tff(fact_101_card__insert__if,axiom,
    ! [X_1: int,A: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
     => ( ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
         => ( hAPP_f957591787ol_nat(finite_card_int,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A)) = hAPP_f957591787ol_nat(finite_card_int,A) ) )
        & ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
         => ( hAPP_f957591787ol_nat(finite_card_int,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A)) = hAPP_nat_nat(suc,hAPP_f957591787ol_nat(finite_card_int,A)) ) ) ) ) ).

tff(fact_102_card__insert__if,axiom,
    ! [X_1: x_a,A: fun_a_bool] :
      ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A))
     => ( ( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_1),A))
         => ( hAPP_fun_a_bool_nat(finite_card_a,hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_1),A)) = hAPP_fun_a_bool_nat(finite_card_a,A) ) )
        & ( ~ hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_1),A))
         => ( hAPP_fun_a_bool_nat(finite_card_a,hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_1),A)) = hAPP_nat_nat(suc,hAPP_fun_a_bool_nat(finite_card_a,A)) ) ) ) ) ).

tff(fact_103_card__insert__disjoint,axiom,
    ! [X_1: fun_int_bool,A: fun_fu819804446l_bool] :
      ( hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,A))
     => ( ~ hBOOL(hAPP_f215623910l_bool(hAPP_f628503027l_bool(member_fun_int_bool,X_1),A))
       => ( hAPP_f521865025ol_nat(finite1876863882t_bool,hAPP_f1948010709l_bool(hAPP_f587666462l_bool(insert_fun_int_bool,X_1),A)) = hAPP_nat_nat(suc,hAPP_f521865025ol_nat(finite1876863882t_bool,A)) ) ) ) ).

tff(fact_104_card__insert__disjoint,axiom,
    ! [X_1: fun_nat_bool,A: fun_fu425979586l_bool] :
      ( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,A))
     => ( ~ hBOOL(hAPP_f1637334154l_bool(hAPP_f1951378235l_bool(member_fun_nat_bool,X_1),A))
       => ( hAPP_f696928925ol_nat(finite346522414t_bool,hAPP_f1246832597l_bool(hAPP_f633452666l_bool(insert_fun_nat_bool,X_1),A)) = hAPP_nat_nat(suc,hAPP_f696928925ol_nat(finite346522414t_bool,A)) ) ) ) ).

tff(fact_105_card__insert__disjoint,axiom,
    ! [X_1: fun_pname_bool,A: fun_fu1430349052l_bool] :
      ( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,A))
     => ( ~ hBOOL(hAPP_f1935102916l_bool(hAPP_f556039215l_bool(member799430823e_bool,X_1),A))
       => ( hAPP_f55526627ol_nat(finite1340463720e_bool,hAPP_f559147733l_bool(hAPP_f1320879424l_bool(insert1325755072e_bool,X_1),A)) = hAPP_nat_nat(suc,hAPP_f55526627ol_nat(finite1340463720e_bool,A)) ) ) ) ).

tff(fact_106_card__insert__disjoint,axiom,
    ! [X_1: fun_a_bool,A: fun_fun_a_bool_bool] :
      ( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,A))
     => ( ~ hBOOL(hAPP_f621171935l_bool(hAPP_f285962445l_bool(member_fun_a_bool,X_1),A))
       => ( hAPP_f2009550088ol_nat(finite1306199131a_bool,hAPP_f2117159681l_bool(hAPP_f1531278575l_bool(insert_fun_a_bool,X_1),A)) = hAPP_nat_nat(suc,hAPP_f2009550088ol_nat(finite1306199131a_bool,A)) ) ) ) ).

tff(fact_107_card__insert__disjoint,axiom,
    ! [X_1: pname,A: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A))
     => ( ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_1),A))
       => ( hAPP_f921600141ol_nat(finite_card_pname,hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_1),A)) = hAPP_nat_nat(suc,hAPP_f921600141ol_nat(finite_card_pname,A)) ) ) ) ).

tff(fact_108_card__insert__disjoint,axiom,
    ! [X_1: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
       => ( hAPP_f22106695ol_nat(finite_card_nat,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)) = hAPP_nat_nat(suc,hAPP_f22106695ol_nat(finite_card_nat,A)) ) ) ) ).

tff(fact_109_card__insert__disjoint,axiom,
    ! [X_1: int,A: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
     => ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
       => ( hAPP_f957591787ol_nat(finite_card_int,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A)) = hAPP_nat_nat(suc,hAPP_f957591787ol_nat(finite_card_int,A)) ) ) ) ).

tff(fact_110_card__insert__disjoint,axiom,
    ! [X_1: x_a,A: fun_a_bool] :
      ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A))
     => ( ~ hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_1),A))
       => ( hAPP_fun_a_bool_nat(finite_card_a,hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_1),A)) = hAPP_nat_nat(suc,hAPP_fun_a_bool_nat(finite_card_a,A)) ) ) ) ).

tff(fact_111_finite__Collect__conjI,axiom,
    ! [Q_3: fun_a_bool,Pa: fun_a_bool] :
      ( ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,collect_a(Pa)))
        | hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,collect_a(Q_3))) )
     => hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,collect_a(cOMBS_a_bool_bool(cOMBB_1972296269bool_a(fconj,Pa),Q_3)))) ) ).

tff(fact_112_finite__Collect__conjI,axiom,
    ! [Q_3: fun_fu819804446l_bool,Pa: fun_fu819804446l_bool] :
      ( ( hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,collect_fun_int_bool(Pa)))
        | hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,collect_fun_int_bool(Q_3))) )
     => hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,collect_fun_int_bool(cOMBS_1912792529l_bool(cOMBB_1974511970t_bool(fconj,Pa),Q_3)))) ) ).

tff(fact_113_finite__Collect__conjI,axiom,
    ! [Q_3: fun_fu425979586l_bool,Pa: fun_fu425979586l_bool] :
      ( ( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,collect_fun_nat_bool(Pa)))
        | hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,collect_fun_nat_bool(Q_3))) )
     => hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,collect_fun_nat_bool(cOMBS_1187019125l_bool(cOMBB_444170502t_bool(fconj,Pa),Q_3)))) ) ).

tff(fact_114_finite__Collect__conjI,axiom,
    ! [Q_3: fun_fu1430349052l_bool,Pa: fun_fu1430349052l_bool] :
      ( ( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,collec1974731493e_bool(Pa)))
        | hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,collec1974731493e_bool(Q_3))) )
     => hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,collec1974731493e_bool(cOMBS_350070575l_bool(cOMBB_2095475776e_bool(fconj,Pa),Q_3)))) ) ).

tff(fact_115_finite__Collect__conjI,axiom,
    ! [Q_3: fun_fun_a_bool_bool,Pa: fun_fun_a_bool_bool] :
      ( ( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,collect_fun_a_bool(Pa)))
        | hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,collect_fun_a_bool(Q_3))) )
     => hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,collect_fun_a_bool(cOMBS_1035972772l_bool(cOMBB_338059395a_bool(fconj,Pa),Q_3)))) ) ).

tff(fact_116_finite__Collect__conjI,axiom,
    ! [Q_3: fun_pname_bool,Pa: fun_pname_bool] :
      ( ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,collect_pname(Pa)))
        | hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,collect_pname(Q_3))) )
     => hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,collect_pname(cOMBS_568398431l_bool(cOMBB_675860798_pname(fconj,Pa),Q_3)))) ) ).

tff(fact_117_finite__Collect__conjI,axiom,
    ! [Q_3: fun_nat_bool,Pa: fun_nat_bool] :
      ( ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(Pa)))
        | hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(Q_3))) )
     => hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(cOMBS_nat_bool_bool(cOMBB_1015721476ol_nat(fconj,Pa),Q_3)))) ) ).

tff(fact_118_finite__Collect__conjI,axiom,
    ! [Q_3: fun_int_bool,Pa: fun_int_bool] :
      ( ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,collect_int(Pa)))
        | hBOOL(hAPP_f448129468l_bool(finite_finite_int,collect_int(Q_3))) )
     => hBOOL(hAPP_f448129468l_bool(finite_finite_int,collect_int(cOMBS_int_bool_bool(cOMBB_1652995168ol_int(fconj,Pa),Q_3)))) ) ).

tff(fact_119_Suc__diff__le,axiom,
    ! [N: nat,M_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M_1))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,M_1)),N) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),N)) ) ) ).

tff(fact_120_finite__Collect__le__nat,axiom,
    ! [K: nat] : hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(hAPP_n1699378549t_bool(cOMBC_nat_nat_bool(ord_less_eq_nat),K)))) ).

tff(fact_121_card__Collect__le__nat,axiom,
    ! [Na: nat] : ( hAPP_f22106695ol_nat(finite_card_nat,collect_nat(hAPP_n1699378549t_bool(cOMBC_nat_nat_bool(ord_less_eq_nat),Na))) = hAPP_nat_nat(suc,Na) ) ).

tff(fact_122_Suc__inject,axiom,
    ! [X: nat,Y: nat] :
      ( ( hAPP_nat_nat(suc,X) = hAPP_nat_nat(suc,Y) )
     => ( X = Y ) ) ).

tff(fact_123_nat_Oinject,axiom,
    ! [Nat_3: nat,Nat_4: nat] :
      ( ( hAPP_nat_nat(suc,Nat_3) = hAPP_nat_nat(suc,Nat_4) )
    <=> ( Nat_3 = Nat_4 ) ) ).

tff(fact_124_Suc__n__not__n,axiom,
    ! [N: nat] : ( hAPP_nat_nat(suc,N) != N ) ).

tff(fact_125_n__not__Suc__n,axiom,
    ! [N: nat] : ( N != hAPP_nat_nat(suc,N) ) ).

tff(fact_126_le__antisym,axiom,
    ! [M_1: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M_1))
       => ( M_1 = N ) ) ) ).

tff(fact_127_le__trans,axiom,
    ! [K_1: nat,I_1: nat,J: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J),K_1))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),K_1)) ) ) ).

tff(fact_128_eq__imp__le,axiom,
    ! [M_1: nat,N: nat] :
      ( ( M_1 = N )
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N)) ) ).

tff(fact_129_nat__le__linear,axiom,
    ! [M_1: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N))
      | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M_1)) ) ).

tff(fact_130_le__refl,axiom,
    ! [N: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),N)) ).

tff(fact_131_diff__commute,axiom,
    ! [I_1: nat,J: nat,K_1: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I_1),J)),K_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I_1),K_1)),J) ) ).

tff(fact_132_finite__Collect__disjI,axiom,
    ! [Pa: fun_a_bool,Q_3: fun_a_bool] :
      ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,collect_a(cOMBS_a_bool_bool(cOMBB_1972296269bool_a(fdisj,Pa),Q_3))))
    <=> ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,collect_a(Pa)))
        & hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,collect_a(Q_3))) ) ) ).

tff(fact_133_finite__Collect__disjI,axiom,
    ! [Pa: fun_fu819804446l_bool,Q_3: fun_fu819804446l_bool] :
      ( hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,collect_fun_int_bool(cOMBS_1912792529l_bool(cOMBB_1974511970t_bool(fdisj,Pa),Q_3))))
    <=> ( hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,collect_fun_int_bool(Pa)))
        & hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,collect_fun_int_bool(Q_3))) ) ) ).

tff(fact_134_finite__Collect__disjI,axiom,
    ! [Pa: fun_fu425979586l_bool,Q_3: fun_fu425979586l_bool] :
      ( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,collect_fun_nat_bool(cOMBS_1187019125l_bool(cOMBB_444170502t_bool(fdisj,Pa),Q_3))))
    <=> ( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,collect_fun_nat_bool(Pa)))
        & hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,collect_fun_nat_bool(Q_3))) ) ) ).

tff(fact_135_finite__Collect__disjI,axiom,
    ! [Pa: fun_fu1430349052l_bool,Q_3: fun_fu1430349052l_bool] :
      ( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,collec1974731493e_bool(cOMBS_350070575l_bool(cOMBB_2095475776e_bool(fdisj,Pa),Q_3))))
    <=> ( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,collec1974731493e_bool(Pa)))
        & hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,collec1974731493e_bool(Q_3))) ) ) ).

tff(fact_136_finite__Collect__disjI,axiom,
    ! [Pa: fun_fun_a_bool_bool,Q_3: fun_fun_a_bool_bool] :
      ( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,collect_fun_a_bool(cOMBS_1035972772l_bool(cOMBB_338059395a_bool(fdisj,Pa),Q_3))))
    <=> ( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,collect_fun_a_bool(Pa)))
        & hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,collect_fun_a_bool(Q_3))) ) ) ).

tff(fact_137_finite__Collect__disjI,axiom,
    ! [Pa: fun_pname_bool,Q_3: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,collect_pname(cOMBS_568398431l_bool(cOMBB_675860798_pname(fdisj,Pa),Q_3))))
    <=> ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,collect_pname(Pa)))
        & hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,collect_pname(Q_3))) ) ) ).

tff(fact_138_finite__Collect__disjI,axiom,
    ! [Pa: fun_nat_bool,Q_3: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(cOMBS_nat_bool_bool(cOMBB_1015721476ol_nat(fdisj,Pa),Q_3))))
    <=> ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(Pa)))
        & hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(Q_3))) ) ) ).

tff(fact_139_finite__Collect__disjI,axiom,
    ! [Pa: fun_int_bool,Q_3: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,collect_int(cOMBS_int_bool_bool(cOMBB_1652995168ol_int(fdisj,Pa),Q_3))))
    <=> ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,collect_int(Pa)))
        & hBOOL(hAPP_f448129468l_bool(finite_finite_int,collect_int(Q_3))) ) ) ).

tff(fact_140_finite__insert,axiom,
    ! [A_2: fun_int_bool,A: fun_fu819804446l_bool] :
      ( hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,hAPP_f1948010709l_bool(hAPP_f587666462l_bool(insert_fun_int_bool,A_2),A)))
    <=> hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,A)) ) ).

tff(fact_141_finite__insert,axiom,
    ! [A_2: fun_nat_bool,A: fun_fu425979586l_bool] :
      ( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,hAPP_f1246832597l_bool(hAPP_f633452666l_bool(insert_fun_nat_bool,A_2),A)))
    <=> hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,A)) ) ).

tff(fact_142_finite__insert,axiom,
    ! [A_2: fun_pname_bool,A: fun_fu1430349052l_bool] :
      ( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,hAPP_f559147733l_bool(hAPP_f1320879424l_bool(insert1325755072e_bool,A_2),A)))
    <=> hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,A)) ) ).

tff(fact_143_finite__insert,axiom,
    ! [A_2: fun_a_bool,A: fun_fun_a_bool_bool] :
      ( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,hAPP_f2117159681l_bool(hAPP_f1531278575l_bool(insert_fun_a_bool,A_2),A)))
    <=> hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,A)) ) ).

tff(fact_144_finite__insert,axiom,
    ! [A_2: pname,A: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_2),A)))
    <=> hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A)) ) ).

tff(fact_145_finite__insert,axiom,
    ! [A_2: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_2),A)))
    <=> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A)) ) ).

tff(fact_146_finite__insert,axiom,
    ! [A_2: int,A: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_2),A)))
    <=> hBOOL(hAPP_f448129468l_bool(finite_finite_int,A)) ) ).

tff(fact_147_finite__insert,axiom,
    ! [A_2: x_a,A: fun_a_bool] :
      ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,A_2),A)))
    <=> hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A)) ) ).

tff(fact_148_finite__subset,axiom,
    ! [A: fun_fu819804446l_bool,B: fun_fu819804446l_bool] :
      ( hBOOL(hAPP_f215623910l_bool(hAPP_f317860125l_bool(ord_le1848167016l_bool,A),B))
     => ( hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,B))
       => hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,A)) ) ) ).

tff(fact_149_finite__subset,axiom,
    ! [A: fun_fu425979586l_bool,B: fun_fu425979586l_bool] :
      ( hBOOL(hAPP_f1637334154l_bool(hAPP_f1772781669l_bool(ord_le1454342156l_bool,A),B))
     => ( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,B))
       => hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,A)) ) ) ).

tff(fact_150_finite__subset,axiom,
    ! [A: fun_fu1430349052l_bool,B: fun_fu1430349052l_bool] :
      ( hBOOL(hAPP_f1935102916l_bool(hAPP_f510955609l_bool(ord_le675606854l_bool,A),B))
     => ( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,B))
       => hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,A)) ) ) ).

tff(fact_151_finite__subset,axiom,
    ! [A: fun_fun_a_bool_bool,B: fun_fun_a_bool_bool] :
      ( hBOOL(hAPP_f621171935l_bool(hAPP_f1434722111l_bool(ord_le1375614389l_bool,A),B))
     => ( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,B))
       => hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,A)) ) ) ).

tff(fact_152_finite__subset,axiom,
    ! [A: fun_a_bool,B: fun_a_bool] :
      ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),B))
     => ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,B))
       => hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A)) ) ) ).

tff(fact_153_finite__subset,axiom,
    ! [A: fun_pname_bool,B: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A),B))
     => ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,B))
       => hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A)) ) ) ).

tff(fact_154_finite__subset,axiom,
    ! [A: fun_nat_bool,B: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B))
     => ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B))
       => hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A)) ) ) ).

tff(fact_155_finite__subset,axiom,
    ! [A: fun_int_bool,B: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),B))
     => ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,B))
       => hBOOL(hAPP_f448129468l_bool(finite_finite_int,A)) ) ) ).

tff(fact_156_rev__finite__subset,axiom,
    ! [A: fun_fu819804446l_bool,B: fun_fu819804446l_bool] :
      ( hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,B))
     => ( hBOOL(hAPP_f215623910l_bool(hAPP_f317860125l_bool(ord_le1848167016l_bool,A),B))
       => hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,A)) ) ) ).

tff(fact_157_rev__finite__subset,axiom,
    ! [A: fun_fu425979586l_bool,B: fun_fu425979586l_bool] :
      ( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,B))
     => ( hBOOL(hAPP_f1637334154l_bool(hAPP_f1772781669l_bool(ord_le1454342156l_bool,A),B))
       => hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,A)) ) ) ).

tff(fact_158_rev__finite__subset,axiom,
    ! [A: fun_fu1430349052l_bool,B: fun_fu1430349052l_bool] :
      ( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,B))
     => ( hBOOL(hAPP_f1935102916l_bool(hAPP_f510955609l_bool(ord_le675606854l_bool,A),B))
       => hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,A)) ) ) ).

tff(fact_159_rev__finite__subset,axiom,
    ! [A: fun_fun_a_bool_bool,B: fun_fun_a_bool_bool] :
      ( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,B))
     => ( hBOOL(hAPP_f621171935l_bool(hAPP_f1434722111l_bool(ord_le1375614389l_bool,A),B))
       => hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,A)) ) ) ).

tff(fact_160_rev__finite__subset,axiom,
    ! [A: fun_a_bool,B: fun_a_bool] :
      ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,B))
     => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),B))
       => hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A)) ) ) ).

tff(fact_161_rev__finite__subset,axiom,
    ! [A: fun_pname_bool,B: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,B))
     => ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A),B))
       => hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A)) ) ) ).

tff(fact_162_rev__finite__subset,axiom,
    ! [A: fun_nat_bool,B: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B))
       => hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A)) ) ) ).

tff(fact_163_rev__finite__subset,axiom,
    ! [A: fun_int_bool,B: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,B))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),B))
       => hBOOL(hAPP_f448129468l_bool(finite_finite_int,A)) ) ) ).

tff(fact_164_Suc__leD,axiom,
    ! [M_1: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,M_1)),N))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N)) ) ).

tff(fact_165_le__SucE,axiom,
    ! [M_1: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),hAPP_nat_nat(suc,N)))
     => ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N))
       => ( M_1 = hAPP_nat_nat(suc,N) ) ) ) ).

tff(fact_166_le__SucI,axiom,
    ! [M_1: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),hAPP_nat_nat(suc,N))) ) ).

tff(fact_167_Suc__le__mono,axiom,
    ! [Na: nat,M: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,Na)),hAPP_nat_nat(suc,M)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Na),M)) ) ).

tff(fact_168_le__Suc__eq,axiom,
    ! [M: nat,Na: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),hAPP_nat_nat(suc,Na)))
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),Na))
        | ( M = hAPP_nat_nat(suc,Na) ) ) ) ).

tff(fact_169_not__less__eq__eq,axiom,
    ! [M: nat,Na: nat] :
      ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),Na))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,Na)),M)) ) ).

tff(fact_170_Suc__n__not__le__n,axiom,
    ! [N: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,N)),N)) ).

tff(fact_171_Suc__diff__diff,axiom,
    ! [M_1: nat,N: nat,K_1: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,M_1)),N)),hAPP_nat_nat(suc,K_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),N)),K_1) ) ).

tff(fact_172_diff__Suc__Suc,axiom,
    ! [M_1: nat,N: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,M_1)),hAPP_nat_nat(suc,N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),N) ) ).

tff(fact_173_le__diff__iff,axiom,
    ! [Na: nat,K: nat,M: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),M))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),Na))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),K)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,Na),K)))
        <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),Na)) ) ) ) ).

tff(fact_174_Nat_Odiff__diff__eq,axiom,
    ! [N: nat,K_1: nat,M_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),M_1))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),N))
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),K_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),K_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),N) ) ) ) ).

tff(fact_175_eq__diff__iff,axiom,
    ! [Na: nat,K: nat,M: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),M))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),Na))
       => ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),K) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,Na),K) )
        <=> ( M = Na ) ) ) ) ).

tff(fact_176_diff__diff__cancel,axiom,
    ! [I_1: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),N))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),I_1)) = I_1 ) ) ).

tff(fact_177_diff__le__mono,axiom,
    ! [L_1: nat,M_1: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),L_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),L_1))) ) ).

tff(fact_178_diff__le__mono2,axiom,
    ! [L_1: nat,M_1: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,L_1),N)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,L_1),M_1))) ) ).

tff(fact_179_diff__le__self,axiom,
    ! [M_1: nat,N: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),N)),M_1)) ).

tff(fact_180_finite__surj,axiom,
    ! [B: fun_a_bool,F: fun_a_a,A: fun_a_bool] :
      ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A))
     => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B),image_a_a(F,A)))
       => hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,B)) ) ) ).

tff(fact_181_finite__surj,axiom,
    ! [B: fun_a_bool,F: fun_fun_int_bool_a,A: fun_fu819804446l_bool] :
      ( hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,A))
     => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B),image_fun_int_bool_a(F,A)))
       => hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,B)) ) ) ).

tff(fact_182_finite__surj,axiom,
    ! [B: fun_a_bool,F: fun_fun_nat_bool_a,A: fun_fu425979586l_bool] :
      ( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,A))
     => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B),image_fun_nat_bool_a(F,A)))
       => hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,B)) ) ) ).

tff(fact_183_finite__surj,axiom,
    ! [B: fun_a_bool,F: fun_fun_pname_bool_a,A: fun_fu1430349052l_bool] :
      ( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,A))
     => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B),image_876012084bool_a(F,A)))
       => hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,B)) ) ) ).

tff(fact_184_finite__surj,axiom,
    ! [B: fun_a_bool,F: fun_fun_a_bool_a,A: fun_fun_a_bool_bool] :
      ( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,A))
     => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B),image_fun_a_bool_a(F,A)))
       => hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,B)) ) ) ).

tff(fact_185_finite__surj,axiom,
    ! [B: fun_fu819804446l_bool,F: fun_pn665169337t_bool,A: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A))
     => ( hBOOL(hAPP_f215623910l_bool(hAPP_f317860125l_bool(ord_le1848167016l_bool,B),image_1512837979t_bool(F,A)))
       => hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,B)) ) ) ).

tff(fact_186_finite__surj,axiom,
    ! [B: fun_fu425979586l_bool,F: fun_pn406123357t_bool,A: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A))
     => ( hBOOL(hAPP_f1637334154l_bool(hAPP_f1772781669l_bool(ord_le1454342156l_bool,B),image_2129980159t_bool(F,A)))
       => hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,B)) ) ) ).

tff(fact_187_finite__surj,axiom,
    ! [B: fun_fu1430349052l_bool,F: fun_pn800050071e_bool,A: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A))
     => ( hBOOL(hAPP_f1935102916l_bool(hAPP_f510955609l_bool(ord_le675606854l_bool,B),image_47868345e_bool(F,A)))
       => hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,B)) ) ) ).

tff(fact_188_finite__surj,axiom,
    ! [B: fun_fun_a_bool_bool,F: fun_pname_fun_a_bool,A: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A))
     => ( hBOOL(hAPP_f621171935l_bool(hAPP_f1434722111l_bool(ord_le1375614389l_bool,B),image_112932426a_bool(F,A)))
       => hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,B)) ) ) ).

tff(fact_189_finite__surj,axiom,
    ! [B: fun_pname_bool,F: fun_pname_pname,A: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A))
     => ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,B),image_pname_pname(F,A)))
       => hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,B)) ) ) ).

tff(fact_190_finite__surj,axiom,
    ! [B: fun_a_bool,F: fun_nat_a,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B),image_nat_a(F,A)))
       => hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,B)) ) ) ).

tff(fact_191_finite__surj,axiom,
    ! [B: fun_fu819804446l_bool,F: fun_nat_fun_int_bool,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => ( hBOOL(hAPP_f215623910l_bool(hAPP_f317860125l_bool(ord_le1848167016l_bool,B),image_1556378401t_bool(F,A)))
       => hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,B)) ) ) ).

tff(fact_192_finite__surj,axiom,
    ! [B: fun_fu425979586l_bool,F: fun_nat_fun_nat_bool,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => ( hBOOL(hAPP_f1637334154l_bool(hAPP_f1772781669l_bool(ord_le1454342156l_bool,B),image_26036933t_bool(F,A)))
       => hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,B)) ) ) ).

tff(fact_193_finite__surj,axiom,
    ! [B: fun_fu1430349052l_bool,F: fun_na936072029e_bool,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => ( hBOOL(hAPP_f1935102916l_bool(hAPP_f510955609l_bool(ord_le675606854l_bool,B),image_1655916159e_bool(F,A)))
       => hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,B)) ) ) ).

tff(fact_194_finite__surj,axiom,
    ! [B: fun_fun_a_bool_bool,F: fun_nat_fun_a_bool,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => ( hBOOL(hAPP_f621171935l_bool(hAPP_f1434722111l_bool(ord_le1375614389l_bool,B),image_nat_fun_a_bool(F,A)))
       => hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,B)) ) ) ).

tff(fact_195_finite__surj,axiom,
    ! [B: fun_pname_bool,F: fun_nat_pname,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,B),image_nat_pname(F,A)))
       => hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,B)) ) ) ).

tff(fact_196_finite__surj,axiom,
    ! [B: fun_a_bool,F: fun_int_a,A: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
     => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B),image_int_a(F,A)))
       => hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,B)) ) ) ).

tff(fact_197_finite__surj,axiom,
    ! [B: fun_fu819804446l_bool,F: fun_int_fun_int_bool,A: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
     => ( hBOOL(hAPP_f215623910l_bool(hAPP_f317860125l_bool(ord_le1848167016l_bool,B),image_275383677t_bool(F,A)))
       => hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,B)) ) ) ).

tff(fact_198_finite__surj,axiom,
    ! [B: fun_fu425979586l_bool,F: fun_int_fun_nat_bool,A: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
     => ( hBOOL(hAPP_f1637334154l_bool(hAPP_f1772781669l_bool(ord_le1454342156l_bool,B),image_892525857t_bool(F,A)))
       => hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,B)) ) ) ).

tff(fact_199_finite__surj,axiom,
    ! [B: fun_fu1430349052l_bool,F: fun_in2065004473e_bool,A: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
     => ( hBOOL(hAPP_f1935102916l_bool(hAPP_f510955609l_bool(ord_le675606854l_bool,B),image_637364955e_bool(F,A)))
       => hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,B)) ) ) ).

tff(fact_200_finite__surj,axiom,
    ! [B: fun_fun_a_bool_bool,F: fun_int_fun_a_bool,A: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
     => ( hBOOL(hAPP_f621171935l_bool(hAPP_f1434722111l_bool(ord_le1375614389l_bool,B),image_int_fun_a_bool(F,A)))
       => hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,B)) ) ) ).

tff(fact_201_finite__surj,axiom,
    ! [B: fun_pname_bool,F: fun_int_pname,A: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
     => ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,B),image_int_pname(F,A)))
       => hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,B)) ) ) ).

tff(fact_202_finite__surj,axiom,
    ! [B: fun_pname_bool,F: fun_a_pname,A: fun_a_bool] :
      ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A))
     => ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,B),image_a_pname(F,A)))
       => hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,B)) ) ) ).

tff(fact_203_finite__surj,axiom,
    ! [B: fun_pname_bool,F: fun_fu953875929_pname,A: fun_fu819804446l_bool] :
      ( hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,A))
     => ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,B),image_62782965_pname(F,A)))
       => hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,B)) ) ) ).

tff(fact_204_finite__surj,axiom,
    ! [B: fun_pname_bool,F: fun_fu665170229_pname,A: fun_fu425979586l_bool] :
      ( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,A))
     => ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,B),image_1921560913_pname(F,A)))
       => hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,B)) ) ) ).

tff(fact_205_finite__surj,axiom,
    ! [B: fun_pname_bool,F: fun_fu1499449723_pname,A: fun_fu1430349052l_bool] :
      ( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,A))
     => ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,B),image_1283814551_pname(F,A)))
       => hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,B)) ) ) ).

tff(fact_206_finite__surj,axiom,
    ! [B: fun_pname_bool,F: fun_fun_a_bool_pname,A: fun_fun_a_bool_bool] :
      ( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,A))
     => ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,B),image_1854862208_pname(F,A)))
       => hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,B)) ) ) ).

tff(fact_207_finite__surj,axiom,
    ! [B: fun_nat_bool,F: fun_a_nat,A: fun_a_bool] :
      ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B),image_a_nat(F,A)))
       => hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B)) ) ) ).

tff(fact_208_finite__surj,axiom,
    ! [B: fun_nat_bool,F: fun_fun_int_bool_nat,A: fun_fu819804446l_bool] :
      ( hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,A))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B),image_1431733819ol_nat(F,A)))
       => hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B)) ) ) ).

tff(fact_209_finite__surj,axiom,
    ! [B: fun_nat_bool,F: fun_fun_nat_bool_nat,A: fun_fu425979586l_bool] :
      ( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,A))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B),image_496248727ol_nat(F,A)))
       => hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B)) ) ) ).

tff(fact_210_finite__surj,axiom,
    ! [B: fun_nat_bool,F: fun_fu1668467777ol_nat,A: fun_fu1430349052l_bool] :
      ( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,A))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B),image_1551609309ol_nat(F,A)))
       => hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B)) ) ) ).

tff(fact_211_finite__surj,axiom,
    ! [B: fun_nat_bool,F: fun_fun_a_bool_nat,A: fun_fun_a_bool_bool] :
      ( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,A))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B),image_fun_a_bool_nat(F,A)))
       => hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B)) ) ) ).

tff(fact_212_finite__surj,axiom,
    ! [B: fun_int_bool,F: fun_a_int,A: fun_a_bool] :
      ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,B),image_a_int(F,A)))
       => hBOOL(hAPP_f448129468l_bool(finite_finite_int,B)) ) ) ).

tff(fact_213_finite__surj,axiom,
    ! [B: fun_int_bool,F: fun_fun_int_bool_int,A: fun_fu819804446l_bool] :
      ( hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,A))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,B),image_2069007511ol_int(F,A)))
       => hBOOL(hAPP_f448129468l_bool(finite_finite_int,B)) ) ) ).

tff(fact_214_finite__surj,axiom,
    ! [B: fun_int_bool,F: fun_fun_nat_bool_int,A: fun_fu425979586l_bool] :
      ( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,A))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,B),image_1133522419ol_int(F,A)))
       => hBOOL(hAPP_f448129468l_bool(finite_finite_int,B)) ) ) ).

tff(fact_215_finite__surj,axiom,
    ! [B: fun_int_bool,F: fun_fu1830380701ol_int,A: fun_fu1430349052l_bool] :
      ( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,A))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,B),image_41399353ol_int(F,A)))
       => hBOOL(hAPP_f448129468l_bool(finite_finite_int,B)) ) ) ).

tff(fact_216_finite__surj,axiom,
    ! [B: fun_int_bool,F: fun_fun_a_bool_int,A: fun_fun_a_bool_bool] :
      ( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,A))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,B),image_fun_a_bool_int(F,A)))
       => hBOOL(hAPP_f448129468l_bool(finite_finite_int,B)) ) ) ).

tff(fact_217_finite__surj,axiom,
    ! [B: fun_a_bool,F: fun_pname_a,A: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A))
     => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B),image_pname_a(F,A)))
       => hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,B)) ) ) ).

tff(fact_218_finite__surj,axiom,
    ! [B: fun_int_bool,F: fun_nat_int,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,B),image_nat_int(F,A)))
       => hBOOL(hAPP_f448129468l_bool(finite_finite_int,B)) ) ) ).

tff(fact_219_finite__subset__image,axiom,
    ! [F: fun_fun_int_bool_a,A: fun_fu819804446l_bool,B: fun_a_bool] :
      ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,B))
     => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B),image_fun_int_bool_a(F,A)))
       => ? [C_16: fun_fu819804446l_bool] :
            ( hBOOL(hAPP_f215623910l_bool(hAPP_f317860125l_bool(ord_le1848167016l_bool,C_16),A))
            & hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,C_16))
            & ( B = image_fun_int_bool_a(F,C_16) ) ) ) ) ).

tff(fact_220_finite__subset__image,axiom,
    ! [F: fun_fun_nat_bool_a,A: fun_fu425979586l_bool,B: fun_a_bool] :
      ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,B))
     => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B),image_fun_nat_bool_a(F,A)))
       => ? [C_16: fun_fu425979586l_bool] :
            ( hBOOL(hAPP_f1637334154l_bool(hAPP_f1772781669l_bool(ord_le1454342156l_bool,C_16),A))
            & hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,C_16))
            & ( B = image_fun_nat_bool_a(F,C_16) ) ) ) ) ).

tff(fact_221_finite__subset__image,axiom,
    ! [F: fun_fun_pname_bool_a,A: fun_fu1430349052l_bool,B: fun_a_bool] :
      ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,B))
     => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B),image_876012084bool_a(F,A)))
       => ? [C_16: fun_fu1430349052l_bool] :
            ( hBOOL(hAPP_f1935102916l_bool(hAPP_f510955609l_bool(ord_le675606854l_bool,C_16),A))
            & hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,C_16))
            & ( B = image_876012084bool_a(F,C_16) ) ) ) ) ).

tff(fact_222_finite__subset__image,axiom,
    ! [F: fun_fun_a_bool_a,A: fun_fun_a_bool_bool,B: fun_a_bool] :
      ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,B))
     => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B),image_fun_a_bool_a(F,A)))
       => ? [C_16: fun_fun_a_bool_bool] :
            ( hBOOL(hAPP_f621171935l_bool(hAPP_f1434722111l_bool(ord_le1375614389l_bool,C_16),A))
            & hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,C_16))
            & ( B = image_fun_a_bool_a(F,C_16) ) ) ) ) ).

tff(fact_223_finite__subset__image,axiom,
    ! [F: fun_a_a,A: fun_a_bool,B: fun_a_bool] :
      ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,B))
     => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B),image_a_a(F,A)))
       => ? [C_16: fun_a_bool] :
            ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,C_16),A))
            & hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,C_16))
            & ( B = image_a_a(F,C_16) ) ) ) ) ).

tff(fact_224_finite__subset__image,axiom,
    ! [F: fun_a_fun_int_bool,A: fun_a_bool,B: fun_fu819804446l_bool] :
      ( hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,B))
     => ( hBOOL(hAPP_f215623910l_bool(hAPP_f317860125l_bool(ord_le1848167016l_bool,B),image_a_fun_int_bool(F,A)))
       => ? [C_16: fun_a_bool] :
            ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,C_16),A))
            & hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,C_16))
            & ( B = image_a_fun_int_bool(F,C_16) ) ) ) ) ).

tff(fact_225_finite__subset__image,axiom,
    ! [F: fun_a_fun_nat_bool,A: fun_a_bool,B: fun_fu425979586l_bool] :
      ( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,B))
     => ( hBOOL(hAPP_f1637334154l_bool(hAPP_f1772781669l_bool(ord_le1454342156l_bool,B),image_a_fun_nat_bool(F,A)))
       => ? [C_16: fun_a_bool] :
            ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,C_16),A))
            & hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,C_16))
            & ( B = image_a_fun_nat_bool(F,C_16) ) ) ) ) ).

tff(fact_226_finite__subset__image,axiom,
    ! [F: fun_a_fun_pname_bool,A: fun_a_bool,B: fun_fu1430349052l_bool] :
      ( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,B))
     => ( hBOOL(hAPP_f1935102916l_bool(hAPP_f510955609l_bool(ord_le675606854l_bool,B),image_819518260e_bool(F,A)))
       => ? [C_16: fun_a_bool] :
            ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,C_16),A))
            & hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,C_16))
            & ( B = image_819518260e_bool(F,C_16) ) ) ) ) ).

tff(fact_227_finite__subset__image,axiom,
    ! [F: fun_a_fun_a_bool,A: fun_a_bool,B: fun_fun_a_bool_bool] :
      ( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,B))
     => ( hBOOL(hAPP_f621171935l_bool(hAPP_f1434722111l_bool(ord_le1375614389l_bool,B),image_a_fun_a_bool(F,A)))
       => ? [C_16: fun_a_bool] :
            ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,C_16),A))
            & hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,C_16))
            & ( B = image_a_fun_a_bool(F,C_16) ) ) ) ) ).

tff(fact_228_finite__subset__image,axiom,
    ! [F: fun_a_pname,A: fun_a_bool,B: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,B))
     => ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,B),image_a_pname(F,A)))
       => ? [C_16: fun_a_bool] :
            ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,C_16),A))
            & hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,C_16))
            & ( B = image_a_pname(F,C_16) ) ) ) ) ).

tff(fact_229_finite__subset__image,axiom,
    ! [F: fun_fu953875929_pname,A: fun_fu819804446l_bool,B: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,B))
     => ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,B),image_62782965_pname(F,A)))
       => ? [C_16: fun_fu819804446l_bool] :
            ( hBOOL(hAPP_f215623910l_bool(hAPP_f317860125l_bool(ord_le1848167016l_bool,C_16),A))
            & hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,C_16))
            & ( B = image_62782965_pname(F,C_16) ) ) ) ) ).

tff(fact_230_finite__subset__image,axiom,
    ! [F: fun_fu665170229_pname,A: fun_fu425979586l_bool,B: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,B))
     => ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,B),image_1921560913_pname(F,A)))
       => ? [C_16: fun_fu425979586l_bool] :
            ( hBOOL(hAPP_f1637334154l_bool(hAPP_f1772781669l_bool(ord_le1454342156l_bool,C_16),A))
            & hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,C_16))
            & ( B = image_1921560913_pname(F,C_16) ) ) ) ) ).

tff(fact_231_finite__subset__image,axiom,
    ! [F: fun_fu1499449723_pname,A: fun_fu1430349052l_bool,B: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,B))
     => ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,B),image_1283814551_pname(F,A)))
       => ? [C_16: fun_fu1430349052l_bool] :
            ( hBOOL(hAPP_f1935102916l_bool(hAPP_f510955609l_bool(ord_le675606854l_bool,C_16),A))
            & hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,C_16))
            & ( B = image_1283814551_pname(F,C_16) ) ) ) ) ).

tff(fact_232_finite__subset__image,axiom,
    ! [F: fun_fun_a_bool_pname,A: fun_fun_a_bool_bool,B: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,B))
     => ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,B),image_1854862208_pname(F,A)))
       => ? [C_16: fun_fun_a_bool_bool] :
            ( hBOOL(hAPP_f621171935l_bool(hAPP_f1434722111l_bool(ord_le1375614389l_bool,C_16),A))
            & hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,C_16))
            & ( B = image_1854862208_pname(F,C_16) ) ) ) ) ).

tff(fact_233_finite__subset__image,axiom,
    ! [F: fun_a_nat,A: fun_a_bool,B: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B),image_a_nat(F,A)))
       => ? [C_16: fun_a_bool] :
            ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,C_16),A))
            & hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,C_16))
            & ( B = image_a_nat(F,C_16) ) ) ) ) ).

tff(fact_234_finite__subset__image,axiom,
    ! [F: fun_fun_int_bool_nat,A: fun_fu819804446l_bool,B: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B),image_1431733819ol_nat(F,A)))
       => ? [C_16: fun_fu819804446l_bool] :
            ( hBOOL(hAPP_f215623910l_bool(hAPP_f317860125l_bool(ord_le1848167016l_bool,C_16),A))
            & hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,C_16))
            & ( B = image_1431733819ol_nat(F,C_16) ) ) ) ) ).

tff(fact_235_finite__subset__image,axiom,
    ! [F: fun_fun_nat_bool_nat,A: fun_fu425979586l_bool,B: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B),image_496248727ol_nat(F,A)))
       => ? [C_16: fun_fu425979586l_bool] :
            ( hBOOL(hAPP_f1637334154l_bool(hAPP_f1772781669l_bool(ord_le1454342156l_bool,C_16),A))
            & hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,C_16))
            & ( B = image_496248727ol_nat(F,C_16) ) ) ) ) ).

tff(fact_236_finite__subset__image,axiom,
    ! [F: fun_fu1668467777ol_nat,A: fun_fu1430349052l_bool,B: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B),image_1551609309ol_nat(F,A)))
       => ? [C_16: fun_fu1430349052l_bool] :
            ( hBOOL(hAPP_f1935102916l_bool(hAPP_f510955609l_bool(ord_le675606854l_bool,C_16),A))
            & hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,C_16))
            & ( B = image_1551609309ol_nat(F,C_16) ) ) ) ) ).

tff(fact_237_finite__subset__image,axiom,
    ! [F: fun_fun_a_bool_nat,A: fun_fun_a_bool_bool,B: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B),image_fun_a_bool_nat(F,A)))
       => ? [C_16: fun_fun_a_bool_bool] :
            ( hBOOL(hAPP_f621171935l_bool(hAPP_f1434722111l_bool(ord_le1375614389l_bool,C_16),A))
            & hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,C_16))
            & ( B = image_fun_a_bool_nat(F,C_16) ) ) ) ) ).

tff(fact_238_finite__subset__image,axiom,
    ! [F: fun_pname_nat,A: fun_pname_bool,B: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B),image_pname_nat(F,A)))
       => ? [C_16: fun_pname_bool] :
            ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,C_16),A))
            & hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,C_16))
            & ( B = image_pname_nat(F,C_16) ) ) ) ) ).

tff(fact_239_finite__subset__image,axiom,
    ! [F: fun_a_int,A: fun_a_bool,B: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,B))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,B),image_a_int(F,A)))
       => ? [C_16: fun_a_bool] :
            ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,C_16),A))
            & hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,C_16))
            & ( B = image_a_int(F,C_16) ) ) ) ) ).

tff(fact_240_finite__subset__image,axiom,
    ! [F: fun_fun_int_bool_int,A: fun_fu819804446l_bool,B: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,B))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,B),image_2069007511ol_int(F,A)))
       => ? [C_16: fun_fu819804446l_bool] :
            ( hBOOL(hAPP_f215623910l_bool(hAPP_f317860125l_bool(ord_le1848167016l_bool,C_16),A))
            & hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,C_16))
            & ( B = image_2069007511ol_int(F,C_16) ) ) ) ) ).

tff(fact_241_finite__subset__image,axiom,
    ! [F: fun_fun_nat_bool_int,A: fun_fu425979586l_bool,B: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,B))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,B),image_1133522419ol_int(F,A)))
       => ? [C_16: fun_fu425979586l_bool] :
            ( hBOOL(hAPP_f1637334154l_bool(hAPP_f1772781669l_bool(ord_le1454342156l_bool,C_16),A))
            & hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,C_16))
            & ( B = image_1133522419ol_int(F,C_16) ) ) ) ) ).

tff(fact_242_finite__subset__image,axiom,
    ! [F: fun_fu1830380701ol_int,A: fun_fu1430349052l_bool,B: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,B))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,B),image_41399353ol_int(F,A)))
       => ? [C_16: fun_fu1430349052l_bool] :
            ( hBOOL(hAPP_f1935102916l_bool(hAPP_f510955609l_bool(ord_le675606854l_bool,C_16),A))
            & hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,C_16))
            & ( B = image_41399353ol_int(F,C_16) ) ) ) ) ).

tff(fact_243_finite__subset__image,axiom,
    ! [F: fun_fun_a_bool_int,A: fun_fun_a_bool_bool,B: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,B))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,B),image_fun_a_bool_int(F,A)))
       => ? [C_16: fun_fun_a_bool_bool] :
            ( hBOOL(hAPP_f621171935l_bool(hAPP_f1434722111l_bool(ord_le1375614389l_bool,C_16),A))
            & hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,C_16))
            & ( B = image_fun_a_bool_int(F,C_16) ) ) ) ) ).

tff(fact_244_finite__subset__image,axiom,
    ! [F: fun_pname_int,A: fun_pname_bool,B: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,B))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,B),image_pname_int(F,A)))
       => ? [C_16: fun_pname_bool] :
            ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,C_16),A))
            & hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,C_16))
            & ( B = image_pname_int(F,C_16) ) ) ) ) ).

tff(fact_245_finite__subset__image,axiom,
    ! [F: fun_pn665169337t_bool,A: fun_pname_bool,B: fun_fu819804446l_bool] :
      ( hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,B))
     => ( hBOOL(hAPP_f215623910l_bool(hAPP_f317860125l_bool(ord_le1848167016l_bool,B),image_1512837979t_bool(F,A)))
       => ? [C_16: fun_pname_bool] :
            ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,C_16),A))
            & hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,C_16))
            & ( B = image_1512837979t_bool(F,C_16) ) ) ) ) ).

tff(fact_246_finite__subset__image,axiom,
    ! [F: fun_pn406123357t_bool,A: fun_pname_bool,B: fun_fu425979586l_bool] :
      ( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,B))
     => ( hBOOL(hAPP_f1637334154l_bool(hAPP_f1772781669l_bool(ord_le1454342156l_bool,B),image_2129980159t_bool(F,A)))
       => ? [C_16: fun_pname_bool] :
            ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,C_16),A))
            & hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,C_16))
            & ( B = image_2129980159t_bool(F,C_16) ) ) ) ) ).

tff(fact_247_finite__subset__image,axiom,
    ! [F: fun_pn800050071e_bool,A: fun_pname_bool,B: fun_fu1430349052l_bool] :
      ( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,B))
     => ( hBOOL(hAPP_f1935102916l_bool(hAPP_f510955609l_bool(ord_le675606854l_bool,B),image_47868345e_bool(F,A)))
       => ? [C_16: fun_pname_bool] :
            ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,C_16),A))
            & hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,C_16))
            & ( B = image_47868345e_bool(F,C_16) ) ) ) ) ).

tff(fact_248_finite__subset__image,axiom,
    ! [F: fun_pname_fun_a_bool,A: fun_pname_bool,B: fun_fun_a_bool_bool] :
      ( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,B))
     => ( hBOOL(hAPP_f621171935l_bool(hAPP_f1434722111l_bool(ord_le1375614389l_bool,B),image_112932426a_bool(F,A)))
       => ? [C_16: fun_pname_bool] :
            ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,C_16),A))
            & hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,C_16))
            & ( B = image_112932426a_bool(F,C_16) ) ) ) ) ).

tff(fact_249_finite__subset__image,axiom,
    ! [F: fun_pname_pname,A: fun_pname_bool,B: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,B))
     => ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,B),image_pname_pname(F,A)))
       => ? [C_16: fun_pname_bool] :
            ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,C_16),A))
            & hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,C_16))
            & ( B = image_pname_pname(F,C_16) ) ) ) ) ).

tff(fact_250_finite__subset__image,axiom,
    ! [F: fun_nat_a,A: fun_nat_bool,B: fun_a_bool] :
      ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,B))
     => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B),image_nat_a(F,A)))
       => ? [C_16: fun_nat_bool] :
            ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,C_16),A))
            & hBOOL(hAPP_f54304608l_bool(finite_finite_nat,C_16))
            & ( B = image_nat_a(F,C_16) ) ) ) ) ).

tff(fact_251_finite__subset__image,axiom,
    ! [F: fun_nat_fun_int_bool,A: fun_nat_bool,B: fun_fu819804446l_bool] :
      ( hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,B))
     => ( hBOOL(hAPP_f215623910l_bool(hAPP_f317860125l_bool(ord_le1848167016l_bool,B),image_1556378401t_bool(F,A)))
       => ? [C_16: fun_nat_bool] :
            ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,C_16),A))
            & hBOOL(hAPP_f54304608l_bool(finite_finite_nat,C_16))
            & ( B = image_1556378401t_bool(F,C_16) ) ) ) ) ).

tff(fact_252_finite__subset__image,axiom,
    ! [F: fun_nat_fun_nat_bool,A: fun_nat_bool,B: fun_fu425979586l_bool] :
      ( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,B))
     => ( hBOOL(hAPP_f1637334154l_bool(hAPP_f1772781669l_bool(ord_le1454342156l_bool,B),image_26036933t_bool(F,A)))
       => ? [C_16: fun_nat_bool] :
            ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,C_16),A))
            & hBOOL(hAPP_f54304608l_bool(finite_finite_nat,C_16))
            & ( B = image_26036933t_bool(F,C_16) ) ) ) ) ).

tff(fact_253_finite__subset__image,axiom,
    ! [F: fun_na936072029e_bool,A: fun_nat_bool,B: fun_fu1430349052l_bool] :
      ( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,B))
     => ( hBOOL(hAPP_f1935102916l_bool(hAPP_f510955609l_bool(ord_le675606854l_bool,B),image_1655916159e_bool(F,A)))
       => ? [C_16: fun_nat_bool] :
            ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,C_16),A))
            & hBOOL(hAPP_f54304608l_bool(finite_finite_nat,C_16))
            & ( B = image_1655916159e_bool(F,C_16) ) ) ) ) ).

tff(fact_254_finite__subset__image,axiom,
    ! [F: fun_nat_fun_a_bool,A: fun_nat_bool,B: fun_fun_a_bool_bool] :
      ( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,B))
     => ( hBOOL(hAPP_f621171935l_bool(hAPP_f1434722111l_bool(ord_le1375614389l_bool,B),image_nat_fun_a_bool(F,A)))
       => ? [C_16: fun_nat_bool] :
            ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,C_16),A))
            & hBOOL(hAPP_f54304608l_bool(finite_finite_nat,C_16))
            & ( B = image_nat_fun_a_bool(F,C_16) ) ) ) ) ).

tff(fact_255_finite__subset__image,axiom,
    ! [F: fun_nat_pname,A: fun_nat_bool,B: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,B))
     => ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,B),image_nat_pname(F,A)))
       => ? [C_16: fun_nat_bool] :
            ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,C_16),A))
            & hBOOL(hAPP_f54304608l_bool(finite_finite_nat,C_16))
            & ( B = image_nat_pname(F,C_16) ) ) ) ) ).

tff(fact_256_finite__subset__image,axiom,
    ! [F: fun_int_a,A: fun_int_bool,B: fun_a_bool] :
      ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,B))
     => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B),image_int_a(F,A)))
       => ? [C_16: fun_int_bool] :
            ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,C_16),A))
            & hBOOL(hAPP_f448129468l_bool(finite_finite_int,C_16))
            & ( B = image_int_a(F,C_16) ) ) ) ) ).

tff(fact_257_finite__subset__image,axiom,
    ! [F: fun_int_fun_int_bool,A: fun_int_bool,B: fun_fu819804446l_bool] :
      ( hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,B))
     => ( hBOOL(hAPP_f215623910l_bool(hAPP_f317860125l_bool(ord_le1848167016l_bool,B),image_275383677t_bool(F,A)))
       => ? [C_16: fun_int_bool] :
            ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,C_16),A))
            & hBOOL(hAPP_f448129468l_bool(finite_finite_int,C_16))
            & ( B = image_275383677t_bool(F,C_16) ) ) ) ) ).

tff(fact_258_finite__subset__image,axiom,
    ! [F: fun_int_fun_nat_bool,A: fun_int_bool,B: fun_fu425979586l_bool] :
      ( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,B))
     => ( hBOOL(hAPP_f1637334154l_bool(hAPP_f1772781669l_bool(ord_le1454342156l_bool,B),image_892525857t_bool(F,A)))
       => ? [C_16: fun_int_bool] :
            ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,C_16),A))
            & hBOOL(hAPP_f448129468l_bool(finite_finite_int,C_16))
            & ( B = image_892525857t_bool(F,C_16) ) ) ) ) ).

tff(fact_259_finite__subset__image,axiom,
    ! [F: fun_in2065004473e_bool,A: fun_int_bool,B: fun_fu1430349052l_bool] :
      ( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,B))
     => ( hBOOL(hAPP_f1935102916l_bool(hAPP_f510955609l_bool(ord_le675606854l_bool,B),image_637364955e_bool(F,A)))
       => ? [C_16: fun_int_bool] :
            ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,C_16),A))
            & hBOOL(hAPP_f448129468l_bool(finite_finite_int,C_16))
            & ( B = image_637364955e_bool(F,C_16) ) ) ) ) ).

tff(fact_260_finite__subset__image,axiom,
    ! [F: fun_int_fun_a_bool,A: fun_int_bool,B: fun_fun_a_bool_bool] :
      ( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,B))
     => ( hBOOL(hAPP_f621171935l_bool(hAPP_f1434722111l_bool(ord_le1375614389l_bool,B),image_int_fun_a_bool(F,A)))
       => ? [C_16: fun_int_bool] :
            ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,C_16),A))
            & hBOOL(hAPP_f448129468l_bool(finite_finite_int,C_16))
            & ( B = image_int_fun_a_bool(F,C_16) ) ) ) ) ).

tff(fact_261_finite__subset__image,axiom,
    ! [F: fun_int_pname,A: fun_int_bool,B: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,B))
     => ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,B),image_int_pname(F,A)))
       => ? [C_16: fun_int_bool] :
            ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,C_16),A))
            & hBOOL(hAPP_f448129468l_bool(finite_finite_int,C_16))
            & ( B = image_int_pname(F,C_16) ) ) ) ) ).

tff(fact_262_finite__subset__image,axiom,
    ! [F: fun_pname_a,A: fun_pname_bool,B: fun_a_bool] :
      ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,B))
     => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B),image_pname_a(F,A)))
       => ? [C_16: fun_pname_bool] :
            ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,C_16),A))
            & hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,C_16))
            & ( B = image_pname_a(F,C_16) ) ) ) ) ).

tff(fact_263_finite__subset__image,axiom,
    ! [F: fun_nat_int,A: fun_nat_bool,B: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,B))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,B),image_nat_int(F,A)))
       => ? [C_16: fun_nat_bool] :
            ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,C_16),A))
            & hBOOL(hAPP_f54304608l_bool(finite_finite_nat,C_16))
            & ( B = image_nat_int(F,C_16) ) ) ) ) ).

tff(fact_264_lift__Suc__mono__le,axiom,
    ! [Na: nat,N_3: nat,F: fun_na936072029e_bool] :
      ( ! [N_1: nat] : hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,hAPP_n1025906991e_bool(F,N_1)),hAPP_n1025906991e_bool(F,hAPP_nat_nat(suc,N_1))))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Na),N_3))
       => hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,hAPP_n1025906991e_bool(F,Na)),hAPP_n1025906991e_bool(F,N_3))) ) ) ).

tff(fact_265_lift__Suc__mono__le,axiom,
    ! [Na: nat,N_3: nat,F: fun_nat_bool] :
      ( ! [N_1: nat] : hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_nat_bool(F,N_1)),hAPP_nat_bool(F,hAPP_nat_nat(suc,N_1))))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Na),N_3))
       => hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_nat_bool(F,Na)),hAPP_nat_bool(F,N_3))) ) ) ).

tff(fact_266_lift__Suc__mono__le,axiom,
    ! [Na: nat,N_3: nat,F: fun_nat_fun_a_bool] :
      ( ! [N_1: nat] : hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,hAPP_nat_fun_a_bool(F,N_1)),hAPP_nat_fun_a_bool(F,hAPP_nat_nat(suc,N_1))))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Na),N_3))
       => hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,hAPP_nat_fun_a_bool(F,Na)),hAPP_nat_fun_a_bool(F,N_3))) ) ) ).

tff(fact_267_lift__Suc__mono__le,axiom,
    ! [Na: nat,N_3: nat,F: fun_nat_nat] :
      ( ! [N_1: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(F,N_1)),hAPP_nat_nat(F,hAPP_nat_nat(suc,N_1))))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Na),N_3))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(F,Na)),hAPP_nat_nat(F,N_3))) ) ) ).

tff(fact_268_lift__Suc__mono__le,axiom,
    ! [Na: nat,N_3: nat,F: fun_nat_int] :
      ( ! [N_1: nat] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(F,N_1)),hAPP_nat_int(F,hAPP_nat_nat(suc,N_1))))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Na),N_3))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(F,Na)),hAPP_nat_int(F,N_3))) ) ) ).

tff(fact_269_lift__Suc__mono__le,axiom,
    ! [Na: nat,N_3: nat,F: fun_nat_fun_nat_bool] :
      ( ! [N_1: nat] : hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,hAPP_n1699378549t_bool(F,N_1)),hAPP_n1699378549t_bool(F,hAPP_nat_nat(suc,N_1))))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Na),N_3))
       => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,hAPP_n1699378549t_bool(F,Na)),hAPP_n1699378549t_bool(F,N_3))) ) ) ).

tff(fact_270_lift__Suc__mono__le,axiom,
    ! [Na: nat,N_3: nat,F: fun_nat_fun_int_bool] :
      ( ! [N_1: nat] : hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,hAPP_n1082236369t_bool(F,N_1)),hAPP_n1082236369t_bool(F,hAPP_nat_nat(suc,N_1))))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Na),N_3))
       => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,hAPP_n1082236369t_bool(F,Na)),hAPP_n1082236369t_bool(F,N_3))) ) ) ).

tff(fact_271_pigeonhole__infinite,axiom,
    ! [F: fun_nat_fun_int_bool,A: fun_nat_bool] :
      ( ~ hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => ( hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,image_1556378401t_bool(F,A)))
       => ? [X_2: nat] :
            ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A))
            & ~ hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(cOMBS_nat_bool_bool(cOMBB_1015721476ol_nat(fconj,hAPP_f800510211t_bool(cOMBC_226598744l_bool(member_nat),A)),hAPP_f274826591t_bool(cOMBC_620423604l_bool(cOMBB_1123824708ol_nat(fequal_fun_int_bool,F)),hAPP_n1082236369t_bool(F,X_2)))))) ) ) ) ).

tff(fact_272_pigeonhole__infinite,axiom,
    ! [F: fun_nat_fun_nat_bool,A: fun_nat_bool] :
      ( ~ hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => ( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,image_26036933t_bool(F,A)))
       => ? [X_2: nat] :
            ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A))
            & ~ hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(cOMBS_nat_bool_bool(cOMBB_1015721476ol_nat(fconj,hAPP_f800510211t_bool(cOMBC_226598744l_bool(member_nat),A)),hAPP_f800510211t_bool(cOMBC_226598744l_bool(cOMBB_2026977092ol_nat(fequal_fun_nat_bool,F)),hAPP_n1699378549t_bool(F,X_2)))))) ) ) ) ).

tff(fact_273_pigeonhole__infinite,axiom,
    ! [F: fun_na936072029e_bool,A: fun_nat_bool] :
      ( ~ hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => ( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,image_1655916159e_bool(F,A)))
       => ? [X_2: nat] :
            ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A))
            & ~ hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(cOMBS_nat_bool_bool(cOMBB_1015721476ol_nat(fconj,hAPP_f800510211t_bool(cOMBC_226598744l_bool(member_nat),A)),hAPP_f1066163005t_bool(cOMBC_386238098l_bool(cOMBB_2117322052ol_nat(fequal533582459e_bool,F)),hAPP_n1025906991e_bool(F,X_2)))))) ) ) ) ).

tff(fact_274_pigeonhole__infinite,axiom,
    ! [F: fun_nat_fun_a_bool,A: fun_nat_bool] :
      ( ~ hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => ( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,image_nat_fun_a_bool(F,A)))
       => ? [X_2: nat] :
            ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A))
            & ~ hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(cOMBS_nat_bool_bool(cOMBB_1015721476ol_nat(fconj,hAPP_f800510211t_bool(cOMBC_226598744l_bool(member_nat),A)),hAPP_f282463732t_bool(cOMBC_1928494297l_bool(cOMBB_1823939024ol_nat(fequal_fun_a_bool,F)),hAPP_nat_fun_a_bool(F,X_2)))))) ) ) ) ).

tff(fact_275_pigeonhole__infinite,axiom,
    ! [F: fun_int_a,A: fun_int_bool] :
      ( ~ hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
     => ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,image_int_a(F,A)))
       => ? [X_2: int] :
            ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),A))
            & ~ hBOOL(hAPP_f448129468l_bool(finite_finite_int,collect_int(cOMBS_int_bool_bool(cOMBB_1652995168ol_int(fconj,hAPP_f1805168059t_bool(cOMBC_94739984l_bool(member_int),A)),hAPP_a_fun_int_bool(cOMBC_int_a_bool(cOMBB_375279840ol_int(fequal_a,F)),hAPP_int_a(F,X_2)))))) ) ) ) ).

tff(fact_276_pigeonhole__infinite,axiom,
    ! [F: fun_int_fun_int_bool,A: fun_int_bool] :
      ( ~ hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
     => ( hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,image_275383677t_bool(F,A)))
       => ? [X_2: int] :
            ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),A))
            & ~ hBOOL(hAPP_f448129468l_bool(finite_finite_int,collect_int(cOMBS_int_bool_bool(cOMBB_1652995168ol_int(fconj,hAPP_f1805168059t_bool(cOMBC_94739984l_bool(member_int),A)),hAPP_f1805168059t_bool(cOMBC_94739984l_bool(cOMBB_1761098400ol_int(fequal_fun_int_bool,F)),hAPP_i1948725293t_bool(F,X_2)))))) ) ) ) ).

tff(fact_277_pigeonhole__infinite,axiom,
    ! [F: fun_int_fun_nat_bool,A: fun_int_bool] :
      ( ~ hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
     => ( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,image_892525857t_bool(F,A)))
       => ? [X_2: int] :
            ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),A))
            & ~ hBOOL(hAPP_f448129468l_bool(finite_finite_int,collect_int(cOMBS_int_bool_bool(cOMBB_1652995168ol_int(fconj,hAPP_f1805168059t_bool(cOMBC_94739984l_bool(member_int),A)),hAPP_f183368031t_bool(cOMBC_1848398772l_bool(cOMBB_516767136ol_int(fequal_fun_nat_bool,F)),hAPP_i418383825t_bool(F,X_2)))))) ) ) ) ).

tff(fact_278_pigeonhole__infinite,axiom,
    ! [F: fun_in2065004473e_bool,A: fun_int_bool] :
      ( ~ hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
     => ( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,image_637364955e_bool(F,A)))
       => ? [X_2: int] :
            ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),A))
            & ~ hBOOL(hAPP_f448129468l_bool(finite_finite_int,collect_int(cOMBS_int_bool_bool(cOMBB_1652995168ol_int(fconj,hAPP_f1805168059t_bool(cOMBC_94739984l_bool(member_int),A)),hAPP_f449020825t_bool(cOMBC_316919022l_bool(cOMBB_607112096ol_int(fequal533582459e_bool,F)),hAPP_i7355787e_bool(F,X_2)))))) ) ) ) ).

tff(fact_279_pigeonhole__infinite,axiom,
    ! [F: fun_int_fun_a_bool,A: fun_int_bool] :
      ( ~ hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
     => ( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,image_int_fun_a_bool(F,A)))
       => ? [X_2: int] :
            ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),A))
            & ~ hBOOL(hAPP_f448129468l_bool(finite_finite_int,collect_int(cOMBS_int_bool_bool(cOMBB_1652995168ol_int(fconj,hAPP_f1805168059t_bool(cOMBC_94739984l_bool(member_int),A)),hAPP_f1812805200t_bool(cOMBC_1753430397l_bool(cOMBB_313729068ol_int(fequal_fun_a_bool,F)),hAPP_int_fun_a_bool(F,X_2)))))) ) ) ) ).

tff(fact_280_pigeonhole__infinite,axiom,
    ! [F: fun_pname_a,A: fun_pname_bool] :
      ( ~ hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A))
     => ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,image_pname_a(F,A)))
       => ? [X_2: pname] :
            ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),A))
            & ~ hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,collect_pname(cOMBS_568398431l_bool(cOMBB_675860798_pname(fconj,hAPP_f759274231e_bool(cOMBC_1058051404l_bool(member_pname),A)),hAPP_a93125764e_bool(cOMBC_pname_a_bool(cOMBB_1897541054_pname(fequal_a,F)),hAPP_pname_a(F,X_2)))))) ) ) ) ).

tff(fact_281_pigeonhole__infinite,axiom,
    ! [F: fun_nat_int,A: fun_nat_bool] :
      ( ~ hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,image_nat_int(F,A)))
       => ? [X_2: nat] :
            ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A))
            & ~ hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(cOMBS_nat_bool_bool(cOMBB_1015721476ol_nat(fconj,hAPP_f800510211t_bool(cOMBC_226598744l_bool(member_nat),A)),hAPP_i418383825t_bool(cOMBC_nat_int_bool(cOMBB_1628441366ol_nat(fequal_int,F)),hAPP_nat_int(F,X_2)))))) ) ) ) ).

tff(fact_282_image__eqI,axiom,
    ! [A: fun_nat_bool,B_3: int,F: fun_nat_int,X_1: nat] :
      ( ( B_3 = hAPP_nat_int(F,X_1) )
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
       => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B_3),image_nat_int(F,A))) ) ) ).

tff(fact_283_image__eqI,axiom,
    ! [A: fun_pname_bool,B_3: x_a,F: fun_pname_a,X_1: pname] :
      ( ( B_3 = hAPP_pname_a(F,X_1) )
     => ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_1),A))
       => hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,B_3),image_pname_a(F,A))) ) ) ).

tff(fact_284_equalityI,axiom,
    ! [A: fun_int_bool,B: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),B))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,B),A))
       => ( A = B ) ) ) ).

tff(fact_285_equalityI,axiom,
    ! [A: fun_nat_bool,B: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B),A))
       => ( A = B ) ) ) ).

tff(fact_286_equalityI,axiom,
    ! [A: fun_a_bool,B: fun_a_bool] :
      ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),B))
     => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B),A))
       => ( A = B ) ) ) ).

tff(fact_287_subsetD,axiom,
    ! [C_6: int,A: fun_int_bool,B: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),B))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,C_6),A))
       => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,C_6),B)) ) ) ).

tff(fact_288_subsetD,axiom,
    ! [C_6: nat,A: fun_nat_bool,B: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_6),A))
       => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_6),B)) ) ) ).

tff(fact_289_subsetD,axiom,
    ! [C_6: x_a,A: fun_a_bool,B: fun_a_bool] :
      ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),B))
     => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,C_6),A))
       => hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,C_6),B)) ) ) ).

tff(fact_290_subsetD,axiom,
    ! [C_6: pname,A: fun_pname_bool,B: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A),B))
     => ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C_6),A))
       => hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C_6),B)) ) ) ).

tff(fact_291_insertCI,axiom,
    ! [B_3: x_a,A_2: x_a,B: fun_a_bool] :
      ( ( ~ hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,A_2),B))
       => ( A_2 = B_3 ) )
     => hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,A_2),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,B_3),B))) ) ).

tff(fact_292_insertCI,axiom,
    ! [B_3: int,A_2: int,B: fun_int_bool] :
      ( ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_2),B))
       => ( A_2 = B_3 ) )
     => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_2),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,B_3),B))) ) ).

tff(fact_293_insertCI,axiom,
    ! [B_3: nat,A_2: nat,B: fun_nat_bool] :
      ( ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_2),B))
       => ( A_2 = B_3 ) )
     => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_2),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,B_3),B))) ) ).

tff(fact_294_insertCI,axiom,
    ! [B_3: pname,A_2: pname,B: fun_pname_bool] :
      ( ( ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_2),B))
       => ( A_2 = B_3 ) )
     => hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_2),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,B_3),B))) ) ).

tff(fact_295_insertE,axiom,
    ! [A_2: x_a,B_3: x_a,A: fun_a_bool] :
      ( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,A_2),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,B_3),A)))
     => ( ( A_2 != B_3 )
       => hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,A_2),A)) ) ) ).

tff(fact_296_insertE,axiom,
    ! [A_2: int,B_3: int,A: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_2),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,B_3),A)))
     => ( ( A_2 != B_3 )
       => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_2),A)) ) ) ).

tff(fact_297_insertE,axiom,
    ! [A_2: nat,B_3: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_2),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,B_3),A)))
     => ( ( A_2 != B_3 )
       => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_2),A)) ) ) ).

tff(fact_298_insertE,axiom,
    ! [A_2: pname,B_3: pname,A: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_2),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,B_3),A)))
     => ( ( A_2 != B_3 )
       => hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_2),A)) ) ) ).

tff(fact_299_zero__induct__lemma,axiom,
    ! [I_2: nat,Pa: fun_nat_bool,K: nat] :
      ( hBOOL(hAPP_nat_bool(Pa,K))
     => ( ! [N_1: nat] :
            ( hBOOL(hAPP_nat_bool(Pa,hAPP_nat_nat(suc,N_1)))
           => hBOOL(hAPP_nat_bool(Pa,N_1)) )
       => hBOOL(hAPP_nat_bool(Pa,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,K),I_2))) ) ) ).

tff(fact_300_Suc__le__D,axiom,
    ! [N: nat,M_4: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,N)),M_4))
     => ? [M_2: nat] : ( M_4 = hAPP_nat_nat(suc,M_2) ) ) ).

tff(fact_301_order__refl,axiom,
    ! [X_1: fun_int_bool] : hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,X_1),X_1)) ).

tff(fact_302_order__refl,axiom,
    ! [X_1: fun_nat_bool] : hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,X_1),X_1)) ).

tff(fact_303_order__refl,axiom,
    ! [X_11: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_11),X_11)) ).

tff(fact_304_order__refl,axiom,
    ! [X_11: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_11),X_11)) ).

tff(fact_305_order__refl,axiom,
    ! [X_1: fun_a_bool] : hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,X_1),X_1)) ).

tff(fact_306_linorder__linear,axiom,
    ! [X_10: int,Y_9: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_10),Y_9))
      | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_9),X_10)) ) ).

tff(fact_307_linorder__linear,axiom,
    ! [X_10: nat,Y_9: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_10),Y_9))
      | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_9),X_10)) ) ).

tff(fact_308_order__eq__iff,axiom,
    ! [X_1: fun_int_bool,Y_1: fun_int_bool] :
      ( ( X_1 = Y_1 )
    <=> ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,X_1),Y_1))
        & hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Y_1),X_1)) ) ) ).

tff(fact_309_order__eq__iff,axiom,
    ! [X_1: fun_nat_bool,Y_1: fun_nat_bool] :
      ( ( X_1 = Y_1 )
    <=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,X_1),Y_1))
        & hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Y_1),X_1)) ) ) ).

tff(fact_310_order__eq__iff,axiom,
    ! [X_1: int,Y_1: int] :
      ( ( X_1 = Y_1 )
    <=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),Y_1))
        & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_1),X_1)) ) ) ).

tff(fact_311_order__eq__iff,axiom,
    ! [X_1: nat,Y_1: nat] :
      ( ( X_1 = Y_1 )
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_1))
        & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_1),X_1)) ) ) ).

tff(fact_312_order__eq__iff,axiom,
    ! [X_1: fun_a_bool,Y_1: fun_a_bool] :
      ( ( X_1 = Y_1 )
    <=> ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,X_1),Y_1))
        & hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,Y_1),X_1)) ) ) ).

tff(fact_313_order__eq__refl,axiom,
    ! [X_1: fun_int_bool,Y_1: fun_int_bool] :
      ( ( X_1 = Y_1 )
     => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,X_1),Y_1)) ) ).

tff(fact_314_order__eq__refl,axiom,
    ! [X_1: fun_nat_bool,Y_1: fun_nat_bool] :
      ( ( X_1 = Y_1 )
     => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,X_1),Y_1)) ) ).

tff(fact_315_order__eq__refl,axiom,
    ! [X_9: int,Y_8: int] :
      ( ( X_9 = Y_8 )
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_9),Y_8)) ) ).

tff(fact_316_order__eq__refl,axiom,
    ! [X_9: nat,Y_8: nat] :
      ( ( X_9 = Y_8 )
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_9),Y_8)) ) ).

tff(fact_317_order__eq__refl,axiom,
    ! [X_1: fun_a_bool,Y_1: fun_a_bool] :
      ( ( X_1 = Y_1 )
     => hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,X_1),Y_1)) ) ).

tff(fact_318_order__antisym__conv,axiom,
    ! [Y_1: fun_int_bool,X_1: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Y_1),X_1))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,X_1),Y_1))
      <=> ( X_1 = Y_1 ) ) ) ).

tff(fact_319_order__antisym__conv,axiom,
    ! [Y_1: fun_nat_bool,X_1: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Y_1),X_1))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,X_1),Y_1))
      <=> ( X_1 = Y_1 ) ) ) ).

tff(fact_320_order__antisym__conv,axiom,
    ! [Y_1: int,X_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_1),X_1))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),Y_1))
      <=> ( X_1 = Y_1 ) ) ) ).

tff(fact_321_order__antisym__conv,axiom,
    ! [Y_1: nat,X_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_1),X_1))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_1))
      <=> ( X_1 = Y_1 ) ) ) ).

tff(fact_322_order__antisym__conv,axiom,
    ! [Y_1: fun_a_bool,X_1: fun_a_bool] :
      ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,Y_1),X_1))
     => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,X_1),Y_1))
      <=> ( X_1 = Y_1 ) ) ) ).

tff(fact_323_ord__eq__le__trans,axiom,
    ! [C_6: fun_int_bool,A_2: fun_int_bool,B_3: fun_int_bool] :
      ( ( A_2 = B_3 )
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,B_3),C_6))
       => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A_2),C_6)) ) ) ).

tff(fact_324_ord__eq__le__trans,axiom,
    ! [C_6: fun_nat_bool,A_2: fun_nat_bool,B_3: fun_nat_bool] :
      ( ( A_2 = B_3 )
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B_3),C_6))
       => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_2),C_6)) ) ) ).

tff(fact_325_ord__eq__le__trans,axiom,
    ! [C_15: int,A_21: int,B_19: int] :
      ( ( A_21 = B_19 )
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_19),C_15))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_21),C_15)) ) ) ).

tff(fact_326_ord__eq__le__trans,axiom,
    ! [C_15: nat,A_21: nat,B_19: nat] :
      ( ( A_21 = B_19 )
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_19),C_15))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_21),C_15)) ) ) ).

tff(fact_327_ord__eq__le__trans,axiom,
    ! [C_6: fun_a_bool,A_2: fun_a_bool,B_3: fun_a_bool] :
      ( ( A_2 = B_3 )
     => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B_3),C_6))
       => hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A_2),C_6)) ) ) ).

tff(fact_328_xt1_I3_J,axiom,
    ! [C_6: fun_int_bool,A_2: fun_int_bool,B_3: fun_int_bool] :
      ( ( A_2 = B_3 )
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,C_6),B_3))
       => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,C_6),A_2)) ) ) ).

tff(fact_329_xt1_I3_J,axiom,
    ! [C_6: fun_nat_bool,A_2: fun_nat_bool,B_3: fun_nat_bool] :
      ( ( A_2 = B_3 )
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,C_6),B_3))
       => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,C_6),A_2)) ) ) ).

tff(fact_330_xt1_I3_J,axiom,
    ! [C_14: int,A_20: int,B_18: int] :
      ( ( A_20 = B_18 )
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_14),B_18))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_14),A_20)) ) ) ).

tff(fact_331_xt1_I3_J,axiom,
    ! [C_14: nat,A_20: nat,B_18: nat] :
      ( ( A_20 = B_18 )
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_14),B_18))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_14),A_20)) ) ) ).

tff(fact_332_xt1_I3_J,axiom,
    ! [C_6: fun_a_bool,A_2: fun_a_bool,B_3: fun_a_bool] :
      ( ( A_2 = B_3 )
     => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,C_6),B_3))
       => hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,C_6),A_2)) ) ) ).

tff(fact_333_ord__le__eq__trans,axiom,
    ! [C_6: fun_int_bool,A_2: fun_int_bool,B_3: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A_2),B_3))
     => ( ( B_3 = C_6 )
       => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A_2),C_6)) ) ) ).

tff(fact_334_ord__le__eq__trans,axiom,
    ! [C_6: fun_nat_bool,A_2: fun_nat_bool,B_3: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_2),B_3))
     => ( ( B_3 = C_6 )
       => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_2),C_6)) ) ) ).

tff(fact_335_ord__le__eq__trans,axiom,
    ! [C_13: int,A_19: int,B_17: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_19),B_17))
     => ( ( B_17 = C_13 )
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_19),C_13)) ) ) ).

tff(fact_336_ord__le__eq__trans,axiom,
    ! [C_13: nat,A_19: nat,B_17: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_19),B_17))
     => ( ( B_17 = C_13 )
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_19),C_13)) ) ) ).

tff(fact_337_ord__le__eq__trans,axiom,
    ! [C_6: fun_a_bool,A_2: fun_a_bool,B_3: fun_a_bool] :
      ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A_2),B_3))
     => ( ( B_3 = C_6 )
       => hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A_2),C_6)) ) ) ).

tff(fact_338_xt1_I4_J,axiom,
    ! [C_6: fun_int_bool,B_3: fun_int_bool,A_2: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,B_3),A_2))
     => ( ( B_3 = C_6 )
       => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,C_6),A_2)) ) ) ).

tff(fact_339_xt1_I4_J,axiom,
    ! [C_6: fun_nat_bool,B_3: fun_nat_bool,A_2: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B_3),A_2))
     => ( ( B_3 = C_6 )
       => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,C_6),A_2)) ) ) ).

tff(fact_340_xt1_I4_J,axiom,
    ! [C_12: int,B_16: int,A_18: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_16),A_18))
     => ( ( B_16 = C_12 )
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_12),A_18)) ) ) ).

tff(fact_341_xt1_I4_J,axiom,
    ! [C_12: nat,B_16: nat,A_18: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_16),A_18))
     => ( ( B_16 = C_12 )
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_12),A_18)) ) ) ).

tff(fact_342_xt1_I4_J,axiom,
    ! [C_6: fun_a_bool,B_3: fun_a_bool,A_2: fun_a_bool] :
      ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B_3),A_2))
     => ( ( B_3 = C_6 )
       => hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,C_6),A_2)) ) ) ).

tff(fact_343_order__antisym,axiom,
    ! [X_1: fun_int_bool,Y_1: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,X_1),Y_1))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Y_1),X_1))
       => ( X_1 = Y_1 ) ) ) ).

tff(fact_344_order__antisym,axiom,
    ! [X_1: fun_nat_bool,Y_1: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,X_1),Y_1))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Y_1),X_1))
       => ( X_1 = Y_1 ) ) ) ).

tff(fact_345_order__antisym,axiom,
    ! [X_8: int,Y_7: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_8),Y_7))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_7),X_8))
       => ( X_8 = Y_7 ) ) ) ).

tff(fact_346_order__antisym,axiom,
    ! [X_8: nat,Y_7: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_8),Y_7))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_7),X_8))
       => ( X_8 = Y_7 ) ) ) ).

tff(fact_347_order__antisym,axiom,
    ! [X_1: fun_a_bool,Y_1: fun_a_bool] :
      ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,X_1),Y_1))
     => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,Y_1),X_1))
       => ( X_1 = Y_1 ) ) ) ).

tff(fact_348_order__trans,axiom,
    ! [Z_2: fun_int_bool,X_1: fun_int_bool,Y_1: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,X_1),Y_1))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Y_1),Z_2))
       => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,X_1),Z_2)) ) ) ).

tff(fact_349_order__trans,axiom,
    ! [Z_2: fun_nat_bool,X_1: fun_nat_bool,Y_1: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,X_1),Y_1))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Y_1),Z_2))
       => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,X_1),Z_2)) ) ) ).

tff(fact_350_order__trans,axiom,
    ! [Z_6: int,X_7: int,Y_6: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_7),Y_6))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_6),Z_6))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_7),Z_6)) ) ) ).

tff(fact_351_order__trans,axiom,
    ! [Z_6: nat,X_7: nat,Y_6: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_7),Y_6))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_6),Z_6))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_7),Z_6)) ) ) ).

tff(fact_352_order__trans,axiom,
    ! [Z_2: fun_a_bool,X_1: fun_a_bool,Y_1: fun_a_bool] :
      ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,X_1),Y_1))
     => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,Y_1),Z_2))
       => hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,X_1),Z_2)) ) ) ).

tff(fact_353_xt1_I5_J,axiom,
    ! [Y_1: fun_int_bool,X_1: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Y_1),X_1))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,X_1),Y_1))
       => ( X_1 = Y_1 ) ) ) ).

tff(fact_354_xt1_I5_J,axiom,
    ! [Y_1: fun_nat_bool,X_1: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Y_1),X_1))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,X_1),Y_1))
       => ( X_1 = Y_1 ) ) ) ).

tff(fact_355_xt1_I5_J,axiom,
    ! [Y_5: int,X_6: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_5),X_6))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_6),Y_5))
       => ( X_6 = Y_5 ) ) ) ).

tff(fact_356_xt1_I5_J,axiom,
    ! [Y_5: nat,X_6: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_5),X_6))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_6),Y_5))
       => ( X_6 = Y_5 ) ) ) ).

tff(fact_357_xt1_I5_J,axiom,
    ! [Y_1: fun_a_bool,X_1: fun_a_bool] :
      ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,Y_1),X_1))
     => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,X_1),Y_1))
       => ( X_1 = Y_1 ) ) ) ).

tff(fact_358_xt1_I6_J,axiom,
    ! [Z_2: fun_int_bool,Y_1: fun_int_bool,X_1: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Y_1),X_1))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Z_2),Y_1))
       => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Z_2),X_1)) ) ) ).

tff(fact_359_xt1_I6_J,axiom,
    ! [Z_2: fun_nat_bool,Y_1: fun_nat_bool,X_1: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Y_1),X_1))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Z_2),Y_1))
       => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Z_2),X_1)) ) ) ).

tff(fact_360_xt1_I6_J,axiom,
    ! [Z_5: int,Y_4: int,X_5: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_4),X_5))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_5),Y_4))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_5),X_5)) ) ) ).

tff(fact_361_xt1_I6_J,axiom,
    ! [Z_5: nat,Y_4: nat,X_5: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_4),X_5))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Z_5),Y_4))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Z_5),X_5)) ) ) ).

tff(fact_362_xt1_I6_J,axiom,
    ! [Z_2: fun_a_bool,Y_1: fun_a_bool,X_1: fun_a_bool] :
      ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,Y_1),X_1))
     => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,Z_2),Y_1))
       => hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,Z_2),X_1)) ) ) ).

tff(fact_363_linorder__le__cases,axiom,
    ! [X_4: int,Y_3: int] :
      ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_4),Y_3))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_3),X_4)) ) ).

tff(fact_364_linorder__le__cases,axiom,
    ! [X_4: nat,Y_3: nat] :
      ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_4),Y_3))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_3),X_4)) ) ).

tff(fact_365_insertI1,axiom,
    ! [A_2: x_a,B: fun_a_bool] : hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,A_2),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,A_2),B))) ).

tff(fact_366_insertI1,axiom,
    ! [A_2: int,B: fun_int_bool] : hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_2),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_2),B))) ).

tff(fact_367_insertI1,axiom,
    ! [A_2: nat,B: fun_nat_bool] : hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_2),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_2),B))) ).

tff(fact_368_insertI1,axiom,
    ! [A_2: pname,B: fun_pname_bool] : hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_2),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_2),B))) ).

tff(fact_369_insert__compr,axiom,
    ! [A_2: x_a,B: fun_a_bool] : ( hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,A_2),B) = collect_a(cOMBS_a_bool_bool(cOMBB_1972296269bool_a(fdisj,hAPP_a_fun_a_bool(cOMBC_a_a_bool(fequal_a),A_2)),hAPP_f2050579477a_bool(cOMBC_1355376034l_bool(member_a),B))) ) ).

tff(fact_370_insert__compr,axiom,
    ! [A_2: int,B: fun_int_bool] : ( hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_2),B) = collect_int(cOMBS_int_bool_bool(cOMBB_1652995168ol_int(fdisj,hAPP_i1948725293t_bool(cOMBC_int_int_bool(fequal_int),A_2)),hAPP_f1805168059t_bool(cOMBC_94739984l_bool(member_int),B))) ) ).

tff(fact_371_insert__compr,axiom,
    ! [A_2: nat,B: fun_nat_bool] : ( hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_2),B) = collect_nat(cOMBS_nat_bool_bool(cOMBB_1015721476ol_nat(fdisj,hAPP_n1699378549t_bool(cOMBC_nat_nat_bool(fequal_nat),A_2)),hAPP_f800510211t_bool(cOMBC_226598744l_bool(member_nat),B))) ) ).

tff(fact_372_insert__compr,axiom,
    ! [A_2: pname,B: fun_pname_bool] : ( hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_2),B) = collect_pname(cOMBS_568398431l_bool(cOMBB_675860798_pname(fdisj,hAPP_p61793385e_bool(cOMBC_1149511130e_bool(fequal_pname),A_2)),hAPP_f759274231e_bool(cOMBC_1058051404l_bool(member_pname),B))) ) ).

tff(fact_373_insert__Collect,axiom,
    ! [A_2: x_a,Pa: fun_a_bool] : ( hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,A_2),collect_a(Pa)) = collect_a(cOMBS_a_bool_bool(cOMBB_1972296269bool_a(fimplies,cOMBB_bool_bool_a(fNot,hAPP_a_fun_a_bool(cOMBC_a_a_bool(fequal_a),A_2))),Pa)) ) ).

tff(fact_374_insert__Collect,axiom,
    ! [A_2: int,Pa: fun_int_bool] : ( hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_2),collect_int(Pa)) = collect_int(cOMBS_int_bool_bool(cOMBB_1652995168ol_int(fimplies,cOMBB_bool_bool_int(fNot,hAPP_i1948725293t_bool(cOMBC_int_int_bool(fequal_int),A_2))),Pa)) ) ).

tff(fact_375_insert__Collect,axiom,
    ! [A_2: nat,Pa: fun_nat_bool] : ( hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_2),collect_nat(Pa)) = collect_nat(cOMBS_nat_bool_bool(cOMBB_1015721476ol_nat(fimplies,cOMBB_bool_bool_nat(fNot,hAPP_n1699378549t_bool(cOMBC_nat_nat_bool(fequal_nat),A_2))),Pa)) ) ).

tff(fact_376_mem__def,axiom,
    ! [X_1: int,A: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
    <=> hBOOL(hAPP_int_bool(A,X_1)) ) ).

tff(fact_377_mem__def,axiom,
    ! [X_1: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
    <=> hBOOL(hAPP_nat_bool(A,X_1)) ) ).

tff(fact_378_mem__def,axiom,
    ! [X_1: x_a,A: fun_a_bool] :
      ( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_1),A))
    <=> hBOOL(hAPP_a_bool(A,X_1)) ) ).

tff(fact_379_mem__def,axiom,
    ! [X_1: pname,A: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_1),A))
    <=> hBOOL(hAPP_pname_bool(A,X_1)) ) ).

tff(fact_380_Collect__def,axiom,
    ! [Pa: fun_int_bool] : ( collect_int(Pa) = Pa ) ).

tff(fact_381_Collect__def,axiom,
    ! [Pa: fun_nat_bool] : ( collect_nat(Pa) = Pa ) ).

tff(fact_382_insert__absorb2,axiom,
    ! [X_1: x_a,A: fun_a_bool] : ( hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_1),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_1),A)) = hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_1),A) ) ).

tff(fact_383_insert__commute,axiom,
    ! [X_1: x_a,Y_1: x_a,A: fun_a_bool] : ( hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_1),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,Y_1),A)) = hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,Y_1),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_1),A)) ) ).

tff(fact_384_insert__iff,axiom,
    ! [A_2: x_a,B_3: x_a,A: fun_a_bool] :
      ( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,A_2),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,B_3),A)))
    <=> ( ( A_2 = B_3 )
        | hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,A_2),A)) ) ) ).

tff(fact_385_insert__iff,axiom,
    ! [A_2: int,B_3: int,A: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_2),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,B_3),A)))
    <=> ( ( A_2 = B_3 )
        | hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_2),A)) ) ) ).

tff(fact_386_insert__iff,axiom,
    ! [A_2: nat,B_3: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_2),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,B_3),A)))
    <=> ( ( A_2 = B_3 )
        | hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_2),A)) ) ) ).

tff(fact_387_insert__iff,axiom,
    ! [A_2: pname,B_3: pname,A: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_2),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,B_3),A)))
    <=> ( ( A_2 = B_3 )
        | hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_2),A)) ) ) ).

tff(fact_388_insert__code,axiom,
    ! [Y_1: x_a,A: fun_a_bool,X_1: x_a] :
      ( hBOOL(hAPP_a_bool(hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,Y_1),A),X_1))
    <=> ( ( Y_1 = X_1 )
        | hBOOL(hAPP_a_bool(A,X_1)) ) ) ).

tff(fact_389_insert__ident,axiom,
    ! [B: fun_a_bool,X_1: x_a,A: fun_a_bool] :
      ( ~ hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_1),A))
     => ( ~ hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_1),B))
       => ( ( hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_1),A) = hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_1),B) )
        <=> ( A = B ) ) ) ) ).

tff(fact_390_insert__ident,axiom,
    ! [B: fun_int_bool,X_1: int,A: fun_int_bool] :
      ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
     => ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),B))
       => ( ( hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),B) )
        <=> ( A = B ) ) ) ) ).

tff(fact_391_insert__ident,axiom,
    ! [B: fun_nat_bool,X_1: nat,A: fun_nat_bool] :
      ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
     => ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),B))
       => ( ( hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),B) )
        <=> ( A = B ) ) ) ) ).

tff(fact_392_insert__ident,axiom,
    ! [B: fun_pname_bool,X_1: pname,A: fun_pname_bool] :
      ( ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_1),A))
     => ( ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_1),B))
       => ( ( hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_1),A) = hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_1),B) )
        <=> ( A = B ) ) ) ) ).

tff(fact_393_insertI2,axiom,
    ! [B_3: x_a,A_2: x_a,B: fun_a_bool] :
      ( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,A_2),B))
     => hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,A_2),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,B_3),B))) ) ).

tff(fact_394_insertI2,axiom,
    ! [B_3: int,A_2: int,B: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_2),B))
     => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_2),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,B_3),B))) ) ).

tff(fact_395_insertI2,axiom,
    ! [B_3: nat,A_2: nat,B: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_2),B))
     => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_2),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,B_3),B))) ) ).

tff(fact_396_insertI2,axiom,
    ! [B_3: pname,A_2: pname,B: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_2),B))
     => hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_2),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,B_3),B))) ) ).

tff(fact_397_insert__absorb,axiom,
    ! [A_2: x_a,A: fun_a_bool] :
      ( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,A_2),A))
     => ( hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,A_2),A) = A ) ) ).

tff(fact_398_insert__absorb,axiom,
    ! [A_2: int,A: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_2),A))
     => ( hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_2),A) = A ) ) ).

tff(fact_399_insert__absorb,axiom,
    ! [A_2: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_2),A))
     => ( hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_2),A) = A ) ) ).

tff(fact_400_insert__absorb,axiom,
    ! [A_2: pname,A: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_2),A))
     => ( hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_2),A) = A ) ) ).

tff(fact_401_subset__refl,axiom,
    ! [A: fun_int_bool] : hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),A)) ).

tff(fact_402_subset__refl,axiom,
    ! [A: fun_nat_bool] : hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),A)) ).

tff(fact_403_subset__refl,axiom,
    ! [A: fun_a_bool] : hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),A)) ).

tff(fact_404_set__eq__subset,axiom,
    ! [A: fun_int_bool,B: fun_int_bool] :
      ( ( A = B )
    <=> ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),B))
        & hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,B),A)) ) ) ).

tff(fact_405_set__eq__subset,axiom,
    ! [A: fun_nat_bool,B: fun_nat_bool] :
      ( ( A = B )
    <=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B))
        & hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B),A)) ) ) ).

tff(fact_406_set__eq__subset,axiom,
    ! [A: fun_a_bool,B: fun_a_bool] :
      ( ( A = B )
    <=> ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),B))
        & hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B),A)) ) ) ).

tff(fact_407_equalityD1,axiom,
    ! [A: fun_int_bool,B: fun_int_bool] :
      ( ( A = B )
     => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),B)) ) ).

tff(fact_408_equalityD1,axiom,
    ! [A: fun_nat_bool,B: fun_nat_bool] :
      ( ( A = B )
     => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B)) ) ).

tff(fact_409_equalityD1,axiom,
    ! [A: fun_a_bool,B: fun_a_bool] :
      ( ( A = B )
     => hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),B)) ) ).

tff(fact_410_equalityD2,axiom,
    ! [A: fun_int_bool,B: fun_int_bool] :
      ( ( A = B )
     => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,B),A)) ) ).

tff(fact_411_equalityD2,axiom,
    ! [A: fun_nat_bool,B: fun_nat_bool] :
      ( ( A = B )
     => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B),A)) ) ).

tff(fact_412_equalityD2,axiom,
    ! [A: fun_a_bool,B: fun_a_bool] :
      ( ( A = B )
     => hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B),A)) ) ).

tff(fact_413_in__mono,axiom,
    ! [X_1: int,A: fun_int_bool,B: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),B))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
       => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),B)) ) ) ).

tff(fact_414_in__mono,axiom,
    ! [X_1: nat,A: fun_nat_bool,B: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
       => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),B)) ) ) ).

tff(fact_415_in__mono,axiom,
    ! [X_1: x_a,A: fun_a_bool,B: fun_a_bool] :
      ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),B))
     => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_1),A))
       => hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_1),B)) ) ) ).

tff(fact_416_in__mono,axiom,
    ! [X_1: pname,A: fun_pname_bool,B: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A),B))
     => ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_1),A))
       => hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_1),B)) ) ) ).

tff(fact_417_set__rev__mp,axiom,
    ! [B: fun_int_bool,X_1: int,A: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),B))
       => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),B)) ) ) ).

tff(fact_418_set__rev__mp,axiom,
    ! [B: fun_nat_bool,X_1: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B))
       => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),B)) ) ) ).

tff(fact_419_set__rev__mp,axiom,
    ! [B: fun_a_bool,X_1: x_a,A: fun_a_bool] :
      ( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_1),A))
     => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),B))
       => hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_1),B)) ) ) ).

tff(fact_420_set__rev__mp,axiom,
    ! [B: fun_pname_bool,X_1: pname,A: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_1),A))
     => ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A),B))
       => hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_1),B)) ) ) ).

tff(fact_421_set__mp,axiom,
    ! [X_1: int,A: fun_int_bool,B: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),B))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
       => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),B)) ) ) ).

tff(fact_422_set__mp,axiom,
    ! [X_1: nat,A: fun_nat_bool,B: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
       => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),B)) ) ) ).

tff(fact_423_set__mp,axiom,
    ! [X_1: x_a,A: fun_a_bool,B: fun_a_bool] :
      ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),B))
     => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_1),A))
       => hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_1),B)) ) ) ).

tff(fact_424_set__mp,axiom,
    ! [X_1: pname,A: fun_pname_bool,B: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A),B))
     => ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_1),A))
       => hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_1),B)) ) ) ).

tff(fact_425_subset__trans,axiom,
    ! [C_11: fun_int_bool,A: fun_int_bool,B: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),B))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,B),C_11))
       => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),C_11)) ) ) ).

tff(fact_426_subset__trans,axiom,
    ! [C_11: fun_nat_bool,A: fun_nat_bool,B: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B),C_11))
       => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),C_11)) ) ) ).

tff(fact_427_subset__trans,axiom,
    ! [C_11: fun_a_bool,A: fun_a_bool,B: fun_a_bool] :
      ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),B))
     => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B),C_11))
       => hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),C_11)) ) ) ).

tff(fact_428_equalityE,axiom,
    ! [A: fun_int_bool,B: fun_int_bool] :
      ( ( A = B )
     => ~ ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),B))
         => ~ hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,B),A)) ) ) ).

tff(fact_429_equalityE,axiom,
    ! [A: fun_nat_bool,B: fun_nat_bool] :
      ( ( A = B )
     => ~ ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B))
         => ~ hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B),A)) ) ) ).

tff(fact_430_equalityE,axiom,
    ! [A: fun_a_bool,B: fun_a_bool] :
      ( ( A = B )
     => ~ ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),B))
         => ~ hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B),A)) ) ) ).

tff(fact_431_image__iff,axiom,
    ! [Z_2: int,F: fun_nat_int,A: fun_nat_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Z_2),image_nat_int(F,A)))
    <=> ? [X_2: nat] :
          ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A))
          & ( Z_2 = hAPP_nat_int(F,X_2) ) ) ) ).

tff(fact_432_image__iff,axiom,
    ! [Z_2: x_a,F: fun_pname_a,A: fun_pname_bool] :
      ( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,Z_2),image_pname_a(F,A)))
    <=> ? [X_2: pname] :
          ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),A))
          & ( Z_2 = hAPP_pname_a(F,X_2) ) ) ) ).

tff(fact_433_imageI,axiom,
    ! [F: fun_nat_int,X_1: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
     => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_nat_int(F,X_1)),image_nat_int(F,A))) ) ).

tff(fact_434_imageI,axiom,
    ! [F: fun_pname_a,X_1: pname,A: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_1),A))
     => hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,hAPP_pname_a(F,X_1)),image_pname_a(F,A))) ) ).

tff(fact_435_rev__image__eqI,axiom,
    ! [B_3: int,F: fun_nat_int,X_1: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
     => ( ( B_3 = hAPP_nat_int(F,X_1) )
       => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B_3),image_nat_int(F,A))) ) ) ).

tff(fact_436_rev__image__eqI,axiom,
    ! [B_3: x_a,F: fun_pname_a,X_1: pname,A: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_1),A))
     => ( ( B_3 = hAPP_pname_a(F,X_1) )
       => hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,B_3),image_pname_a(F,A))) ) ) ).

tff(fact_437_insert__compr__raw,axiom,
    ! [X_2: x_a,Xa: fun_a_bool] : ( hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_2),Xa) = collect_a(cOMBS_a_bool_bool(cOMBB_1972296269bool_a(fdisj,hAPP_a_fun_a_bool(cOMBC_a_a_bool(fequal_a),X_2)),hAPP_f2050579477a_bool(cOMBC_1355376034l_bool(member_a),Xa))) ) ).

tff(fact_438_insert__compr__raw,axiom,
    ! [X_2: int,Xa: fun_int_bool] : ( hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_2),Xa) = collect_int(cOMBS_int_bool_bool(cOMBB_1652995168ol_int(fdisj,hAPP_i1948725293t_bool(cOMBC_int_int_bool(fequal_int),X_2)),hAPP_f1805168059t_bool(cOMBC_94739984l_bool(member_int),Xa))) ) ).

tff(fact_439_insert__compr__raw,axiom,
    ! [X_2: nat,Xa: fun_nat_bool] : ( hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_2),Xa) = collect_nat(cOMBS_nat_bool_bool(cOMBB_1015721476ol_nat(fdisj,hAPP_n1699378549t_bool(cOMBC_nat_nat_bool(fequal_nat),X_2)),hAPP_f800510211t_bool(cOMBC_226598744l_bool(member_nat),Xa))) ) ).

tff(fact_440_insert__compr__raw,axiom,
    ! [X_2: pname,Xa: fun_pname_bool] : ( hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_2),Xa) = collect_pname(cOMBS_568398431l_bool(cOMBB_675860798_pname(fdisj,hAPP_p61793385e_bool(cOMBC_1149511130e_bool(fequal_pname),X_2)),hAPP_f759274231e_bool(cOMBC_1058051404l_bool(member_pname),Xa))) ) ).

tff(fact_441_le__fun__def,axiom,
    ! [F: fun_int_bool,G_1: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,F),G_1))
    <=> ! [X_2: int] : hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_int_bool(F,X_2)),hAPP_int_bool(G_1,X_2))) ) ).

tff(fact_442_le__fun__def,axiom,
    ! [F: fun_nat_bool,G_1: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,F),G_1))
    <=> ! [X_2: nat] : hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_nat_bool(F,X_2)),hAPP_nat_bool(G_1,X_2))) ) ).

tff(fact_443_le__fun__def,axiom,
    ! [F: fun_a_bool,G_1: fun_a_bool] :
      ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,F),G_1))
    <=> ! [X_2: x_a] : hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_a_bool(F,X_2)),hAPP_a_bool(G_1,X_2))) ) ).

tff(fact_444_le__funD,axiom,
    ! [X_1: int,F: fun_int_bool,G_1: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,F),G_1))
     => hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_int_bool(F,X_1)),hAPP_int_bool(G_1,X_1))) ) ).

tff(fact_445_le__funD,axiom,
    ! [X_1: nat,F: fun_nat_bool,G_1: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,F),G_1))
     => hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_nat_bool(F,X_1)),hAPP_nat_bool(G_1,X_1))) ) ).

tff(fact_446_le__funD,axiom,
    ! [X_1: x_a,F: fun_a_bool,G_1: fun_a_bool] :
      ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,F),G_1))
     => hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_a_bool(F,X_1)),hAPP_a_bool(G_1,X_1))) ) ).

tff(fact_447_le__funE,axiom,
    ! [X_1: int,F: fun_int_bool,G_1: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,F),G_1))
     => hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_int_bool(F,X_1)),hAPP_int_bool(G_1,X_1))) ) ).

tff(fact_448_le__funE,axiom,
    ! [X_1: nat,F: fun_nat_bool,G_1: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,F),G_1))
     => hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_nat_bool(F,X_1)),hAPP_nat_bool(G_1,X_1))) ) ).

tff(fact_449_le__funE,axiom,
    ! [X_1: x_a,F: fun_a_bool,G_1: fun_a_bool] :
      ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,F),G_1))
     => hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_a_bool(F,X_1)),hAPP_a_bool(G_1,X_1))) ) ).

tff(fact_450_subset__insertI,axiom,
    ! [B: fun_a_bool,A_2: x_a] : hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,A_2),B))) ).

tff(fact_451_subset__insertI,axiom,
    ! [B: fun_int_bool,A_2: int] : hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,B),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_2),B))) ).

tff(fact_452_subset__insertI,axiom,
    ! [B: fun_nat_bool,A_2: nat] : hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_2),B))) ).

tff(fact_453_insert__subset,axiom,
    ! [X_1: x_a,A: fun_a_bool,B: fun_a_bool] :
      ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_1),A)),B))
    <=> ( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_1),B))
        & hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),B)) ) ) ).

tff(fact_454_insert__subset,axiom,
    ! [X_1: int,A: fun_int_bool,B: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A)),B))
    <=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),B))
        & hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),B)) ) ) ).

tff(fact_455_insert__subset,axiom,
    ! [X_1: nat,A: fun_nat_bool,B: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)),B))
    <=> ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),B))
        & hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B)) ) ) ).

tff(fact_456_insert__subset,axiom,
    ! [X_1: pname,A: fun_pname_bool,B: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_1),A)),B))
    <=> ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_1),B))
        & hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A),B)) ) ) ).

tff(fact_457_subset__insert,axiom,
    ! [B: fun_a_bool,X_1: x_a,A: fun_a_bool] :
      ( ~ hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_1),A))
     => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_1),B)))
      <=> hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),B)) ) ) ).

tff(fact_458_subset__insert,axiom,
    ! [B: fun_int_bool,X_1: int,A: fun_int_bool] :
      ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),B)))
      <=> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),B)) ) ) ).

tff(fact_459_subset__insert,axiom,
    ! [B: fun_nat_bool,X_1: nat,A: fun_nat_bool] :
      ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),B)))
      <=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B)) ) ) ).

tff(fact_460_subset__insert,axiom,
    ! [B: fun_pname_bool,X_1: pname,A: fun_pname_bool] :
      ( ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_1),A))
     => ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_1),B)))
      <=> hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A),B)) ) ) ).

tff(fact_461_subset__insertI2,axiom,
    ! [B_3: x_a,A: fun_a_bool,B: fun_a_bool] :
      ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),B))
     => hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,B_3),B))) ) ).

tff(fact_462_subset__insertI2,axiom,
    ! [B_3: int,A: fun_int_bool,B: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),B))
     => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,B_3),B))) ) ).

tff(fact_463_subset__insertI2,axiom,
    ! [B_3: nat,A: fun_nat_bool,B: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B))
     => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,B_3),B))) ) ).

tff(fact_464_insert__mono,axiom,
    ! [A_2: x_a,C_11: fun_a_bool,D_4: fun_a_bool] :
      ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,C_11),D_4))
     => hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,A_2),C_11)),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,A_2),D_4))) ) ).

tff(fact_465_insert__mono,axiom,
    ! [A_2: int,C_11: fun_int_bool,D_4: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,C_11),D_4))
     => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_2),C_11)),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_2),D_4))) ) ).

tff(fact_466_insert__mono,axiom,
    ! [A_2: nat,C_11: fun_nat_bool,D_4: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,C_11),D_4))
     => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_2),C_11)),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_2),D_4))) ) ).

tff(fact_467_image__insert,axiom,
    ! [F: fun_nat_int,A_2: nat,B: fun_nat_bool] : ( image_nat_int(F,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_2),B)) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,hAPP_nat_int(F,A_2)),image_nat_int(F,B)) ) ).

tff(fact_468_image__insert,axiom,
    ! [F: fun_pname_a,A_2: pname,B: fun_pname_bool] : ( image_pname_a(F,hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_2),B)) = hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,hAPP_pname_a(F,A_2)),image_pname_a(F,B)) ) ).

tff(fact_469_insert__image,axiom,
    ! [F: fun_nat_int,X_1: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
     => ( hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,hAPP_nat_int(F,X_1)),image_nat_int(F,A)) = image_nat_int(F,A) ) ) ).

tff(fact_470_insert__image,axiom,
    ! [F: fun_pname_a,X_1: pname,A: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_1),A))
     => ( hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,hAPP_pname_a(F,X_1)),image_pname_a(F,A)) = image_pname_a(F,A) ) ) ).

tff(fact_471_subset__image__iff,axiom,
    ! [B: fun_int_bool,F: fun_nat_int,A: fun_nat_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,B),image_nat_int(F,A)))
    <=> ? [AA: fun_nat_bool] :
          ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,AA),A))
          & ( B = image_nat_int(F,AA) ) ) ) ).

tff(fact_472_subset__image__iff,axiom,
    ! [B: fun_a_bool,F: fun_pname_a,A: fun_pname_bool] :
      ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B),image_pname_a(F,A)))
    <=> ? [AA: fun_pname_bool] :
          ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,AA),A))
          & ( B = image_pname_a(F,AA) ) ) ) ).

tff(fact_473_image__mono,axiom,
    ! [F: fun_nat_int,A: fun_nat_bool,B: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B))
     => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,image_nat_int(F,A)),image_nat_int(F,B))) ) ).

tff(fact_474_image__mono,axiom,
    ! [F: fun_pname_a,A: fun_pname_bool,B: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A),B))
     => hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,image_pname_a(F,A)),image_pname_a(F,B))) ) ).

tff(fact_475_imageE,axiom,
    ! [B_3: int,F: fun_nat_int,A: fun_nat_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B_3),image_nat_int(F,A)))
     => ~ ! [X_2: nat] :
            ( ( B_3 = hAPP_nat_int(F,X_2) )
           => ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A)) ) ) ).

tff(fact_476_imageE,axiom,
    ! [B_3: x_a,F: fun_pname_a,A: fun_pname_bool] :
      ( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,B_3),image_pname_a(F,A)))
     => ~ ! [X_2: pname] :
            ( ( B_3 = hAPP_pname_a(F,X_2) )
           => ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),A)) ) ) ).

tff(fact_477_subsetI,axiom,
    ! [B: fun_int_bool,A: fun_int_bool] :
      ( ! [X_2: int] :
          ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),A))
         => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),B)) )
     => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),B)) ) ).

tff(fact_478_subsetI,axiom,
    ! [B: fun_nat_bool,A: fun_nat_bool] :
      ( ! [X_2: nat] :
          ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A))
         => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),B)) )
     => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B)) ) ).

tff(fact_479_subsetI,axiom,
    ! [B: fun_a_bool,A: fun_a_bool] :
      ( ! [X_2: x_a] :
          ( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_2),A))
         => hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_2),B)) )
     => hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),B)) ) ).

tff(fact_480_subsetI,axiom,
    ! [B: fun_pname_bool,A: fun_pname_bool] :
      ( ! [X_2: pname] :
          ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),A))
         => hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),B)) )
     => hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A),B)) ) ).

tff(fact_481_image__subsetI,axiom,
    ! [F: fun_nat_int,B: fun_int_bool,A: fun_nat_bool] :
      ( ! [X_2: nat] :
          ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A))
         => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_nat_int(F,X_2)),B)) )
     => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,image_nat_int(F,A)),B)) ) ).

tff(fact_482_image__subsetI,axiom,
    ! [F: fun_pname_a,B: fun_a_bool,A: fun_pname_bool] :
      ( ! [X_2: pname] :
          ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),A))
         => hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,hAPP_pname_a(F,X_2)),B)) )
     => hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,image_pname_a(F,A)),B)) ) ).

tff(fact_483_le__funI,axiom,
    ! [F: fun_int_bool,G_1: fun_int_bool] :
      ( ! [X_2: int] : hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_int_bool(F,X_2)),hAPP_int_bool(G_1,X_2)))
     => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,F),G_1)) ) ).

tff(fact_484_le__funI,axiom,
    ! [F: fun_nat_bool,G_1: fun_nat_bool] :
      ( ! [X_2: nat] : hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_nat_bool(F,X_2)),hAPP_nat_bool(G_1,X_2)))
     => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,F),G_1)) ) ).

tff(fact_485_le__funI,axiom,
    ! [F: fun_a_bool,G_1: fun_a_bool] :
      ( ! [X_2: x_a] : hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_a_bool(F,X_2)),hAPP_a_bool(G_1,X_2)))
     => hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,F),G_1)) ) ).

tff(fact_486_finite__nat__set__iff__bounded__le,axiom,
    ! [N_2: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,N_2))
    <=> ? [M_2: nat] :
        ! [X_2: nat] :
          ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),N_2))
         => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_2),M_2)) ) ) ).

tff(fact_487_assms_I3_J,axiom,
    ! [G: fun_a_bool,C_6: com] :
      ( hBOOL(wt(C_6))
     => ( ! [X_2: pname] :
            ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),u))
           => hBOOL(hAPP_fun_a_bool_bool(p(G),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,hAPP_pname_a(mgt_call,X_2)),bot_bot_fun_a_bool))) )
       => hBOOL(hAPP_fun_a_bool_bool(p(G),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,mgt(C_6)),bot_bot_fun_a_bool))) ) ) ).

tff(fact_488_diff__eq__diff__less__eq,axiom,
    ! [A_2: int,B_3: int,C_6: int,D_1: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_2),B_3) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,C_6),D_1) )
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_2),B_3))
      <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_6),D_1)) ) ) ).

tff(fact_489_less__eq__nat_Osimps_I2_J,axiom,
    ! [M: nat,Na: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,M)),Na))
    <=> hBOOL(hAPP_nat_bool(nat_case_bool(fFalse,hAPP_n1699378549t_bool(ord_less_eq_nat,M)),Na)) ) ).

tff(fact_490_emptyE,axiom,
    ! [A_2: int] : ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_2),bot_bot_fun_int_bool)) ).

tff(fact_491_emptyE,axiom,
    ! [A_2: nat] : ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_2),bot_bot_fun_nat_bool)) ).

tff(fact_492_emptyE,axiom,
    ! [A_2: x_a] : ~ hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,A_2),bot_bot_fun_a_bool)) ).

tff(fact_493_emptyE,axiom,
    ! [A_2: pname] : ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_2),bot_bo844097828e_bool)) ).

tff(fact_494_finite__Diff,axiom,
    ! [B: fun_int_bool,A: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
     => hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),B))) ) ).

tff(fact_495_finite__Diff,axiom,
    ! [B: fun_nat_bool,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B))) ) ).

tff(fact_496_finite__Diff,axiom,
    ! [B: fun_pname_bool,A: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A))
     => hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,A),B))) ) ).

tff(fact_497_finite_OemptyI,axiom,
    hBOOL(hAPP_f448129468l_bool(finite_finite_int,bot_bot_fun_int_bool)) ).

tff(fact_498_finite_OemptyI,axiom,
    hBOOL(hAPP_f54304608l_bool(finite_finite_nat,bot_bot_fun_nat_bool)) ).

tff(fact_499_finite_OemptyI,axiom,
    hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,bot_bo844097828e_bool)) ).

tff(fact_500_finite_OemptyI,axiom,
    hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,bot_bot_fun_a_bool)) ).

tff(fact_501_empty__subsetI,axiom,
    ! [A: fun_nat_bool] : hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,bot_bot_fun_nat_bool),A)) ).

tff(fact_502_empty__subsetI,axiom,
    ! [A: fun_int_bool] : hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,bot_bot_fun_int_bool),A)) ).

tff(fact_503_empty__subsetI,axiom,
    ! [A: fun_a_bool] : hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,bot_bot_fun_a_bool),A)) ).

tff(fact_504_equals0D,axiom,
    ! [A_2: int,A: fun_int_bool] :
      ( ( A = bot_bot_fun_int_bool )
     => ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_2),A)) ) ).

tff(fact_505_equals0D,axiom,
    ! [A_2: nat,A: fun_nat_bool] :
      ( ( A = bot_bot_fun_nat_bool )
     => ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_2),A)) ) ).

tff(fact_506_equals0D,axiom,
    ! [A_2: x_a,A: fun_a_bool] :
      ( ( A = bot_bot_fun_a_bool )
     => ~ hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,A_2),A)) ) ).

tff(fact_507_equals0D,axiom,
    ! [A_2: pname,A: fun_pname_bool] :
      ( ( A = bot_bo844097828e_bool )
     => ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_2),A)) ) ).

tff(fact_508_Collect__empty__eq,axiom,
    ! [Pa: fun_int_bool] :
      ( ( collect_int(Pa) = bot_bot_fun_int_bool )
    <=> ! [X_2: int] : ~ hBOOL(hAPP_int_bool(Pa,X_2)) ) ).

tff(fact_509_Collect__empty__eq,axiom,
    ! [Pa: fun_nat_bool] :
      ( ( collect_nat(Pa) = bot_bot_fun_nat_bool )
    <=> ! [X_2: nat] : ~ hBOOL(hAPP_nat_bool(Pa,X_2)) ) ).

tff(fact_510_Collect__empty__eq,axiom,
    ! [Pa: fun_a_bool] :
      ( ( collect_a(Pa) = bot_bot_fun_a_bool )
    <=> ! [X_2: x_a] : ~ hBOOL(hAPP_a_bool(Pa,X_2)) ) ).

tff(fact_511_Diff__cancel,axiom,
    ! [A: fun_nat_bool] : ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),A) = bot_bot_fun_nat_bool ) ).

tff(fact_512_Diff__cancel,axiom,
    ! [A: fun_int_bool] : ( hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),A) = bot_bot_fun_int_bool ) ).

tff(fact_513_Diff__cancel,axiom,
    ! [A: fun_a_bool] : ( hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,A),A) = bot_bot_fun_a_bool ) ).

tff(fact_514_Diff__empty,axiom,
    ! [A: fun_nat_bool] : ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),bot_bot_fun_nat_bool) = A ) ).

tff(fact_515_Diff__empty,axiom,
    ! [A: fun_int_bool] : ( hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),bot_bot_fun_int_bool) = A ) ).

tff(fact_516_Diff__empty,axiom,
    ! [A: fun_a_bool] : ( hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,A),bot_bot_fun_a_bool) = A ) ).

tff(fact_517_empty__iff,axiom,
    ! [C_6: int] : ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,C_6),bot_bot_fun_int_bool)) ).

tff(fact_518_empty__iff,axiom,
    ! [C_6: nat] : ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_6),bot_bot_fun_nat_bool)) ).

tff(fact_519_empty__iff,axiom,
    ! [C_6: x_a] : ~ hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,C_6),bot_bot_fun_a_bool)) ).

tff(fact_520_empty__iff,axiom,
    ! [C_6: pname] : ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C_6),bot_bo844097828e_bool)) ).

tff(fact_521_empty__Collect__eq,axiom,
    ! [Pa: fun_int_bool] :
      ( ( bot_bot_fun_int_bool = collect_int(Pa) )
    <=> ! [X_2: int] : ~ hBOOL(hAPP_int_bool(Pa,X_2)) ) ).

tff(fact_522_empty__Collect__eq,axiom,
    ! [Pa: fun_nat_bool] :
      ( ( bot_bot_fun_nat_bool = collect_nat(Pa) )
    <=> ! [X_2: nat] : ~ hBOOL(hAPP_nat_bool(Pa,X_2)) ) ).

tff(fact_523_empty__Collect__eq,axiom,
    ! [Pa: fun_a_bool] :
      ( ( bot_bot_fun_a_bool = collect_a(Pa) )
    <=> ! [X_2: x_a] : ~ hBOOL(hAPP_a_bool(Pa,X_2)) ) ).

tff(fact_524_empty__Diff,axiom,
    ! [A: fun_nat_bool] : ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,bot_bot_fun_nat_bool),A) = bot_bot_fun_nat_bool ) ).

tff(fact_525_empty__Diff,axiom,
    ! [A: fun_int_bool] : ( hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,bot_bot_fun_int_bool),A) = bot_bot_fun_int_bool ) ).

tff(fact_526_empty__Diff,axiom,
    ! [A: fun_a_bool] : ( hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,bot_bot_fun_a_bool),A) = bot_bot_fun_a_bool ) ).

tff(fact_527_ex__in__conv,axiom,
    ! [A: fun_int_bool] :
      ( ? [X_2: int] : hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),A))
    <=> ( A != bot_bot_fun_int_bool ) ) ).

tff(fact_528_ex__in__conv,axiom,
    ! [A: fun_nat_bool] :
      ( ? [X_2: nat] : hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A))
    <=> ( A != bot_bot_fun_nat_bool ) ) ).

tff(fact_529_ex__in__conv,axiom,
    ! [A: fun_a_bool] :
      ( ? [X_2: x_a] : hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_2),A))
    <=> ( A != bot_bot_fun_a_bool ) ) ).

tff(fact_530_ex__in__conv,axiom,
    ! [A: fun_pname_bool] :
      ( ? [X_2: pname] : hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),A))
    <=> ( A != bot_bo844097828e_bool ) ) ).

tff(fact_531_all__not__in__conv,axiom,
    ! [A: fun_int_bool] :
      ( ! [X_2: int] : ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),A))
    <=> ( A = bot_bot_fun_int_bool ) ) ).

tff(fact_532_all__not__in__conv,axiom,
    ! [A: fun_nat_bool] :
      ( ! [X_2: nat] : ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A))
    <=> ( A = bot_bot_fun_nat_bool ) ) ).

tff(fact_533_all__not__in__conv,axiom,
    ! [A: fun_a_bool] :
      ( ! [X_2: x_a] : ~ hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_2),A))
    <=> ( A = bot_bot_fun_a_bool ) ) ).

tff(fact_534_all__not__in__conv,axiom,
    ! [A: fun_pname_bool] :
      ( ! [X_2: pname] : ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),A))
    <=> ( A = bot_bo844097828e_bool ) ) ).

tff(fact_535_bot__apply,axiom,
    ! [X_1: nat] :
      ( hBOOL(hAPP_nat_bool(bot_bot_fun_nat_bool,X_1))
    <=> hBOOL(bot_bot_bool) ) ).

tff(fact_536_bot__apply,axiom,
    ! [X_1: int] :
      ( hBOOL(hAPP_int_bool(bot_bot_fun_int_bool,X_1))
    <=> hBOOL(bot_bot_bool) ) ).

tff(fact_537_bot__apply,axiom,
    ! [X_1: x_a] :
      ( hBOOL(hAPP_a_bool(bot_bot_fun_a_bool,X_1))
    <=> hBOOL(bot_bot_bool) ) ).

tff(fact_538_empty__def,axiom,
    bot_bot_fun_int_bool = collect_int(cOMBK_bool_int(fFalse)) ).

tff(fact_539_empty__def,axiom,
    bot_bot_fun_nat_bool = collect_nat(cOMBK_bool_nat(fFalse)) ).

tff(fact_540_empty__def,axiom,
    bot_bot_fun_a_bool = collect_a(cOMBK_bool_a(fFalse)) ).

tff(fact_541_bot__fun__def,axiom,
    ! [X_2: nat] :
      ( hBOOL(hAPP_nat_bool(bot_bot_fun_nat_bool,X_2))
    <=> hBOOL(bot_bot_bool) ) ).

tff(fact_542_bot__fun__def,axiom,
    ! [X_2: int] :
      ( hBOOL(hAPP_int_bool(bot_bot_fun_int_bool,X_2))
    <=> hBOOL(bot_bot_bool) ) ).

tff(fact_543_bot__fun__def,axiom,
    ! [X_2: x_a] :
      ( hBOOL(hAPP_a_bool(bot_bot_fun_a_bool,X_2))
    <=> hBOOL(bot_bot_bool) ) ).

tff(fact_544_insert__Diff,axiom,
    ! [A_2: x_a,A: fun_a_bool] :
      ( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,A_2),A))
     => ( hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,A_2),hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,A),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,A_2),bot_bot_fun_a_bool))) = A ) ) ).

tff(fact_545_insert__Diff,axiom,
    ! [A_2: int,A: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_2),A))
     => ( hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_2),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_2),bot_bot_fun_int_bool))) = A ) ) ).

tff(fact_546_insert__Diff,axiom,
    ! [A_2: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_2),A))
     => ( hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_2),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_2),bot_bot_fun_nat_bool))) = A ) ) ).

tff(fact_547_insert__Diff,axiom,
    ! [A_2: pname,A: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_2),A))
     => ( hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_2),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,A),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_2),bot_bo844097828e_bool))) = A ) ) ).

tff(fact_548_Diff__insert__absorb,axiom,
    ! [X_1: x_a,A: fun_a_bool] :
      ( ~ hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_1),A))
     => ( hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_1),A)),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_1),bot_bot_fun_a_bool)) = A ) ) ).

tff(fact_549_Diff__insert__absorb,axiom,
    ! [X_1: int,A: fun_int_bool] :
      ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
     => ( hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A)),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool)) = A ) ) ).

tff(fact_550_Diff__insert__absorb,axiom,
    ! [X_1: nat,A: fun_nat_bool] :
      ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
     => ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)) = A ) ) ).

tff(fact_551_Diff__insert__absorb,axiom,
    ! [X_1: pname,A: fun_pname_bool] :
      ( ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_1),A))
     => ( hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_1),A)),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_1),bot_bo844097828e_bool)) = A ) ) ).

tff(fact_552_insert__Diff__single,axiom,
    ! [A_2: x_a,A: fun_a_bool] : ( hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,A_2),hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,A),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,A_2),bot_bot_fun_a_bool))) = hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,A_2),A) ) ).

tff(fact_553_insert__Diff__single,axiom,
    ! [A_2: nat,A: fun_nat_bool] : ( hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_2),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_2),bot_bot_fun_nat_bool))) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_2),A) ) ).

tff(fact_554_insert__Diff__single,axiom,
    ! [A_2: int,A: fun_int_bool] : ( hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_2),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_2),bot_bot_fun_int_bool))) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_2),A) ) ).

tff(fact_555_Diff__insert2,axiom,
    ! [A: fun_a_bool,A_2: x_a,B: fun_a_bool] : ( hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,A),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,A_2),B)) = hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,A),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,A_2),bot_bot_fun_a_bool))),B) ) ).

tff(fact_556_Diff__insert2,axiom,
    ! [A: fun_nat_bool,A_2: nat,B: fun_nat_bool] : ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_2),B)) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_2),bot_bot_fun_nat_bool))),B) ) ).

tff(fact_557_Diff__insert2,axiom,
    ! [A: fun_int_bool,A_2: int,B: fun_int_bool] : ( hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_2),B)) = hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_2),bot_bot_fun_int_bool))),B) ) ).

tff(fact_558_Diff__insert,axiom,
    ! [A: fun_a_bool,A_2: x_a,B: fun_a_bool] : ( hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,A),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,A_2),B)) = hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,A),B)),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,A_2),bot_bot_fun_a_bool)) ) ).

tff(fact_559_Diff__insert,axiom,
    ! [A: fun_nat_bool,A_2: nat,B: fun_nat_bool] : ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_2),B)) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B)),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_2),bot_bot_fun_nat_bool)) ) ).

tff(fact_560_Diff__insert,axiom,
    ! [A: fun_int_bool,A_2: int,B: fun_int_bool] : ( hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_2),B)) = hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),B)),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_2),bot_bot_fun_int_bool)) ) ).

tff(fact_561_diff__single__insert,axiom,
    ! [A: fun_a_bool,X_1: x_a,B: fun_a_bool] :
      ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,A),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_1),bot_bot_fun_a_bool))),B))
     => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_1),A))
       => hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_1),B))) ) ) ).

tff(fact_562_diff__single__insert,axiom,
    ! [A: fun_int_bool,X_1: int,B: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool))),B))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
       => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),B))) ) ) ).

tff(fact_563_diff__single__insert,axiom,
    ! [A: fun_nat_bool,X_1: nat,B: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool))),B))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
       => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),B))) ) ) ).

tff(fact_564_diff__single__insert,axiom,
    ! [A: fun_pname_bool,X_1: pname,B: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,A),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_1),bot_bo844097828e_bool))),B))
     => ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_1),A))
       => hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_1),B))) ) ) ).

tff(fact_565_subset__insert__iff,axiom,
    ! [A: fun_a_bool,X_1: x_a,B: fun_a_bool] :
      ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_1),B)))
    <=> ( ( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_1),A))
         => hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,A),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_1),bot_bot_fun_a_bool))),B)) )
        & ( ~ hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_1),A))
         => hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),B)) ) ) ) ).

tff(fact_566_subset__insert__iff,axiom,
    ! [A: fun_int_bool,X_1: int,B: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),B)))
    <=> ( ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
         => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool))),B)) )
        & ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
         => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),B)) ) ) ) ).

tff(fact_567_subset__insert__iff,axiom,
    ! [A: fun_nat_bool,X_1: nat,B: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),B)))
    <=> ( ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
         => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool))),B)) )
        & ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
         => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B)) ) ) ) ).

tff(fact_568_subset__insert__iff,axiom,
    ! [A: fun_pname_bool,X_1: pname,B: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_1),B)))
    <=> ( ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_1),A))
         => hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,A),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_1),bot_bo844097828e_bool))),B)) )
        & ( ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_1),A))
         => hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A),B)) ) ) ) ).

tff(fact_569_finite__Diff2,axiom,
    ! [A: fun_int_bool,B: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,B))
     => ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),B)))
      <=> hBOOL(hAPP_f448129468l_bool(finite_finite_int,A)) ) ) ).

tff(fact_570_finite__Diff2,axiom,
    ! [A: fun_nat_bool,B: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B))
     => ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B)))
      <=> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A)) ) ) ).

tff(fact_571_finite__Diff2,axiom,
    ! [A: fun_pname_bool,B: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,B))
     => ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,A),B)))
      <=> hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A)) ) ) ).

tff(fact_572_insert__Diff1,axiom,
    ! [A: fun_a_bool,X_1: x_a,B: fun_a_bool] :
      ( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_1),B))
     => ( hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_1),A)),B) = hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,A),B) ) ) ).

tff(fact_573_insert__Diff1,axiom,
    ! [A: fun_int_bool,X_1: int,B: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),B))
     => ( hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A)),B) = hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),B) ) ) ).

tff(fact_574_insert__Diff1,axiom,
    ! [A: fun_nat_bool,X_1: nat,B: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),B))
     => ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)),B) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B) ) ) ).

tff(fact_575_insert__Diff1,axiom,
    ! [A: fun_pname_bool,X_1: pname,B: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_1),B))
     => ( hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_1),A)),B) = hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,A),B) ) ) ).

tff(fact_576_insert__Diff__if,axiom,
    ! [A: fun_a_bool,X_1: x_a,B: fun_a_bool] :
      ( ( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_1),B))
       => ( hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_1),A)),B) = hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,A),B) ) )
      & ( ~ hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_1),B))
       => ( hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_1),A)),B) = hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_1),hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,A),B)) ) ) ) ).

tff(fact_577_insert__Diff__if,axiom,
    ! [A: fun_int_bool,X_1: int,B: fun_int_bool] :
      ( ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),B))
       => ( hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A)),B) = hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),B) ) )
      & ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),B))
       => ( hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A)),B) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),B)) ) ) ) ).

tff(fact_578_insert__Diff__if,axiom,
    ! [A: fun_nat_bool,X_1: nat,B: fun_nat_bool] :
      ( ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),B))
       => ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)),B) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B) ) )
      & ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),B))
       => ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)),B) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B)) ) ) ) ).

tff(fact_579_insert__Diff__if,axiom,
    ! [A: fun_pname_bool,X_1: pname,B: fun_pname_bool] :
      ( ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_1),B))
       => ( hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_1),A)),B) = hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,A),B) ) )
      & ( ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_1),B))
       => ( hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_1),A)),B) = hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_1),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,A),B)) ) ) ) ).

tff(fact_580_double__diff,axiom,
    ! [C_11: fun_int_bool,A: fun_int_bool,B: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),B))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,B),C_11))
       => ( hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,B),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,C_11),A)) = A ) ) ) ).

tff(fact_581_double__diff,axiom,
    ! [C_11: fun_nat_bool,A: fun_nat_bool,B: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B),C_11))
       => ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,B),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,C_11),A)) = A ) ) ) ).

tff(fact_582_double__diff,axiom,
    ! [C_11: fun_a_bool,A: fun_a_bool,B: fun_a_bool] :
      ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),B))
     => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B),C_11))
       => ( hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,B),hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,C_11),A)) = A ) ) ) ).

tff(fact_583_Diff__mono,axiom,
    ! [D_4: fun_int_bool,B: fun_int_bool,A: fun_int_bool,C_11: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),C_11))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,D_4),B))
       => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),B)),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,C_11),D_4))) ) ) ).

tff(fact_584_Diff__mono,axiom,
    ! [D_4: fun_nat_bool,B: fun_nat_bool,A: fun_nat_bool,C_11: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),C_11))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,D_4),B))
       => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B)),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,C_11),D_4))) ) ) ).

tff(fact_585_Diff__mono,axiom,
    ! [D_4: fun_a_bool,B: fun_a_bool,A: fun_a_bool,C_11: fun_a_bool] :
      ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),C_11))
     => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,D_4),B))
       => hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,A),B)),hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,C_11),D_4))) ) ) ).

tff(fact_586_Diff__subset,axiom,
    ! [A: fun_int_bool,B: fun_int_bool] : hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),B)),A)) ).

tff(fact_587_Diff__subset,axiom,
    ! [A: fun_nat_bool,B: fun_nat_bool] : hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B)),A)) ).

tff(fact_588_Diff__subset,axiom,
    ! [A: fun_a_bool,B: fun_a_bool] : hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,A),B)),A)) ).

tff(fact_589_singleton__inject,axiom,
    ! [A_2: x_a,B_3: x_a] :
      ( ( hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,A_2),bot_bot_fun_a_bool) = hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,B_3),bot_bot_fun_a_bool) )
     => ( A_2 = B_3 ) ) ).

tff(fact_590_singleton__inject,axiom,
    ! [A_2: nat,B_3: nat] :
      ( ( hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_2),bot_bot_fun_nat_bool) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,B_3),bot_bot_fun_nat_bool) )
     => ( A_2 = B_3 ) ) ).

tff(fact_591_singleton__inject,axiom,
    ! [A_2: int,B_3: int] :
      ( ( hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_2),bot_bot_fun_int_bool) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,B_3),bot_bot_fun_int_bool) )
     => ( A_2 = B_3 ) ) ).

tff(fact_592_singletonE,axiom,
    ! [B_3: x_a,A_2: x_a] :
      ( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,B_3),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,A_2),bot_bot_fun_a_bool)))
     => ( B_3 = A_2 ) ) ).

tff(fact_593_singletonE,axiom,
    ! [B_3: int,A_2: int] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B_3),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_2),bot_bot_fun_int_bool)))
     => ( B_3 = A_2 ) ) ).

tff(fact_594_singletonE,axiom,
    ! [B_3: nat,A_2: nat] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,B_3),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_2),bot_bot_fun_nat_bool)))
     => ( B_3 = A_2 ) ) ).

tff(fact_595_singletonE,axiom,
    ! [B_3: pname,A_2: pname] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,B_3),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_2),bot_bo844097828e_bool)))
     => ( B_3 = A_2 ) ) ).

tff(fact_596_doubleton__eq__iff,axiom,
    ! [A_2: x_a,B_3: x_a,C_6: x_a,D_1: x_a] :
      ( ( hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,A_2),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,B_3),bot_bot_fun_a_bool)) = hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,C_6),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,D_1),bot_bot_fun_a_bool)) )
    <=> ( ( ( A_2 = C_6 )
          & ( B_3 = D_1 ) )
        | ( ( A_2 = D_1 )
          & ( B_3 = C_6 ) ) ) ) ).

tff(fact_597_doubleton__eq__iff,axiom,
    ! [A_2: nat,B_3: nat,C_6: nat,D_1: nat] :
      ( ( hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_2),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,B_3),bot_bot_fun_nat_bool)) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,C_6),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,D_1),bot_bot_fun_nat_bool)) )
    <=> ( ( ( A_2 = C_6 )
          & ( B_3 = D_1 ) )
        | ( ( A_2 = D_1 )
          & ( B_3 = C_6 ) ) ) ) ).

tff(fact_598_doubleton__eq__iff,axiom,
    ! [A_2: int,B_3: int,C_6: int,D_1: int] :
      ( ( hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_2),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,B_3),bot_bot_fun_int_bool)) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,C_6),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,D_1),bot_bot_fun_int_bool)) )
    <=> ( ( ( A_2 = C_6 )
          & ( B_3 = D_1 ) )
        | ( ( A_2 = D_1 )
          & ( B_3 = C_6 ) ) ) ) ).

tff(fact_599_singleton__iff,axiom,
    ! [B_3: x_a,A_2: x_a] :
      ( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,B_3),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,A_2),bot_bot_fun_a_bool)))
    <=> ( B_3 = A_2 ) ) ).

tff(fact_600_singleton__iff,axiom,
    ! [B_3: int,A_2: int] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B_3),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_2),bot_bot_fun_int_bool)))
    <=> ( B_3 = A_2 ) ) ).

tff(fact_601_singleton__iff,axiom,
    ! [B_3: nat,A_2: nat] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,B_3),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_2),bot_bot_fun_nat_bool)))
    <=> ( B_3 = A_2 ) ) ).

tff(fact_602_singleton__iff,axiom,
    ! [B_3: pname,A_2: pname] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,B_3),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_2),bot_bo844097828e_bool)))
    <=> ( B_3 = A_2 ) ) ).

tff(fact_603_insert__not__empty,axiom,
    ! [A_2: x_a,A: fun_a_bool] : ( hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,A_2),A) != bot_bot_fun_a_bool ) ).

tff(fact_604_insert__not__empty,axiom,
    ! [A_2: nat,A: fun_nat_bool] : ( hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_2),A) != bot_bot_fun_nat_bool ) ).

tff(fact_605_insert__not__empty,axiom,
    ! [A_2: int,A: fun_int_bool] : ( hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_2),A) != bot_bot_fun_int_bool ) ).

tff(fact_606_empty__not__insert,axiom,
    ! [A_2: x_a,A: fun_a_bool] : ( bot_bot_fun_a_bool != hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,A_2),A) ) ).

tff(fact_607_empty__not__insert,axiom,
    ! [A_2: nat,A: fun_nat_bool] : ( bot_bot_fun_nat_bool != hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_2),A) ) ).

tff(fact_608_empty__not__insert,axiom,
    ! [A_2: int,A: fun_int_bool] : ( bot_bot_fun_int_bool != hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_2),A) ) ).

tff(fact_609_subset__empty,axiom,
    ! [A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),bot_bot_fun_nat_bool))
    <=> ( A = bot_bot_fun_nat_bool ) ) ).

tff(fact_610_subset__empty,axiom,
    ! [A: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),bot_bot_fun_int_bool))
    <=> ( A = bot_bot_fun_int_bool ) ) ).

tff(fact_611_subset__empty,axiom,
    ! [A: fun_a_bool] :
      ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),bot_bot_fun_a_bool))
    <=> ( A = bot_bot_fun_a_bool ) ) ).

tff(fact_612_image__is__empty,axiom,
    ! [F: fun_nat_int,A: fun_nat_bool] :
      ( ( image_nat_int(F,A) = bot_bot_fun_int_bool )
    <=> ( A = bot_bot_fun_nat_bool ) ) ).

tff(fact_613_image__is__empty,axiom,
    ! [F: fun_pname_a,A: fun_pname_bool] :
      ( ( image_pname_a(F,A) = bot_bot_fun_a_bool )
    <=> ( A = bot_bo844097828e_bool ) ) ).

tff(fact_614_image__empty,axiom,
    ! [F: fun_nat_int] : ( image_nat_int(F,bot_bot_fun_nat_bool) = bot_bot_fun_int_bool ) ).

tff(fact_615_image__empty,axiom,
    ! [F: fun_pname_a] : ( image_pname_a(F,bot_bo844097828e_bool) = bot_bot_fun_a_bool ) ).

tff(fact_616_empty__is__image,axiom,
    ! [F: fun_nat_int,A: fun_nat_bool] :
      ( ( bot_bot_fun_int_bool = image_nat_int(F,A) )
    <=> ( A = bot_bot_fun_nat_bool ) ) ).

tff(fact_617_empty__is__image,axiom,
    ! [F: fun_pname_a,A: fun_pname_bool] :
      ( ( bot_bot_fun_a_bool = image_pname_a(F,A) )
    <=> ( A = bot_bo844097828e_bool ) ) ).

tff(fact_618_le__bot,axiom,
    ! [A_2: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_2),bot_bot_fun_nat_bool))
     => ( A_2 = bot_bot_fun_nat_bool ) ) ).

tff(fact_619_le__bot,axiom,
    ! [A_2: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A_2),bot_bot_fun_int_bool))
     => ( A_2 = bot_bot_fun_int_bool ) ) ).

tff(fact_620_le__bot,axiom,
    ! [A_17: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_17),bot_bot_nat))
     => ( A_17 = bot_bot_nat ) ) ).

tff(fact_621_le__bot,axiom,
    ! [A_2: fun_a_bool] :
      ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A_2),bot_bot_fun_a_bool))
     => ( A_2 = bot_bot_fun_a_bool ) ) ).

tff(fact_622_bot__unique,axiom,
    ! [A_2: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_2),bot_bot_fun_nat_bool))
    <=> ( A_2 = bot_bot_fun_nat_bool ) ) ).

tff(fact_623_bot__unique,axiom,
    ! [A_2: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A_2),bot_bot_fun_int_bool))
    <=> ( A_2 = bot_bot_fun_int_bool ) ) ).

tff(fact_624_bot__unique,axiom,
    ! [A_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_2),bot_bot_nat))
    <=> ( A_2 = bot_bot_nat ) ) ).

tff(fact_625_bot__unique,axiom,
    ! [A_2: fun_a_bool] :
      ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A_2),bot_bot_fun_a_bool))
    <=> ( A_2 = bot_bot_fun_a_bool ) ) ).

tff(fact_626_bot__least,axiom,
    ! [A_2: fun_nat_bool] : hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,bot_bot_fun_nat_bool),A_2)) ).

tff(fact_627_bot__least,axiom,
    ! [A_2: fun_int_bool] : hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,bot_bot_fun_int_bool),A_2)) ).

tff(fact_628_bot__least,axiom,
    ! [A_16: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,bot_bot_nat),A_16)) ).

tff(fact_629_bot__least,axiom,
    ! [A_2: fun_a_bool] : hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,bot_bot_fun_a_bool),A_2)) ).

tff(fact_630_Collect__conv__if,axiom,
    ! [Pa: fun_a_bool,A_2: x_a] :
      ( ( hBOOL(hAPP_a_bool(Pa,A_2))
       => ( collect_a(cOMBS_a_bool_bool(cOMBB_1972296269bool_a(fconj,hAPP_a_fun_a_bool(cOMBC_a_a_bool(fequal_a),A_2)),Pa)) = hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,A_2),bot_bot_fun_a_bool) ) )
      & ( ~ hBOOL(hAPP_a_bool(Pa,A_2))
       => ( collect_a(cOMBS_a_bool_bool(cOMBB_1972296269bool_a(fconj,hAPP_a_fun_a_bool(cOMBC_a_a_bool(fequal_a),A_2)),Pa)) = bot_bot_fun_a_bool ) ) ) ).

tff(fact_631_Collect__conv__if,axiom,
    ! [Pa: fun_int_bool,A_2: int] :
      ( ( hBOOL(hAPP_int_bool(Pa,A_2))
       => ( collect_int(cOMBS_int_bool_bool(cOMBB_1652995168ol_int(fconj,hAPP_i1948725293t_bool(cOMBC_int_int_bool(fequal_int),A_2)),Pa)) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_2),bot_bot_fun_int_bool) ) )
      & ( ~ hBOOL(hAPP_int_bool(Pa,A_2))
       => ( collect_int(cOMBS_int_bool_bool(cOMBB_1652995168ol_int(fconj,hAPP_i1948725293t_bool(cOMBC_int_int_bool(fequal_int),A_2)),Pa)) = bot_bot_fun_int_bool ) ) ) ).

tff(fact_632_Collect__conv__if,axiom,
    ! [Pa: fun_nat_bool,A_2: nat] :
      ( ( hBOOL(hAPP_nat_bool(Pa,A_2))
       => ( collect_nat(cOMBS_nat_bool_bool(cOMBB_1015721476ol_nat(fconj,hAPP_n1699378549t_bool(cOMBC_nat_nat_bool(fequal_nat),A_2)),Pa)) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_2),bot_bot_fun_nat_bool) ) )
      & ( ~ hBOOL(hAPP_nat_bool(Pa,A_2))
       => ( collect_nat(cOMBS_nat_bool_bool(cOMBB_1015721476ol_nat(fconj,hAPP_n1699378549t_bool(cOMBC_nat_nat_bool(fequal_nat),A_2)),Pa)) = bot_bot_fun_nat_bool ) ) ) ).

tff(fact_633_Collect__conv__if2,axiom,
    ! [Pa: fun_a_bool,A_2: x_a] :
      ( ( hBOOL(hAPP_a_bool(Pa,A_2))
       => ( collect_a(cOMBS_a_bool_bool(cOMBB_1972296269bool_a(fconj,hAPP_a_fun_a_bool(fequal_a,A_2)),Pa)) = hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,A_2),bot_bot_fun_a_bool) ) )
      & ( ~ hBOOL(hAPP_a_bool(Pa,A_2))
       => ( collect_a(cOMBS_a_bool_bool(cOMBB_1972296269bool_a(fconj,hAPP_a_fun_a_bool(fequal_a,A_2)),Pa)) = bot_bot_fun_a_bool ) ) ) ).

tff(fact_634_Collect__conv__if2,axiom,
    ! [Pa: fun_int_bool,A_2: int] :
      ( ( hBOOL(hAPP_int_bool(Pa,A_2))
       => ( collect_int(cOMBS_int_bool_bool(cOMBB_1652995168ol_int(fconj,hAPP_i1948725293t_bool(fequal_int,A_2)),Pa)) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_2),bot_bot_fun_int_bool) ) )
      & ( ~ hBOOL(hAPP_int_bool(Pa,A_2))
       => ( collect_int(cOMBS_int_bool_bool(cOMBB_1652995168ol_int(fconj,hAPP_i1948725293t_bool(fequal_int,A_2)),Pa)) = bot_bot_fun_int_bool ) ) ) ).

tff(fact_635_Collect__conv__if2,axiom,
    ! [Pa: fun_nat_bool,A_2: nat] :
      ( ( hBOOL(hAPP_nat_bool(Pa,A_2))
       => ( collect_nat(cOMBS_nat_bool_bool(cOMBB_1015721476ol_nat(fconj,hAPP_n1699378549t_bool(fequal_nat,A_2)),Pa)) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_2),bot_bot_fun_nat_bool) ) )
      & ( ~ hBOOL(hAPP_nat_bool(Pa,A_2))
       => ( collect_nat(cOMBS_nat_bool_bool(cOMBB_1015721476ol_nat(fconj,hAPP_n1699378549t_bool(fequal_nat,A_2)),Pa)) = bot_bot_fun_nat_bool ) ) ) ).

tff(fact_636_singleton__conv,axiom,
    ! [A_2: x_a] : ( collect_a(hAPP_a_fun_a_bool(cOMBC_a_a_bool(fequal_a),A_2)) = hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,A_2),bot_bot_fun_a_bool) ) ).

tff(fact_637_singleton__conv,axiom,
    ! [A_2: int] : ( collect_int(hAPP_i1948725293t_bool(cOMBC_int_int_bool(fequal_int),A_2)) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_2),bot_bot_fun_int_bool) ) ).

tff(fact_638_singleton__conv,axiom,
    ! [A_2: nat] : ( collect_nat(hAPP_n1699378549t_bool(cOMBC_nat_nat_bool(fequal_nat),A_2)) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_2),bot_bot_fun_nat_bool) ) ).

tff(fact_639_singleton__conv2,axiom,
    ! [A_2: x_a] : ( collect_a(hAPP_a_fun_a_bool(fequal_a,A_2)) = hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,A_2),bot_bot_fun_a_bool) ) ).

tff(fact_640_singleton__conv2,axiom,
    ! [A_2: int] : ( collect_int(hAPP_i1948725293t_bool(fequal_int,A_2)) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_2),bot_bot_fun_int_bool) ) ).

tff(fact_641_singleton__conv2,axiom,
    ! [A_2: nat] : ( collect_nat(hAPP_n1699378549t_bool(fequal_nat,A_2)) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_2),bot_bot_fun_nat_bool) ) ).

tff(fact_642_card__Suc__Diff1,axiom,
    ! [X_1: x_a,A: fun_a_bool] :
      ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A))
     => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_1),A))
       => ( hAPP_nat_nat(suc,hAPP_fun_a_bool_nat(finite_card_a,hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,A),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_1),bot_bot_fun_a_bool)))) = hAPP_fun_a_bool_nat(finite_card_a,A) ) ) ) ).

tff(fact_643_card__Suc__Diff1,axiom,
    ! [X_1: int,A: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
       => ( hAPP_nat_nat(suc,hAPP_f957591787ol_nat(finite_card_int,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool)))) = hAPP_f957591787ol_nat(finite_card_int,A) ) ) ) ).

tff(fact_644_card__Suc__Diff1,axiom,
    ! [X_1: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
       => ( hAPP_nat_nat(suc,hAPP_f22106695ol_nat(finite_card_nat,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)))) = hAPP_f22106695ol_nat(finite_card_nat,A) ) ) ) ).

tff(fact_645_card__Suc__Diff1,axiom,
    ! [X_1: pname,A: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A))
     => ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_1),A))
       => ( hAPP_nat_nat(suc,hAPP_f921600141ol_nat(finite_card_pname,hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,A),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_1),bot_bo844097828e_bool)))) = hAPP_f921600141ol_nat(finite_card_pname,A) ) ) ) ).

tff(fact_646_card__insert,axiom,
    ! [X_1: x_a,A: fun_a_bool] :
      ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A))
     => ( hAPP_fun_a_bool_nat(finite_card_a,hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_1),A)) = hAPP_nat_nat(suc,hAPP_fun_a_bool_nat(finite_card_a,hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,A),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_1),bot_bot_fun_a_bool)))) ) ) ).

tff(fact_647_card__insert,axiom,
    ! [X_1: int,A: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
     => ( hAPP_f957591787ol_nat(finite_card_int,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A)) = hAPP_nat_nat(suc,hAPP_f957591787ol_nat(finite_card_int,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool)))) ) ) ).

tff(fact_648_card__insert,axiom,
    ! [X_1: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => ( hAPP_f22106695ol_nat(finite_card_nat,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)) = hAPP_nat_nat(suc,hAPP_f22106695ol_nat(finite_card_nat,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)))) ) ) ).

tff(fact_649_card__insert,axiom,
    ! [X_1: pname,A: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A))
     => ( hAPP_f921600141ol_nat(finite_card_pname,hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_1),A)) = hAPP_nat_nat(suc,hAPP_f921600141ol_nat(finite_card_pname,hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,A),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_1),bot_bo844097828e_bool)))) ) ) ).

tff(fact_650_card__Diff1__le,axiom,
    ! [X_1: x_a,A: fun_a_bool] :
      ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_fun_a_bool_nat(finite_card_a,hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,A),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_1),bot_bot_fun_a_bool)))),hAPP_fun_a_bool_nat(finite_card_a,A))) ) ).

tff(fact_651_card__Diff1__le,axiom,
    ! [X_1: int,A: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f957591787ol_nat(finite_card_int,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool)))),hAPP_f957591787ol_nat(finite_card_int,A))) ) ).

tff(fact_652_card__Diff1__le,axiom,
    ! [X_1: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f22106695ol_nat(finite_card_nat,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)))),hAPP_f22106695ol_nat(finite_card_nat,A))) ) ).

tff(fact_653_card__Diff1__le,axiom,
    ! [X_1: pname,A: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f921600141ol_nat(finite_card_pname,hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,A),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_1),bot_bo844097828e_bool)))),hAPP_f921600141ol_nat(finite_card_pname,A))) ) ).

tff(fact_654_finite__Diff__insert,axiom,
    ! [A: fun_a_bool,A_2: x_a,B: fun_a_bool] :
      ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,A),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,A_2),B))))
    <=> hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,A),B))) ) ).

tff(fact_655_finite__Diff__insert,axiom,
    ! [A: fun_int_bool,A_2: int,B: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_2),B))))
    <=> hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),B))) ) ).

tff(fact_656_finite__Diff__insert,axiom,
    ! [A: fun_nat_bool,A_2: nat,B: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_2),B))))
    <=> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B))) ) ).

tff(fact_657_finite__Diff__insert,axiom,
    ! [A: fun_pname_bool,A_2: pname,B: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,A),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_2),B))))
    <=> hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,A),B))) ) ).

tff(fact_658_image__diff__subset,axiom,
    ! [F: fun_nat_int,A: fun_nat_bool,B: fun_nat_bool] : hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,image_nat_int(F,A)),image_nat_int(F,B))),image_nat_int(F,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B)))) ).

tff(fact_659_image__diff__subset,axiom,
    ! [F: fun_pname_a,A: fun_pname_bool,B: fun_pname_bool] : hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,image_pname_a(F,A)),image_pname_a(F,B))),image_pname_a(F,hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,A),B)))) ).

tff(fact_660_subset__singletonD,axiom,
    ! [A: fun_a_bool,X_1: x_a] :
      ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_1),bot_bot_fun_a_bool)))
     => ( ( A = bot_bot_fun_a_bool )
        | ( A = hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_1),bot_bot_fun_a_bool) ) ) ) ).

tff(fact_661_subset__singletonD,axiom,
    ! [A: fun_nat_bool,X_1: nat] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)))
     => ( ( A = bot_bot_fun_nat_bool )
        | ( A = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool) ) ) ) ).

tff(fact_662_subset__singletonD,axiom,
    ! [A: fun_int_bool,X_1: int] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool)))
     => ( ( A = bot_bot_fun_int_bool )
        | ( A = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool) ) ) ) ).

tff(fact_663_nat__case__Suc,axiom,
    ! [F1: nat,F2: fun_nat_nat,Nat_3: nat] : ( hAPP_nat_nat(nat_case_nat(F1,F2),hAPP_nat_nat(suc,Nat_3)) = hAPP_nat_nat(F2,Nat_3) ) ).

tff(fact_664_nat__case__Suc,axiom,
    ! [F1: bool,F2: fun_nat_bool,Nat_3: nat] :
      ( hBOOL(hAPP_nat_bool(nat_case_bool(F1,F2),hAPP_nat_nat(suc,Nat_3)))
    <=> hBOOL(hAPP_nat_bool(F2,Nat_3)) ) ).

tff(fact_665_image__constant,axiom,
    ! [C_6: int,X_1: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
     => ( image_nat_int(cOMBK_int_nat(C_6),A) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,C_6),bot_bot_fun_int_bool) ) ) ).

tff(fact_666_image__constant,axiom,
    ! [C_6: x_a,X_1: pname,A: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_1),A))
     => ( image_pname_a(cOMBK_a_pname(C_6),A) = hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,C_6),bot_bot_fun_a_bool) ) ) ).

tff(fact_667_image__constant__conv,axiom,
    ! [C_6: int,A: fun_nat_bool] :
      ( ( ( A = bot_bot_fun_nat_bool )
       => ( image_nat_int(cOMBK_int_nat(C_6),A) = bot_bot_fun_int_bool ) )
      & ( ( A != bot_bot_fun_nat_bool )
       => ( image_nat_int(cOMBK_int_nat(C_6),A) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,C_6),bot_bot_fun_int_bool) ) ) ) ).

tff(fact_668_image__constant__conv,axiom,
    ! [C_6: x_a,A: fun_pname_bool] :
      ( ( ( A = bot_bo844097828e_bool )
       => ( image_pname_a(cOMBK_a_pname(C_6),A) = bot_bot_fun_a_bool ) )
      & ( ( A != bot_bo844097828e_bool )
       => ( image_pname_a(cOMBK_a_pname(C_6),A) = hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,C_6),bot_bot_fun_a_bool) ) ) ) ).

tff(fact_669_diff__eq__diff__eq,axiom,
    ! [A_2: int,B_3: int,C_6: int,D_1: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_2),B_3) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,C_6),D_1) )
     => ( ( A_2 = B_3 )
      <=> ( C_6 = D_1 ) ) ) ).

tff(fact_670_card__Diff__subset,axiom,
    ! [A: fun_int_bool,B: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,B))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,B),A))
       => ( hAPP_f957591787ol_nat(finite_card_int,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),B)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_f957591787ol_nat(finite_card_int,A)),hAPP_f957591787ol_nat(finite_card_int,B)) ) ) ) ).

tff(fact_671_card__Diff__subset,axiom,
    ! [A: fun_nat_bool,B: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B),A))
       => ( hAPP_f22106695ol_nat(finite_card_nat,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_f22106695ol_nat(finite_card_nat,A)),hAPP_f22106695ol_nat(finite_card_nat,B)) ) ) ) ).

tff(fact_672_card__Diff__subset,axiom,
    ! [A: fun_a_bool,B: fun_a_bool] :
      ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,B))
     => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B),A))
       => ( hAPP_fun_a_bool_nat(finite_card_a,hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,A),B)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_fun_a_bool_nat(finite_card_a,A)),hAPP_fun_a_bool_nat(finite_card_a,B)) ) ) ) ).

tff(fact_673_card__Diff__subset,axiom,
    ! [A: fun_pname_bool,B: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,B))
     => ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,B),A))
       => ( hAPP_f921600141ol_nat(finite_card_pname,hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,A),B)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_f921600141ol_nat(finite_card_pname,A)),hAPP_f921600141ol_nat(finite_card_pname,B)) ) ) ) ).

tff(fact_674_diff__card__le__card__Diff,axiom,
    ! [A: fun_int_bool,B: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,B))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_f957591787ol_nat(finite_card_int,A)),hAPP_f957591787ol_nat(finite_card_int,B))),hAPP_f957591787ol_nat(finite_card_int,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),B)))) ) ).

tff(fact_675_diff__card__le__card__Diff,axiom,
    ! [A: fun_nat_bool,B: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_f22106695ol_nat(finite_card_nat,A)),hAPP_f22106695ol_nat(finite_card_nat,B))),hAPP_f22106695ol_nat(finite_card_nat,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B)))) ) ).

tff(fact_676_diff__card__le__card__Diff,axiom,
    ! [A: fun_a_bool,B: fun_a_bool] :
      ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,B))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_fun_a_bool_nat(finite_card_a,A)),hAPP_fun_a_bool_nat(finite_card_a,B))),hAPP_fun_a_bool_nat(finite_card_a,hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,A),B)))) ) ).

tff(fact_677_diff__card__le__card__Diff,axiom,
    ! [A: fun_pname_bool,B: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,B))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_f921600141ol_nat(finite_card_pname,A)),hAPP_f921600141ol_nat(finite_card_pname,B))),hAPP_f921600141ol_nat(finite_card_pname,hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,A),B)))) ) ).

tff(fact_678_finite__subset__induct,axiom,
    ! [Pa: fun_fun_a_bool_bool,A: fun_a_bool,F_1: fun_a_bool] :
      ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,F_1))
     => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,F_1),A))
       => ( hBOOL(hAPP_fun_a_bool_bool(Pa,bot_bot_fun_a_bool))
         => ( ! [A_14: x_a,F_2: fun_a_bool] :
                ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,F_2))
               => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,A_14),A))
                 => ( ~ hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,A_14),F_2))
                   => ( hBOOL(hAPP_fun_a_bool_bool(Pa,F_2))
                     => hBOOL(hAPP_fun_a_bool_bool(Pa,hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,A_14),F_2))) ) ) ) )
           => hBOOL(hAPP_fun_a_bool_bool(Pa,F_1)) ) ) ) ) ).

tff(fact_679_finite__subset__induct,axiom,
    ! [Pa: fun_fu819804446l_bool,A: fun_int_bool,F_1: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,F_1))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,F_1),A))
       => ( hBOOL(hAPP_f448129468l_bool(Pa,bot_bot_fun_int_bool))
         => ( ! [A_14: int,F_2: fun_int_bool] :
                ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,F_2))
               => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_14),A))
                 => ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_14),F_2))
                   => ( hBOOL(hAPP_f448129468l_bool(Pa,F_2))
                     => hBOOL(hAPP_f448129468l_bool(Pa,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_14),F_2))) ) ) ) )
           => hBOOL(hAPP_f448129468l_bool(Pa,F_1)) ) ) ) ) ).

tff(fact_680_finite__subset__induct,axiom,
    ! [Pa: fun_fu425979586l_bool,A: fun_nat_bool,F_1: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F_1))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,F_1),A))
       => ( hBOOL(hAPP_f54304608l_bool(Pa,bot_bot_fun_nat_bool))
         => ( ! [A_14: nat,F_2: fun_nat_bool] :
                ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F_2))
               => ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_14),A))
                 => ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_14),F_2))
                   => ( hBOOL(hAPP_f54304608l_bool(Pa,F_2))
                     => hBOOL(hAPP_f54304608l_bool(Pa,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_14),F_2))) ) ) ) )
           => hBOOL(hAPP_f54304608l_bool(Pa,F_1)) ) ) ) ) ).

tff(fact_681_finite__subset__induct,axiom,
    ! [Pa: fun_fu1430349052l_bool,A: fun_pname_bool,F_1: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,F_1))
     => ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,F_1),A))
       => ( hBOOL(hAPP_f1664156314l_bool(Pa,bot_bo844097828e_bool))
         => ( ! [A_14: pname,F_2: fun_pname_bool] :
                ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,F_2))
               => ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_14),A))
                 => ( ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_14),F_2))
                   => ( hBOOL(hAPP_f1664156314l_bool(Pa,F_2))
                     => hBOOL(hAPP_f1664156314l_bool(Pa,hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_14),F_2))) ) ) ) )
           => hBOOL(hAPP_f1664156314l_bool(Pa,F_1)) ) ) ) ) ).

tff(fact_682_assms_I2_J,axiom,
    ! [Pn: pname,G: fun_a_bool] :
      ( hBOOL(hAPP_fun_a_bool_bool(p(hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,hAPP_pname_a(mgt_call,Pn)),G)),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,mgt(the_com(body(Pn)))),bot_bot_fun_a_bool)))
     => hBOOL(hAPP_fun_a_bool_bool(p(G),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,hAPP_pname_a(mgt_call,Pn)),bot_bot_fun_a_bool))) ) ).

tff(fact_683_finite__empty__induct,axiom,
    ! [Pa: fun_fun_a_bool_bool,A: fun_a_bool] :
      ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A))
     => ( hBOOL(hAPP_fun_a_bool_bool(Pa,A))
       => ( ! [A_14: x_a,A_15: fun_a_bool] :
              ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A_15))
             => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,A_14),A_15))
               => ( hBOOL(hAPP_fun_a_bool_bool(Pa,A_15))
                 => hBOOL(hAPP_fun_a_bool_bool(Pa,hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,A_15),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,A_14),bot_bot_fun_a_bool)))) ) ) )
         => hBOOL(hAPP_fun_a_bool_bool(Pa,bot_bot_fun_a_bool)) ) ) ) ).

tff(fact_684_finite__empty__induct,axiom,
    ! [Pa: fun_fu819804446l_bool,A: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
     => ( hBOOL(hAPP_f448129468l_bool(Pa,A))
       => ( ! [A_14: int,A_15: fun_int_bool] :
              ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A_15))
             => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_14),A_15))
               => ( hBOOL(hAPP_f448129468l_bool(Pa,A_15))
                 => hBOOL(hAPP_f448129468l_bool(Pa,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A_15),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_14),bot_bot_fun_int_bool)))) ) ) )
         => hBOOL(hAPP_f448129468l_bool(Pa,bot_bot_fun_int_bool)) ) ) ) ).

tff(fact_685_finite__empty__induct,axiom,
    ! [Pa: fun_fu425979586l_bool,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => ( hBOOL(hAPP_f54304608l_bool(Pa,A))
       => ( ! [A_14: nat,A_15: fun_nat_bool] :
              ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_15))
             => ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_14),A_15))
               => ( hBOOL(hAPP_f54304608l_bool(Pa,A_15))
                 => hBOOL(hAPP_f54304608l_bool(Pa,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A_15),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_14),bot_bot_fun_nat_bool)))) ) ) )
         => hBOOL(hAPP_f54304608l_bool(Pa,bot_bot_fun_nat_bool)) ) ) ) ).

tff(fact_686_finite__empty__induct,axiom,
    ! [Pa: fun_fu1430349052l_bool,A: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A))
     => ( hBOOL(hAPP_f1664156314l_bool(Pa,A))
       => ( ! [A_14: pname,A_15: fun_pname_bool] :
              ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A_15))
             => ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_14),A_15))
               => ( hBOOL(hAPP_f1664156314l_bool(Pa,A_15))
                 => hBOOL(hAPP_f1664156314l_bool(Pa,hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,A_15),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_14),bot_bo844097828e_bool)))) ) ) )
         => hBOOL(hAPP_f1664156314l_bool(Pa,bot_bo844097828e_bool)) ) ) ) ).

tff(fact_687_finite__induct,axiom,
    ! [Pa: fun_fun_a_bool_bool,F_1: fun_a_bool] :
      ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,F_1))
     => ( hBOOL(hAPP_fun_a_bool_bool(Pa,bot_bot_fun_a_bool))
       => ( ! [X_2: x_a,F_2: fun_a_bool] :
              ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,F_2))
             => ( ~ hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_2),F_2))
               => ( hBOOL(hAPP_fun_a_bool_bool(Pa,F_2))
                 => hBOOL(hAPP_fun_a_bool_bool(Pa,hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_2),F_2))) ) ) )
         => hBOOL(hAPP_fun_a_bool_bool(Pa,F_1)) ) ) ) ).

tff(fact_688_finite__induct,axiom,
    ! [Pa: fun_fu819804446l_bool,F_1: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,F_1))
     => ( hBOOL(hAPP_f448129468l_bool(Pa,bot_bot_fun_int_bool))
       => ( ! [X_2: int,F_2: fun_int_bool] :
              ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,F_2))
             => ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),F_2))
               => ( hBOOL(hAPP_f448129468l_bool(Pa,F_2))
                 => hBOOL(hAPP_f448129468l_bool(Pa,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_2),F_2))) ) ) )
         => hBOOL(hAPP_f448129468l_bool(Pa,F_1)) ) ) ) ).

tff(fact_689_finite__induct,axiom,
    ! [Pa: fun_fu425979586l_bool,F_1: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F_1))
     => ( hBOOL(hAPP_f54304608l_bool(Pa,bot_bot_fun_nat_bool))
       => ( ! [X_2: nat,F_2: fun_nat_bool] :
              ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F_2))
             => ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),F_2))
               => ( hBOOL(hAPP_f54304608l_bool(Pa,F_2))
                 => hBOOL(hAPP_f54304608l_bool(Pa,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_2),F_2))) ) ) )
         => hBOOL(hAPP_f54304608l_bool(Pa,F_1)) ) ) ) ).

tff(fact_690_finite__induct,axiom,
    ! [Pa: fun_fu1430349052l_bool,F_1: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,F_1))
     => ( hBOOL(hAPP_f1664156314l_bool(Pa,bot_bo844097828e_bool))
       => ( ! [X_2: pname,F_2: fun_pname_bool] :
              ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,F_2))
             => ( ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),F_2))
               => ( hBOOL(hAPP_f1664156314l_bool(Pa,F_2))
                 => hBOOL(hAPP_f1664156314l_bool(Pa,hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_2),F_2))) ) ) )
         => hBOOL(hAPP_f1664156314l_bool(Pa,F_1)) ) ) ) ).

tff(fact_691_finite_Osimps,axiom,
    ! [A_2: fun_a_bool] :
      ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A_2))
    <=> ( ( A_2 = bot_bot_fun_a_bool )
        | ? [A_15: fun_a_bool,A_14: x_a] :
            ( ( A_2 = hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,A_14),A_15) )
            & hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A_15)) ) ) ) ).

tff(fact_692_finite_Osimps,axiom,
    ! [A_2: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A_2))
    <=> ( ( A_2 = bot_bot_fun_int_bool )
        | ? [A_15: fun_int_bool,A_14: int] :
            ( ( A_2 = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_14),A_15) )
            & hBOOL(hAPP_f448129468l_bool(finite_finite_int,A_15)) ) ) ) ).

tff(fact_693_finite_Osimps,axiom,
    ! [A_2: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_2))
    <=> ( ( A_2 = bot_bot_fun_nat_bool )
        | ? [A_15: fun_nat_bool,A_14: nat] :
            ( ( A_2 = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_14),A_15) )
            & hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_15)) ) ) ) ).

tff(fact_694_finite_Osimps,axiom,
    ! [A_2: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A_2))
    <=> ( ( A_2 = bot_bo844097828e_bool )
        | ? [A_15: fun_pname_bool,A_14: pname] :
            ( ( A_2 = hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_14),A_15) )
            & hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A_15)) ) ) ) ).

tff(fact_695_finite__less__ub,axiom,
    ! [U: nat,F: fun_nat_nat] :
      ( ! [N_1: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),hAPP_nat_nat(F,N_1)))
     => hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(hAPP_n1699378549t_bool(cOMBC_nat_nat_bool(cOMBB_800536526ol_nat(ord_less_eq_nat,F)),U)))) ) ).

tff(fact_696_the__elem__eq,axiom,
    ! [X_1: x_a] : ( hAPP_fun_a_bool_a(the_elem_a,hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_1),bot_bot_fun_a_bool)) = X_1 ) ).

tff(fact_697_the__elem__eq,axiom,
    ! [X_1: nat] : ( hAPP_f22106695ol_nat(the_elem_nat,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)) = X_1 ) ).

tff(fact_698_the__elem__eq,axiom,
    ! [X_1: int] : ( hAPP_f1594865479ol_int(the_elem_int,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool)) = X_1 ) ).

tff(fact_699_nonempty__iff,axiom,
    ! [A: fun_a_bool] :
      ( ( A != bot_bot_fun_a_bool )
    <=> ? [X_2: x_a,B_15: fun_a_bool] :
          ( ( A = hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_2),B_15) )
          & ~ hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_2),B_15)) ) ) ).

tff(fact_700_nonempty__iff,axiom,
    ! [A: fun_int_bool] :
      ( ( A != bot_bot_fun_int_bool )
    <=> ? [X_2: int,B_15: fun_int_bool] :
          ( ( A = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_2),B_15) )
          & ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),B_15)) ) ) ).

tff(fact_701_nonempty__iff,axiom,
    ! [A: fun_nat_bool] :
      ( ( A != bot_bot_fun_nat_bool )
    <=> ? [X_2: nat,B_15: fun_nat_bool] :
          ( ( A = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_2),B_15) )
          & ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),B_15)) ) ) ).

tff(fact_702_nonempty__iff,axiom,
    ! [A: fun_pname_bool] :
      ( ( A != bot_bo844097828e_bool )
    <=> ? [X_2: pname,B_15: fun_pname_bool] :
          ( ( A = hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_2),B_15) )
          & ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),B_15)) ) ) ).

tff(fact_703_assms_I4_J,axiom,
    ! [Pn: pname] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,Pn),u))
     => hBOOL(wt(the_com(body(Pn)))) ) ).

tff(fact_704_DiffE,axiom,
    ! [C_6: int,A: fun_int_bool,B: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,C_6),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),B)))
     => ~ ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,C_6),A))
         => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,C_6),B)) ) ) ).

tff(fact_705_DiffE,axiom,
    ! [C_6: nat,A: fun_nat_bool,B: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_6),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B)))
     => ~ ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_6),A))
         => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_6),B)) ) ) ).

tff(fact_706_DiffE,axiom,
    ! [C_6: x_a,A: fun_a_bool,B: fun_a_bool] :
      ( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,C_6),hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,A),B)))
     => ~ ( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,C_6),A))
         => hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,C_6),B)) ) ) ).

tff(fact_707_DiffE,axiom,
    ! [C_6: pname,A: fun_pname_bool,B: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C_6),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,A),B)))
     => ~ ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C_6),A))
         => hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C_6),B)) ) ) ).

tff(fact_708_DiffI,axiom,
    ! [B: fun_int_bool,C_6: int,A: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,C_6),A))
     => ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,C_6),B))
       => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,C_6),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),B))) ) ) ).

tff(fact_709_DiffI,axiom,
    ! [B: fun_nat_bool,C_6: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_6),A))
     => ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_6),B))
       => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_6),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B))) ) ) ).

tff(fact_710_DiffI,axiom,
    ! [B: fun_a_bool,C_6: x_a,A: fun_a_bool] :
      ( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,C_6),A))
     => ( ~ hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,C_6),B))
       => hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,C_6),hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,A),B))) ) ) ).

tff(fact_711_DiffI,axiom,
    ! [B: fun_pname_bool,C_6: pname,A: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C_6),A))
     => ( ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C_6),B))
       => hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C_6),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,A),B))) ) ) ).

tff(fact_712_DiffD2,axiom,
    ! [C_6: int,A: fun_int_bool,B: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,C_6),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),B)))
     => ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,C_6),B)) ) ).

tff(fact_713_DiffD2,axiom,
    ! [C_6: nat,A: fun_nat_bool,B: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_6),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B)))
     => ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_6),B)) ) ).

tff(fact_714_DiffD2,axiom,
    ! [C_6: x_a,A: fun_a_bool,B: fun_a_bool] :
      ( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,C_6),hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,A),B)))
     => ~ hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,C_6),B)) ) ).

tff(fact_715_DiffD2,axiom,
    ! [C_6: pname,A: fun_pname_bool,B: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C_6),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,A),B)))
     => ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C_6),B)) ) ).

tff(fact_716_DiffD1,axiom,
    ! [C_6: int,A: fun_int_bool,B: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,C_6),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),B)))
     => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,C_6),A)) ) ).

tff(fact_717_DiffD1,axiom,
    ! [C_6: nat,A: fun_nat_bool,B: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_6),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B)))
     => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_6),A)) ) ).

tff(fact_718_DiffD1,axiom,
    ! [C_6: x_a,A: fun_a_bool,B: fun_a_bool] :
      ( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,C_6),hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,A),B)))
     => hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,C_6),A)) ) ).

tff(fact_719_DiffD1,axiom,
    ! [C_6: pname,A: fun_pname_bool,B: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C_6),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,A),B)))
     => hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C_6),A)) ) ).

tff(fact_720_Diff__iff,axiom,
    ! [C_6: int,A: fun_int_bool,B: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,C_6),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),B)))
    <=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,C_6),A))
        & ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,C_6),B)) ) ) ).

tff(fact_721_Diff__iff,axiom,
    ! [C_6: nat,A: fun_nat_bool,B: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_6),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B)))
    <=> ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_6),A))
        & ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_6),B)) ) ) ).

tff(fact_722_Diff__iff,axiom,
    ! [C_6: x_a,A: fun_a_bool,B: fun_a_bool] :
      ( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,C_6),hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,A),B)))
    <=> ( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,C_6),A))
        & ~ hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,C_6),B)) ) ) ).

tff(fact_723_Diff__iff,axiom,
    ! [C_6: pname,A: fun_pname_bool,B: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C_6),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,A),B)))
    <=> ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C_6),A))
        & ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C_6),B)) ) ) ).

tff(fact_724_set__diff__eq,axiom,
    ! [A: fun_int_bool,B: fun_int_bool] : ( hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),B) = collect_int(cOMBS_int_bool_bool(cOMBB_1652995168ol_int(fconj,hAPP_f1805168059t_bool(cOMBC_94739984l_bool(member_int),A)),cOMBB_bool_bool_int(fNot,hAPP_f1805168059t_bool(cOMBC_94739984l_bool(member_int),B)))) ) ).

tff(fact_725_set__diff__eq,axiom,
    ! [A: fun_nat_bool,B: fun_nat_bool] : ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B) = collect_nat(cOMBS_nat_bool_bool(cOMBB_1015721476ol_nat(fconj,hAPP_f800510211t_bool(cOMBC_226598744l_bool(member_nat),A)),cOMBB_bool_bool_nat(fNot,hAPP_f800510211t_bool(cOMBC_226598744l_bool(member_nat),B)))) ) ).

tff(fact_726_set__diff__eq,axiom,
    ! [A: fun_a_bool,B: fun_a_bool] : ( hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,A),B) = collect_a(cOMBS_a_bool_bool(cOMBB_1972296269bool_a(fconj,hAPP_f2050579477a_bool(cOMBC_1355376034l_bool(member_a),A)),cOMBB_bool_bool_a(fNot,hAPP_f2050579477a_bool(cOMBC_1355376034l_bool(member_a),B)))) ) ).

tff(fact_727_set__diff__eq,axiom,
    ! [A: fun_pname_bool,B: fun_pname_bool] : ( hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,A),B) = collect_pname(cOMBS_568398431l_bool(cOMBB_675860798_pname(fconj,hAPP_f759274231e_bool(cOMBC_1058051404l_bool(member_pname),A)),cOMBB_647938656_pname(fNot,hAPP_f759274231e_bool(cOMBC_1058051404l_bool(member_pname),B)))) ) ).

tff(fact_728_folding__one_Oinsert__remove,axiom,
    ! [X_1: x_a,A: fun_a_bool,F: fun_a_fun_a_a,F_1: fun_fun_a_bool_a] :
      ( hBOOL(finite_folding_one_a(F,F_1))
     => ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A))
       => ( ( ( hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,A),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_1),bot_bot_fun_a_bool)) = bot_bot_fun_a_bool )
           => ( hAPP_fun_a_bool_a(F_1,hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_1),A)) = X_1 ) )
          & ( ( hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,A),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_1),bot_bot_fun_a_bool)) != bot_bot_fun_a_bool )
           => ( hAPP_fun_a_bool_a(F_1,hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_1),A)) = hAPP_a_a(hAPP_a_fun_a_a(F,X_1),hAPP_fun_a_bool_a(F_1,hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,A),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_1),bot_bot_fun_a_bool)))) ) ) ) ) ) ).

tff(fact_729_folding__one_Oinsert__remove,axiom,
    ! [X_1: int,A: fun_int_bool,F: fun_int_fun_int_int,F_1: fun_fun_int_bool_int] :
      ( hBOOL(finite1626084323ne_int(F,F_1))
     => ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
       => ( ( ( hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool)) = bot_bot_fun_int_bool )
           => ( hAPP_f1594865479ol_int(F_1,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A)) = X_1 ) )
          & ( ( hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool)) != bot_bot_fun_int_bool )
           => ( hAPP_f1594865479ol_int(F_1,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A)) = hAPP_int_int(hAPP_int_fun_int_int(F,X_1),hAPP_f1594865479ol_int(F_1,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool)))) ) ) ) ) ) ).

tff(fact_730_folding__one_Oinsert__remove,axiom,
    ! [X_1: nat,A: fun_nat_bool,F: fun_nat_fun_nat_nat,F_1: fun_fun_nat_bool_nat] :
      ( hBOOL(finite988810631ne_nat(F,F_1))
     => ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
       => ( ( ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)) = bot_bot_fun_nat_bool )
           => ( hAPP_f22106695ol_nat(F_1,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)) = X_1 ) )
          & ( ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)) != bot_bot_fun_nat_bool )
           => ( hAPP_f22106695ol_nat(F_1,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,X_1),hAPP_f22106695ol_nat(F_1,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)))) ) ) ) ) ) ).

tff(fact_731_folding__one_Oinsert__remove,axiom,
    ! [X_1: pname,A: fun_pname_bool,F: fun_pn1208287926_pname,F_1: fun_fu1499449723_pname] :
      ( hBOOL(finite1282449217_pname(F,F_1))
     => ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A))
       => ( ( ( hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,A),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_1),bot_bo844097828e_bool)) = bot_bo844097828e_bool )
           => ( hAPP_f1297739591_pname(F_1,hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_1),A)) = X_1 ) )
          & ( ( hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,A),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_1),bot_bo844097828e_bool)) != bot_bo844097828e_bool )
           => ( hAPP_f1297739591_pname(F_1,hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_1),A)) = hAPP_pname_pname(hAPP_p1170154830_pname(F,X_1),hAPP_f1297739591_pname(F_1,hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,A),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_1),bot_bo844097828e_bool)))) ) ) ) ) ) ).

tff(fact_732_folding__one_Oremove,axiom,
    ! [X_1: x_a,A: fun_a_bool,F: fun_a_fun_a_a,F_1: fun_fun_a_bool_a] :
      ( hBOOL(finite_folding_one_a(F,F_1))
     => ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A))
       => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_1),A))
         => ( ( ( hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,A),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_1),bot_bot_fun_a_bool)) = bot_bot_fun_a_bool )
             => ( hAPP_fun_a_bool_a(F_1,A) = X_1 ) )
            & ( ( hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,A),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_1),bot_bot_fun_a_bool)) != bot_bot_fun_a_bool )
             => ( hAPP_fun_a_bool_a(F_1,A) = hAPP_a_a(hAPP_a_fun_a_a(F,X_1),hAPP_fun_a_bool_a(F_1,hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,A),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_1),bot_bot_fun_a_bool)))) ) ) ) ) ) ) ).

tff(fact_733_folding__one_Oremove,axiom,
    ! [X_1: int,A: fun_int_bool,F: fun_int_fun_int_int,F_1: fun_fun_int_bool_int] :
      ( hBOOL(finite1626084323ne_int(F,F_1))
     => ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
       => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
         => ( ( ( hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool)) = bot_bot_fun_int_bool )
             => ( hAPP_f1594865479ol_int(F_1,A) = X_1 ) )
            & ( ( hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool)) != bot_bot_fun_int_bool )
             => ( hAPP_f1594865479ol_int(F_1,A) = hAPP_int_int(hAPP_int_fun_int_int(F,X_1),hAPP_f1594865479ol_int(F_1,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool)))) ) ) ) ) ) ) ).

tff(fact_734_folding__one_Oremove,axiom,
    ! [X_1: nat,A: fun_nat_bool,F: fun_nat_fun_nat_nat,F_1: fun_fun_nat_bool_nat] :
      ( hBOOL(finite988810631ne_nat(F,F_1))
     => ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
       => ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
         => ( ( ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)) = bot_bot_fun_nat_bool )
             => ( hAPP_f22106695ol_nat(F_1,A) = X_1 ) )
            & ( ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)) != bot_bot_fun_nat_bool )
             => ( hAPP_f22106695ol_nat(F_1,A) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,X_1),hAPP_f22106695ol_nat(F_1,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)))) ) ) ) ) ) ) ).

tff(fact_735_folding__one_Oremove,axiom,
    ! [X_1: pname,A: fun_pname_bool,F: fun_pn1208287926_pname,F_1: fun_fu1499449723_pname] :
      ( hBOOL(finite1282449217_pname(F,F_1))
     => ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A))
       => ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_1),A))
         => ( ( ( hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,A),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_1),bot_bo844097828e_bool)) = bot_bo844097828e_bool )
             => ( hAPP_f1297739591_pname(F_1,A) = X_1 ) )
            & ( ( hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,A),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_1),bot_bo844097828e_bool)) != bot_bo844097828e_bool )
             => ( hAPP_f1297739591_pname(F_1,A) = hAPP_pname_pname(hAPP_p1170154830_pname(F,X_1),hAPP_f1297739591_pname(F_1,hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,A),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_1),bot_bo844097828e_bool)))) ) ) ) ) ) ) ).

tff(fact_736_card__Diff__singleton__if,axiom,
    ! [X_1: x_a,A: fun_a_bool] :
      ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A))
     => ( ( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_1),A))
         => ( hAPP_fun_a_bool_nat(finite_card_a,hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,A),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_1),bot_bot_fun_a_bool))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_fun_a_bool_nat(finite_card_a,A)),one_one_nat) ) )
        & ( ~ hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_1),A))
         => ( hAPP_fun_a_bool_nat(finite_card_a,hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,A),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_1),bot_bot_fun_a_bool))) = hAPP_fun_a_bool_nat(finite_card_a,A) ) ) ) ) ).

tff(fact_737_card__Diff__singleton__if,axiom,
    ! [X_1: int,A: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
     => ( ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
         => ( hAPP_f957591787ol_nat(finite_card_int,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_f957591787ol_nat(finite_card_int,A)),one_one_nat) ) )
        & ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
         => ( hAPP_f957591787ol_nat(finite_card_int,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool))) = hAPP_f957591787ol_nat(finite_card_int,A) ) ) ) ) ).

tff(fact_738_card__Diff__singleton__if,axiom,
    ! [X_1: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => ( ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
         => ( hAPP_f22106695ol_nat(finite_card_nat,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_f22106695ol_nat(finite_card_nat,A)),one_one_nat) ) )
        & ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
         => ( hAPP_f22106695ol_nat(finite_card_nat,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool))) = hAPP_f22106695ol_nat(finite_card_nat,A) ) ) ) ) ).

tff(fact_739_card__Diff__singleton__if,axiom,
    ! [X_1: pname,A: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A))
     => ( ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_1),A))
         => ( hAPP_f921600141ol_nat(finite_card_pname,hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,A),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_1),bot_bo844097828e_bool))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_f921600141ol_nat(finite_card_pname,A)),one_one_nat) ) )
        & ( ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_1),A))
         => ( hAPP_f921600141ol_nat(finite_card_pname,hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,A),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_1),bot_bo844097828e_bool))) = hAPP_f921600141ol_nat(finite_card_pname,A) ) ) ) ) ).

tff(fact_740_card__Diff__singleton,axiom,
    ! [X_1: x_a,A: fun_a_bool] :
      ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A))
     => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_1),A))
       => ( hAPP_fun_a_bool_nat(finite_card_a,hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,A),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_1),bot_bot_fun_a_bool))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_fun_a_bool_nat(finite_card_a,A)),one_one_nat) ) ) ) ).

tff(fact_741_card__Diff__singleton,axiom,
    ! [X_1: int,A: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
       => ( hAPP_f957591787ol_nat(finite_card_int,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_f957591787ol_nat(finite_card_int,A)),one_one_nat) ) ) ) ).

tff(fact_742_card__Diff__singleton,axiom,
    ! [X_1: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
       => ( hAPP_f22106695ol_nat(finite_card_nat,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_f22106695ol_nat(finite_card_nat,A)),one_one_nat) ) ) ) ).

tff(fact_743_card__Diff__singleton,axiom,
    ! [X_1: pname,A: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A))
     => ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_1),A))
       => ( hAPP_f921600141ol_nat(finite_card_pname,hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,A),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_1),bot_bo844097828e_bool))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_f921600141ol_nat(finite_card_pname,A)),one_one_nat) ) ) ) ).

tff(fact_744_one__reorient,axiom,
    ! [X_1: int] :
      ( ( one_one_int = X_1 )
    <=> ( X_1 = one_one_int ) ) ).

tff(fact_745_one__reorient,axiom,
    ! [X_1: nat] :
      ( ( one_one_nat = X_1 )
    <=> ( X_1 = one_one_nat ) ) ).

tff(fact_746_diff__Suc__1,axiom,
    ! [N: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,N)),one_one_nat) = N ) ).

tff(fact_747_diff__Suc__eq__diff__pred,axiom,
    ! [M_1: nat,N: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),hAPP_nat_nat(suc,N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),one_one_nat)),N) ) ).

tff(fact_748_folding__one_Osingleton,axiom,
    ! [X_1: x_a,F: fun_a_fun_a_a,F_1: fun_fun_a_bool_a] :
      ( hBOOL(finite_folding_one_a(F,F_1))
     => ( hAPP_fun_a_bool_a(F_1,hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_1),bot_bot_fun_a_bool)) = X_1 ) ) ).

tff(fact_749_folding__one_Osingleton,axiom,
    ! [X_1: nat,F: fun_nat_fun_nat_nat,F_1: fun_fun_nat_bool_nat] :
      ( hBOOL(finite988810631ne_nat(F,F_1))
     => ( hAPP_f22106695ol_nat(F_1,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)) = X_1 ) ) ).

tff(fact_750_folding__one_Osingleton,axiom,
    ! [X_1: int,F: fun_int_fun_int_int,F_1: fun_fun_int_bool_int] :
      ( hBOOL(finite1626084323ne_int(F,F_1))
     => ( hAPP_f1594865479ol_int(F_1,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool)) = X_1 ) ) ).

tff(fact_751_card__Diff__insert,axiom,
    ! [B: fun_a_bool,A_2: x_a,A: fun_a_bool] :
      ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A))
     => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,A_2),A))
       => ( ~ hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,A_2),B))
         => ( hAPP_fun_a_bool_nat(finite_card_a,hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,A),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,A_2),B))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_fun_a_bool_nat(finite_card_a,hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,A),B))),one_one_nat) ) ) ) ) ).

tff(fact_752_card__Diff__insert,axiom,
    ! [B: fun_int_bool,A_2: int,A: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_2),A))
       => ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_2),B))
         => ( hAPP_f957591787ol_nat(finite_card_int,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_2),B))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_f957591787ol_nat(finite_card_int,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),B))),one_one_nat) ) ) ) ) ).

tff(fact_753_card__Diff__insert,axiom,
    ! [B: fun_nat_bool,A_2: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_2),A))
       => ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_2),B))
         => ( hAPP_f22106695ol_nat(finite_card_nat,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_2),B))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_f22106695ol_nat(finite_card_nat,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B))),one_one_nat) ) ) ) ) ).

tff(fact_754_card__Diff__insert,axiom,
    ! [B: fun_pname_bool,A_2: pname,A: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A))
     => ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_2),A))
       => ( ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_2),B))
         => ( hAPP_f921600141ol_nat(finite_card_pname,hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,A),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_2),B))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_f921600141ol_nat(finite_card_pname,hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,A),B))),one_one_nat) ) ) ) ) ).

tff(fact_755_folding__one_Oinsert,axiom,
    ! [X_1: x_a,A: fun_a_bool,F: fun_a_fun_a_a,F_1: fun_fun_a_bool_a] :
      ( hBOOL(finite_folding_one_a(F,F_1))
     => ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A))
       => ( ~ hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_1),A))
         => ( ( A != bot_bot_fun_a_bool )
           => ( hAPP_fun_a_bool_a(F_1,hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_1),A)) = hAPP_a_a(hAPP_a_fun_a_a(F,X_1),hAPP_fun_a_bool_a(F_1,A)) ) ) ) ) ) ).

tff(fact_756_folding__one_Oinsert,axiom,
    ! [X_1: int,A: fun_int_bool,F: fun_int_fun_int_int,F_1: fun_fun_int_bool_int] :
      ( hBOOL(finite1626084323ne_int(F,F_1))
     => ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
       => ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
         => ( ( A != bot_bot_fun_int_bool )
           => ( hAPP_f1594865479ol_int(F_1,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A)) = hAPP_int_int(hAPP_int_fun_int_int(F,X_1),hAPP_f1594865479ol_int(F_1,A)) ) ) ) ) ) ).

tff(fact_757_folding__one_Oinsert,axiom,
    ! [X_1: nat,A: fun_nat_bool,F: fun_nat_fun_nat_nat,F_1: fun_fun_nat_bool_nat] :
      ( hBOOL(finite988810631ne_nat(F,F_1))
     => ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
       => ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
         => ( ( A != bot_bot_fun_nat_bool )
           => ( hAPP_f22106695ol_nat(F_1,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,X_1),hAPP_f22106695ol_nat(F_1,A)) ) ) ) ) ) ).

tff(fact_758_folding__one_Oinsert,axiom,
    ! [X_1: pname,A: fun_pname_bool,F: fun_pn1208287926_pname,F_1: fun_fu1499449723_pname] :
      ( hBOOL(finite1282449217_pname(F,F_1))
     => ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A))
       => ( ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_1),A))
         => ( ( A != bot_bo844097828e_bool )
           => ( hAPP_f1297739591_pname(F_1,hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_1),A)) = hAPP_pname_pname(hAPP_p1170154830_pname(F,X_1),hAPP_f1297739591_pname(F_1,A)) ) ) ) ) ) ).

tff(fact_759_folding__one_Oclosed,axiom,
    ! [A: fun_a_bool,F: fun_a_fun_a_a,F_1: fun_fun_a_bool_a] :
      ( hBOOL(finite_folding_one_a(F,F_1))
     => ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A))
       => ( ( A != bot_bot_fun_a_bool )
         => ( ! [X_2: x_a,Y_2: x_a] : hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,hAPP_a_a(hAPP_a_fun_a_a(F,X_2),Y_2)),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_2),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,Y_2),bot_bot_fun_a_bool))))
           => hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,hAPP_fun_a_bool_a(F_1,A)),A)) ) ) ) ) ).

tff(fact_760_folding__one_Oclosed,axiom,
    ! [A: fun_int_bool,F: fun_int_fun_int_int,F_1: fun_fun_int_bool_int] :
      ( hBOOL(finite1626084323ne_int(F,F_1))
     => ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
       => ( ( A != bot_bot_fun_int_bool )
         => ( ! [X_2: int,Y_2: int] : hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(F,X_2),Y_2)),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_2),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,Y_2),bot_bot_fun_int_bool))))
           => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_f1594865479ol_int(F_1,A)),A)) ) ) ) ) ).

tff(fact_761_folding__one_Oclosed,axiom,
    ! [A: fun_nat_bool,F: fun_nat_fun_nat_nat,F_1: fun_fun_nat_bool_nat] :
      ( hBOOL(finite988810631ne_nat(F,F_1))
     => ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
       => ( ( A != bot_bot_fun_nat_bool )
         => ( ! [X_2: nat,Y_2: nat] : hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,X_2),Y_2)),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_2),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Y_2),bot_bot_fun_nat_bool))))
           => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,hAPP_f22106695ol_nat(F_1,A)),A)) ) ) ) ) ).

tff(fact_762_folding__one_Oclosed,axiom,
    ! [A: fun_pname_bool,F: fun_pn1208287926_pname,F_1: fun_fu1499449723_pname] :
      ( hBOOL(finite1282449217_pname(F,F_1))
     => ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A))
       => ( ( A != bot_bo844097828e_bool )
         => ( ! [X_2: pname,Y_2: pname] : hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,hAPP_pname_pname(hAPP_p1170154830_pname(F,X_2),Y_2)),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_2),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,Y_2),bot_bo844097828e_bool))))
           => hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,hAPP_f1297739591_pname(F_1,A)),A)) ) ) ) ) ).

tff(fact_763_card_Oremove,axiom,
    ! [X_1: x_a,A: fun_a_bool] :
      ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A))
     => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_1),A))
       => ( hAPP_fun_a_bool_nat(finite_card_a,A) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,one_one_nat),hAPP_fun_a_bool_nat(finite_card_a,hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,A),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_1),bot_bot_fun_a_bool)))) ) ) ) ).

tff(fact_764_card_Oremove,axiom,
    ! [X_1: int,A: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
       => ( hAPP_f957591787ol_nat(finite_card_int,A) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,one_one_nat),hAPP_f957591787ol_nat(finite_card_int,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool)))) ) ) ) ).

tff(fact_765_card_Oremove,axiom,
    ! [X_1: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
       => ( hAPP_f22106695ol_nat(finite_card_nat,A) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,one_one_nat),hAPP_f22106695ol_nat(finite_card_nat,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)))) ) ) ) ).

tff(fact_766_card_Oremove,axiom,
    ! [X_1: pname,A: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A))
     => ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_1),A))
       => ( hAPP_f921600141ol_nat(finite_card_pname,A) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,one_one_nat),hAPP_f921600141ol_nat(finite_card_pname,hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,A),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_1),bot_bo844097828e_bool)))) ) ) ) ).

tff(fact_767_card_Oinsert__remove,axiom,
    ! [X_1: x_a,A: fun_a_bool] :
      ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A))
     => ( hAPP_fun_a_bool_nat(finite_card_a,hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_1),A)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,one_one_nat),hAPP_fun_a_bool_nat(finite_card_a,hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,A),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_1),bot_bot_fun_a_bool)))) ) ) ).

tff(fact_768_card_Oinsert__remove,axiom,
    ! [X_1: int,A: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
     => ( hAPP_f957591787ol_nat(finite_card_int,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,one_one_nat),hAPP_f957591787ol_nat(finite_card_int,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool)))) ) ) ).

tff(fact_769_card_Oinsert__remove,axiom,
    ! [X_1: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => ( hAPP_f22106695ol_nat(finite_card_nat,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,one_one_nat),hAPP_f22106695ol_nat(finite_card_nat,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)))) ) ) ).

tff(fact_770_card_Oinsert__remove,axiom,
    ! [X_1: pname,A: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A))
     => ( hAPP_f921600141ol_nat(finite_card_pname,hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_1),A)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,one_one_nat),hAPP_f921600141ol_nat(finite_card_pname,hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,A),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_1),bot_bo844097828e_bool)))) ) ) ).

tff(fact_771_folding__one__idem_Osubset__idem,axiom,
    ! [B: fun_int_bool,A: fun_int_bool,F: fun_int_fun_int_int,F_1: fun_fun_int_bool_int] :
      ( hBOOL(finite1432773856em_int(F,F_1))
     => ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
       => ( ( B != bot_bot_fun_int_bool )
         => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,B),A))
           => ( hAPP_int_int(hAPP_int_fun_int_int(F,hAPP_f1594865479ol_int(F_1,B)),hAPP_f1594865479ol_int(F_1,A)) = hAPP_f1594865479ol_int(F_1,A) ) ) ) ) ) ).

tff(fact_772_folding__one__idem_Osubset__idem,axiom,
    ! [B: fun_nat_bool,A: fun_nat_bool,F: fun_nat_fun_nat_nat,F_1: fun_fun_nat_bool_nat] :
      ( hBOOL(finite795500164em_nat(F,F_1))
     => ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
       => ( ( B != bot_bot_fun_nat_bool )
         => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B),A))
           => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,hAPP_f22106695ol_nat(F_1,B)),hAPP_f22106695ol_nat(F_1,A)) = hAPP_f22106695ol_nat(F_1,A) ) ) ) ) ) ).

tff(fact_773_folding__one__idem_Osubset__idem,axiom,
    ! [B: fun_pname_bool,A: fun_pname_bool,F: fun_pn1208287926_pname,F_1: fun_fu1499449723_pname] :
      ( hBOOL(finite89670078_pname(F,F_1))
     => ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A))
       => ( ( B != bot_bo844097828e_bool )
         => ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,B),A))
           => ( hAPP_pname_pname(hAPP_p1170154830_pname(F,hAPP_f1297739591_pname(F_1,B)),hAPP_f1297739591_pname(F_1,A)) = hAPP_f1297739591_pname(F_1,A) ) ) ) ) ) ).

tff(fact_774_folding__one__idem_Osubset__idem,axiom,
    ! [B: fun_a_bool,A: fun_a_bool,F: fun_a_fun_a_a,F_1: fun_fun_a_bool_a] :
      ( hBOOL(finite1819937229idem_a(F,F_1))
     => ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A))
       => ( ( B != bot_bot_fun_a_bool )
         => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B),A))
           => ( hAPP_a_a(hAPP_a_fun_a_a(F,hAPP_fun_a_bool_a(F_1,B)),hAPP_fun_a_bool_a(F_1,A)) = hAPP_fun_a_bool_a(F_1,A) ) ) ) ) ) ).

tff(fact_775_folding__one__idem_Oinsert__idem,axiom,
    ! [X_1: x_a,A: fun_a_bool,F: fun_a_fun_a_a,F_1: fun_fun_a_bool_a] :
      ( hBOOL(finite1819937229idem_a(F,F_1))
     => ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A))
       => ( ( A != bot_bot_fun_a_bool )
         => ( hAPP_fun_a_bool_a(F_1,hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_1),A)) = hAPP_a_a(hAPP_a_fun_a_a(F,X_1),hAPP_fun_a_bool_a(F_1,A)) ) ) ) ) ).

tff(fact_776_folding__one__idem_Oinsert__idem,axiom,
    ! [X_1: int,A: fun_int_bool,F: fun_int_fun_int_int,F_1: fun_fun_int_bool_int] :
      ( hBOOL(finite1432773856em_int(F,F_1))
     => ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
       => ( ( A != bot_bot_fun_int_bool )
         => ( hAPP_f1594865479ol_int(F_1,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A)) = hAPP_int_int(hAPP_int_fun_int_int(F,X_1),hAPP_f1594865479ol_int(F_1,A)) ) ) ) ) ).

tff(fact_777_folding__one__idem_Oinsert__idem,axiom,
    ! [X_1: nat,A: fun_nat_bool,F: fun_nat_fun_nat_nat,F_1: fun_fun_nat_bool_nat] :
      ( hBOOL(finite795500164em_nat(F,F_1))
     => ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
       => ( ( A != bot_bot_fun_nat_bool )
         => ( hAPP_f22106695ol_nat(F_1,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,X_1),hAPP_f22106695ol_nat(F_1,A)) ) ) ) ) ).

tff(fact_778_folding__one__idem_Oinsert__idem,axiom,
    ! [X_1: pname,A: fun_pname_bool,F: fun_pn1208287926_pname,F_1: fun_fu1499449723_pname] :
      ( hBOOL(finite89670078_pname(F,F_1))
     => ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A))
       => ( ( A != bot_bo844097828e_bool )
         => ( hAPP_f1297739591_pname(F_1,hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_1),A)) = hAPP_pname_pname(hAPP_p1170154830_pname(F,X_1),hAPP_f1297739591_pname(F_1,A)) ) ) ) ) ).

tff(fact_779_finite__ne__induct,axiom,
    ! [Pa: fun_fun_a_bool_bool,F_1: fun_a_bool] :
      ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,F_1))
     => ( ( F_1 != bot_bot_fun_a_bool )
       => ( ! [X_2: x_a] : hBOOL(hAPP_fun_a_bool_bool(Pa,hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_2),bot_bot_fun_a_bool)))
         => ( ! [X_2: x_a,F_2: fun_a_bool] :
                ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,F_2))
               => ( ( F_2 != bot_bot_fun_a_bool )
                 => ( ~ hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_2),F_2))
                   => ( hBOOL(hAPP_fun_a_bool_bool(Pa,F_2))
                     => hBOOL(hAPP_fun_a_bool_bool(Pa,hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_2),F_2))) ) ) ) )
           => hBOOL(hAPP_fun_a_bool_bool(Pa,F_1)) ) ) ) ) ).

tff(fact_780_finite__ne__induct,axiom,
    ! [Pa: fun_fu819804446l_bool,F_1: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,F_1))
     => ( ( F_1 != bot_bot_fun_int_bool )
       => ( ! [X_2: int] : hBOOL(hAPP_f448129468l_bool(Pa,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_2),bot_bot_fun_int_bool)))
         => ( ! [X_2: int,F_2: fun_int_bool] :
                ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,F_2))
               => ( ( F_2 != bot_bot_fun_int_bool )
                 => ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),F_2))
                   => ( hBOOL(hAPP_f448129468l_bool(Pa,F_2))
                     => hBOOL(hAPP_f448129468l_bool(Pa,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_2),F_2))) ) ) ) )
           => hBOOL(hAPP_f448129468l_bool(Pa,F_1)) ) ) ) ) ).

tff(fact_781_finite__ne__induct,axiom,
    ! [Pa: fun_fu425979586l_bool,F_1: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F_1))
     => ( ( F_1 != bot_bot_fun_nat_bool )
       => ( ! [X_2: nat] : hBOOL(hAPP_f54304608l_bool(Pa,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_2),bot_bot_fun_nat_bool)))
         => ( ! [X_2: nat,F_2: fun_nat_bool] :
                ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F_2))
               => ( ( F_2 != bot_bot_fun_nat_bool )
                 => ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),F_2))
                   => ( hBOOL(hAPP_f54304608l_bool(Pa,F_2))
                     => hBOOL(hAPP_f54304608l_bool(Pa,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_2),F_2))) ) ) ) )
           => hBOOL(hAPP_f54304608l_bool(Pa,F_1)) ) ) ) ) ).

tff(fact_782_finite__ne__induct,axiom,
    ! [Pa: fun_fu1430349052l_bool,F_1: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,F_1))
     => ( ( F_1 != bot_bo844097828e_bool )
       => ( ! [X_2: pname] : hBOOL(hAPP_f1664156314l_bool(Pa,hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_2),bot_bo844097828e_bool)))
         => ( ! [X_2: pname,F_2: fun_pname_bool] :
                ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,F_2))
               => ( ( F_2 != bot_bo844097828e_bool )
                 => ( ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),F_2))
                   => ( hBOOL(hAPP_f1664156314l_bool(Pa,F_2))
                     => hBOOL(hAPP_f1664156314l_bool(Pa,hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_2),F_2))) ) ) ) )
           => hBOOL(hAPP_f1664156314l_bool(Pa,F_1)) ) ) ) ) ).

tff(fact_783_the__elem__def,axiom,
    ! [X_3: fun_a_bool] : ( hAPP_fun_a_bool_a(the_elem_a,X_3) = hAPP_fun_a_bool_a(the_a,cOMBB_1273333599bool_a(hAPP_f1631501043l_bool(fequal_fun_a_bool,X_3),hAPP_f1038053735a_bool(cOMBC_394484100a_bool(insert_a),bot_bot_fun_a_bool))) ) ).

tff(fact_784_the__elem__def,axiom,
    ! [X_3: fun_nat_bool] : ( hAPP_f22106695ol_nat(the_elem_nat,X_3) = hAPP_f22106695ol_nat(the_nat,cOMBB_955900739ol_nat(hAPP_f103356543l_bool(fequal_fun_nat_bool,X_3),hAPP_f1570313510t_bool(cOMBC_178881787t_bool(insert_nat),bot_bot_fun_nat_bool))) ) ).

tff(fact_785_the__elem__def,axiom,
    ! [X_3: fun_int_bool] : ( hAPP_f1594865479ol_int(the_elem_int,X_3) = hAPP_f1594865479ol_int(the_int,cOMBB_1418110531ol_int(hAPP_f284875647l_bool(fequal_fun_int_bool,X_3),hAPP_f1048215610t_bool(cOMBC_1683390479t_bool(insert_int),bot_bot_fun_int_bool))) ) ).

tff(fact_786_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A_13: int,B_14: int,C_10: int] : ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_13),B_14)),C_10) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_13),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_14),C_10)) ) ).

tff(fact_787_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A_13: nat,B_14: nat,C_10: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_13),B_14)),C_10) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_13),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_14),C_10)) ) ).

tff(fact_788_add__left__cancel,axiom,
    ! [A_2: int,B_3: int,C_6: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_2),B_3) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_2),C_6) )
    <=> ( B_3 = C_6 ) ) ).

tff(fact_789_add__left__cancel,axiom,
    ! [A_2: nat,B_3: nat,C_6: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_2),B_3) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_2),C_6) )
    <=> ( B_3 = C_6 ) ) ).

tff(fact_790_add__right__cancel,axiom,
    ! [B_3: int,A_2: int,C_6: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_3),A_2) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_6),A_2) )
    <=> ( B_3 = C_6 ) ) ).

tff(fact_791_add__right__cancel,axiom,
    ! [B_3: nat,A_2: nat,C_6: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_3),A_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_6),A_2) )
    <=> ( B_3 = C_6 ) ) ).

tff(fact_792_add__left__imp__eq,axiom,
    ! [A_12: int,B_13: int,C_9: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_12),B_13) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_12),C_9) )
     => ( B_13 = C_9 ) ) ).

tff(fact_793_add__left__imp__eq,axiom,
    ! [A_12: nat,B_13: nat,C_9: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_12),B_13) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_12),C_9) )
     => ( B_13 = C_9 ) ) ).

tff(fact_794_add__imp__eq,axiom,
    ! [A_11: int,B_12: int,C_8: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_11),B_12) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_11),C_8) )
     => ( B_12 = C_8 ) ) ).

tff(fact_795_add__imp__eq,axiom,
    ! [A_11: nat,B_12: nat,C_8: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_11),B_12) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_11),C_8) )
     => ( B_12 = C_8 ) ) ).

tff(fact_796_add__right__imp__eq,axiom,
    ! [B_11: int,A_10: int,C_7: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_11),A_10) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_7),A_10) )
     => ( B_11 = C_7 ) ) ).

tff(fact_797_add__right__imp__eq,axiom,
    ! [B_11: nat,A_10: nat,C_7: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_11),A_10) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_7),A_10) )
     => ( B_11 = C_7 ) ) ).

tff(fact_798_nat__add__commute,axiom,
    ! [M_1: nat,N: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),N) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),M_1) ) ).

tff(fact_799_nat__add__left__commute,axiom,
    ! [X: nat,Y: nat,Z: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y),Z)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X),Z)) ) ).

tff(fact_800_nat__add__assoc,axiom,
    ! [M_1: nat,N: nat,K_1: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),N)),K_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),K_1)) ) ).

tff(fact_801_nat__add__left__cancel,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K),M) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K),Na) )
    <=> ( M = Na ) ) ).

tff(fact_802_nat__add__right__cancel,axiom,
    ! [M: nat,K: nat,Na: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),K) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Na),K) )
    <=> ( M = Na ) ) ).

tff(fact_803_add__le__cancel__right,axiom,
    ! [A_2: int,C_6: int,B_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_2),C_6)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_3),C_6)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_2),B_3)) ) ).

tff(fact_804_add__le__cancel__right,axiom,
    ! [A_2: nat,C_6: nat,B_3: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_2),C_6)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_3),C_6)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_2),B_3)) ) ).

tff(fact_805_add__le__cancel__left,axiom,
    ! [C_6: int,A_2: int,B_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_6),A_2)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_6),B_3)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_2),B_3)) ) ).

tff(fact_806_add__le__cancel__left,axiom,
    ! [C_6: nat,A_2: nat,B_3: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_6),A_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_6),B_3)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_2),B_3)) ) ).

tff(fact_807_add__right__mono,axiom,
    ! [C_5: int,A_9: int,B_10: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_9),B_10))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_9),C_5)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_10),C_5))) ) ).

tff(fact_808_add__right__mono,axiom,
    ! [C_5: nat,A_9: nat,B_10: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_9),B_10))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_9),C_5)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_10),C_5))) ) ).

tff(fact_809_add__left__mono,axiom,
    ! [C_4: int,A_8: int,B_9: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_8),B_9))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_4),A_8)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_4),B_9))) ) ).

tff(fact_810_add__left__mono,axiom,
    ! [C_4: nat,A_8: nat,B_9: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_8),B_9))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_4),A_8)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_4),B_9))) ) ).

tff(fact_811_add__mono,axiom,
    ! [C_3: int,D_3: int,A_7: int,B_8: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_7),B_8))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_3),D_3))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_7),C_3)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_8),D_3))) ) ) ).

tff(fact_812_add__mono,axiom,
    ! [C_3: nat,D_3: nat,A_7: nat,B_8: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_7),B_8))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_3),D_3))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_7),C_3)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_8),D_3))) ) ) ).

tff(fact_813_add__le__imp__le__right,axiom,
    ! [A_6: int,C_2: int,B_7: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_6),C_2)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_7),C_2)))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_6),B_7)) ) ).

tff(fact_814_add__le__imp__le__right,axiom,
    ! [A_6: nat,C_2: nat,B_7: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_6),C_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_7),C_2)))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_6),B_7)) ) ).

tff(fact_815_add__le__imp__le__left,axiom,
    ! [C_1: int,A_5: int,B_6: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_1),A_5)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_1),B_6)))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_5),B_6)) ) ).

tff(fact_816_add__le__imp__le__left,axiom,
    ! [C_1: nat,A_5: nat,B_6: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_1),A_5)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_1),B_6)))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_5),B_6)) ) ).

tff(fact_817_diff__add__cancel,axiom,
    ! [A_4: int,B_5: int] : ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_4),B_5)),B_5) = A_4 ) ).

tff(fact_818_add__diff__cancel,axiom,
    ! [A_3: int,B_4: int] : ( hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_3),B_4)),B_4) = A_3 ) ).

tff(fact_819_add__Suc__right,axiom,
    ! [M_1: nat,N: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),hAPP_nat_nat(suc,N)) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),N)) ) ).

tff(fact_820_add__Suc,axiom,
    ! [M_1: nat,N: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(suc,M_1)),N) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),N)) ) ).

tff(fact_821_add__Suc__shift,axiom,
    ! [M_1: nat,N: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(suc,M_1)),N) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),hAPP_nat_nat(suc,N)) ) ).

tff(fact_822_le__add2,axiom,
    ! [N: nat,M_1: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),N))) ).

tff(fact_823_le__add1,axiom,
    ! [N: nat,M_1: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),M_1))) ).

tff(fact_824_le__iff__add,axiom,
    ! [M: nat,Na: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),Na))
    <=> ? [K_2: nat] : ( Na = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),K_2) ) ) ).

tff(fact_825_nat__add__left__cancel__le,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K),Na)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),Na)) ) ).

tff(fact_826_trans__le__add1,axiom,
    ! [M_1: nat,I_1: nat,J: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J),M_1))) ) ).

tff(fact_827_trans__le__add2,axiom,
    ! [M_1: nat,I_1: nat,J: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),J))) ) ).

tff(fact_828_add__le__mono1,axiom,
    ! [K_1: nat,I_1: nat,J: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_1),K_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J),K_1))) ) ).

tff(fact_829_add__le__mono,axiom,
    ! [K_1: nat,L_1: nat,I_1: nat,J: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),L_1))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_1),K_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J),L_1))) ) ) ).

tff(fact_830_add__leD2,axiom,
    ! [M_1: nat,K_1: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),K_1)),N))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),N)) ) ).

tff(fact_831_add__leD1,axiom,
    ! [M_1: nat,K_1: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),K_1)),N))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N)) ) ).

tff(fact_832_add__leE,axiom,
    ! [M_1: nat,K_1: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),K_1)),N))
     => ~ ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N))
         => ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),N)) ) ) ).

tff(fact_833_diff__add__inverse2,axiom,
    ! [M_1: nat,N: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),N)),N) = M_1 ) ).

tff(fact_834_diff__add__inverse,axiom,
    ! [N: nat,M_1: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),M_1)),N) = M_1 ) ).

tff(fact_835_diff__diff__left,axiom,
    ! [I_1: nat,J: nat,K_1: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I_1),J)),K_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J),K_1)) ) ).

tff(fact_836_diff__cancel,axiom,
    ! [K_1: nat,M_1: nat,N: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_1),M_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_1),N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),N) ) ).

tff(fact_837_diff__cancel2,axiom,
    ! [M_1: nat,K_1: nat,N: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),K_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),K_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),N) ) ).

tff(fact_838_diff__diff__right,axiom,
    ! [I_1: nat,K_1: nat,J: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),J))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J),K_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_1),K_1)),J) ) ) ).

tff(fact_839_le__diff__conv,axiom,
    ! [J_1: nat,K: nat,I_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_1),K)),I_2))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_2),K))) ) ).

tff(fact_840_le__add__diff,axiom,
    ! [M_1: nat,K_1: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),N))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),M_1)),K_1))) ) ).

tff(fact_841_le__add__diff__inverse,axiom,
    ! [N: nat,M_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M_1))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),N)) = M_1 ) ) ).

tff(fact_842_add__diff__assoc,axiom,
    ! [I_1: nat,K_1: nat,J: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),J))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J),K_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_1),J)),K_1) ) ) ).

tff(fact_843_le__diff__conv2,axiom,
    ! [I_2: nat,K: nat,J_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),J_1))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_1),K)))
      <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_2),K)),J_1)) ) ) ).

tff(fact_844_le__add__diff__inverse2,axiom,
    ! [N: nat,M_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M_1))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),N)),N) = M_1 ) ) ).

tff(fact_845_le__imp__diff__is__add,axiom,
    ! [K: nat,I_2: nat,J_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),J_1))
     => ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_1),I_2) = K )
      <=> ( J_1 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K),I_2) ) ) ) ).

tff(fact_846_diff__add__assoc,axiom,
    ! [I_1: nat,K_1: nat,J: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),J))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_1),J)),K_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J),K_1)) ) ) ).

tff(fact_847_add__diff__assoc2,axiom,
    ! [I_1: nat,K_1: nat,J: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),J))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J),K_1)),I_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J),I_1)),K_1) ) ) ).

tff(fact_848_diff__add__assoc2,axiom,
    ! [I_1: nat,K_1: nat,J: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),J))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J),I_1)),K_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J),K_1)),I_1) ) ) ).

tff(fact_849_Suc__eq__plus1__left,axiom,
    ! [N: nat] : ( hAPP_nat_nat(suc,N) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,one_one_nat),N) ) ).

tff(fact_850_Suc__eq__plus1,axiom,
    ! [N: nat] : ( hAPP_nat_nat(suc,N) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat) ) ).

tff(fact_851_diff__Suc__diff__eq1,axiom,
    ! [M_1: nat,K_1: nat,J: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),J))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J),K_1))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),K_1)),hAPP_nat_nat(suc,J)) ) ) ).

tff(fact_852_diff__Suc__diff__eq2,axiom,
    ! [M_1: nat,K_1: nat,J: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),J))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J),K_1))),M_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,J)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_1),M_1)) ) ) ).

tff(fact_853_termination__basic__simps_I4_J,axiom,
    ! [Y: nat,X: nat,Z: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X),Z))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y),Z))) ) ).

tff(fact_854_termination__basic__simps_I3_J,axiom,
    ! [Z: nat,X: nat,Y: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X),Y))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y),Z))) ) ).

tff(fact_855_lessI,axiom,
    ! [N: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),hAPP_nat_nat(suc,N))) ).

tff(fact_856_Suc__mono,axiom,
    ! [M_1: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,M_1)),hAPP_nat_nat(suc,N))) ) ).

tff(fact_857_finite__Collect__less__nat,axiom,
    ! [K: nat] : hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(hAPP_n1699378549t_bool(cOMBC_nat_nat_bool(ord_less_nat),K)))) ).

tff(fact_858_less__not__refl,axiom,
    ! [N: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),N)) ).

tff(fact_859_nat__neq__iff,axiom,
    ! [M: nat,Na: nat] :
      ( ( M != Na )
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),Na))
        | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Na),M)) ) ) ).

tff(fact_860_linorder__neqE__nat,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
     => ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),Y))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y),X)) ) ) ).

tff(fact_861_less__irrefl__nat,axiom,
    ! [N: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),N)) ).

tff(fact_862_less__not__refl2,axiom,
    ! [N: nat,M_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),M_1))
     => ( M_1 != N ) ) ).

tff(fact_863_less__not__refl3,axiom,
    ! [S: nat,T: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,S),T))
     => ( S != T ) ) ).

tff(fact_864_nat__less__cases,axiom,
    ! [Pa: fun_nat_fun_nat_bool,M: nat,Na: nat] :
      ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),Na))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(Pa,Na),M)) )
     => ( ( ( M = Na )
         => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(Pa,Na),M)) )
       => ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Na),M))
           => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(Pa,Na),M)) )
         => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(Pa,Na),M)) ) ) ) ).

tff(fact_865_not__less__eq,axiom,
    ! [M: nat,Na: nat] :
      ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),Na))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Na),hAPP_nat_nat(suc,M))) ) ).

tff(fact_866_less__Suc__eq,axiom,
    ! [M: nat,Na: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),hAPP_nat_nat(suc,Na)))
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),Na))
        | ( M = Na ) ) ) ).

tff(fact_867_Suc__less__eq,axiom,
    ! [M: nat,Na: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,M)),hAPP_nat_nat(suc,Na)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),Na)) ) ).

tff(fact_868_not__less__less__Suc__eq,axiom,
    ! [Na: nat,M: nat] :
      ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Na),M))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Na),hAPP_nat_nat(suc,M)))
      <=> ( Na = M ) ) ) ).

tff(fact_869_less__antisym,axiom,
    ! [N: nat,M_1: nat] :
      ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),M_1))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),hAPP_nat_nat(suc,M_1)))
       => ( M_1 = N ) ) ) ).

tff(fact_870_less__SucI,axiom,
    ! [M_1: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),hAPP_nat_nat(suc,N))) ) ).

tff(fact_871_Suc__lessI,axiom,
    ! [M_1: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N))
     => ( ( hAPP_nat_nat(suc,M_1) != N )
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,M_1)),N)) ) ) ).

tff(fact_872_less__trans__Suc,axiom,
    ! [K_1: nat,I_1: nat,J: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),J))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,J),K_1))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,I_1)),K_1)) ) ) ).

tff(fact_873_less__SucE,axiom,
    ! [M_1: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),hAPP_nat_nat(suc,N)))
     => ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N))
       => ( M_1 = N ) ) ) ).

tff(fact_874_Suc__lessD,axiom,
    ! [M_1: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,M_1)),N))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N)) ) ).

tff(fact_875_Suc__less__SucD,axiom,
    ! [M_1: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,M_1)),hAPP_nat_nat(suc,N)))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N)) ) ).

tff(fact_876_not__add__less1,axiom,
    ! [I_1: nat,J: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_1),J)),I_1)) ).

tff(fact_877_not__add__less2,axiom,
    ! [J: nat,I_1: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J),I_1)),I_1)) ).

tff(fact_878_nat__add__left__cancel__less,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K),Na)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),Na)) ) ).

tff(fact_879_trans__less__add1,axiom,
    ! [M_1: nat,I_1: nat,J: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),J))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J),M_1))) ) ).

tff(fact_880_trans__less__add2,axiom,
    ! [M_1: nat,I_1: nat,J: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),J))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),J))) ) ).

tff(fact_881_add__less__mono1,axiom,
    ! [K_1: nat,I_1: nat,J: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),J))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_1),K_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J),K_1))) ) ).

tff(fact_882_add__less__mono,axiom,
    ! [K_1: nat,L_1: nat,I_1: nat,J: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),J))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,K_1),L_1))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_1),K_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J),L_1))) ) ) ).

tff(fact_883_less__add__eq__less,axiom,
    ! [M_1: nat,N: nat,K_1: nat,L_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,K_1),L_1))
     => ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),L_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_1),N) )
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N)) ) ) ).

tff(fact_884_add__lessD1,axiom,
    ! [I_1: nat,J: nat,K_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_1),J)),K_1))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),K_1)) ) ).

tff(fact_885_termination__basic__simps_I1_J,axiom,
    ! [Z: nat,X: nat,Y: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),Y))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y),Z))) ) ).

tff(fact_886_termination__basic__simps_I2_J,axiom,
    ! [Y: nat,X: nat,Z: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),Z))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y),Z))) ) ).

tff(fact_887_nat__less__le,axiom,
    ! [M: nat,Na: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),Na))
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),Na))
        & ( M != Na ) ) ) ).

tff(fact_888_le__eq__less__or__eq,axiom,
    ! [M: nat,Na: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),Na))
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),Na))
        | ( M = Na ) ) ) ).

tff(fact_889_less__imp__le__nat,axiom,
    ! [M_1: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N)) ) ).

tff(fact_890_le__neq__implies__less,axiom,
    ! [M_1: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N))
     => ( ( M_1 != N )
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N)) ) ) ).

tff(fact_891_less__or__eq__imp__le,axiom,
    ! [M_1: nat,N: nat] :
      ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N))
        | ( M_1 = N ) )
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N)) ) ).

tff(fact_892_termination__basic__simps_I5_J,axiom,
    ! [X: nat,Y: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),Y))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X),Y)) ) ).

tff(fact_893_less__imp__diff__less,axiom,
    ! [N: nat,J: nat,K_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,J),K_1))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J),N)),K_1)) ) ).

tff(fact_894_diff__less__mono2,axiom,
    ! [L_1: nat,M_1: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),L_1))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,L_1),N)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,L_1),M_1))) ) ) ).

tff(fact_895_finite__nat__set__iff__bounded,axiom,
    ! [N_2: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,N_2))
    <=> ? [M_2: nat] :
        ! [X_2: nat] :
          ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),N_2))
         => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_2),M_2)) ) ) ).

tff(fact_896_card__Collect__less__nat,axiom,
    ! [Na: nat] : ( hAPP_f22106695ol_nat(finite_card_nat,collect_nat(hAPP_n1699378549t_bool(cOMBC_nat_nat_bool(ord_less_nat),Na))) = Na ) ).

tff(fact_897_finite__M__bounded__by__nat,axiom,
    ! [Pa: fun_nat_bool,I_2: nat] : hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(cOMBS_nat_bool_bool(cOMBB_1015721476ol_nat(fconj,Pa),hAPP_n1699378549t_bool(cOMBC_nat_nat_bool(ord_less_nat),I_2))))) ).

tff(fact_898_less__add__Suc1,axiom,
    ! [I_1: nat,M_1: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_1),M_1)))) ).

tff(fact_899_less__add__Suc2,axiom,
    ! [I_1: nat,M_1: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),I_1)))) ).

tff(fact_900_less__iff__Suc__add,axiom,
    ! [M: nat,Na: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),Na))
    <=> ? [K_2: nat] : ( Na = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),K_2)) ) ) ).

tff(fact_901_less__eq__Suc__le,axiom,
    ! [Na: nat,M: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Na),M))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,Na)),M)) ) ).

tff(fact_902_less__Suc__eq__le,axiom,
    ! [M: nat,Na: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),hAPP_nat_nat(suc,Na)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),Na)) ) ).

tff(fact_903_Suc__le__eq,axiom,
    ! [M: nat,Na: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,M)),Na))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),Na)) ) ).

tff(fact_904_le__imp__less__Suc,axiom,
    ! [M_1: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),hAPP_nat_nat(suc,N))) ) ).

tff(fact_905_Suc__leI,axiom,
    ! [M_1: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,M_1)),N)) ) ).

tff(fact_906_le__less__Suc__eq,axiom,
    ! [M: nat,Na: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),Na))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Na),hAPP_nat_nat(suc,M)))
      <=> ( Na = M ) ) ) ).

tff(fact_907_Suc__le__lessD,axiom,
    ! [M_1: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,M_1)),N))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N)) ) ).

tff(fact_908_diff__less__Suc,axiom,
    ! [M_1: nat,N: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),N)),hAPP_nat_nat(suc,M_1))) ).

tff(fact_909_less__diff__conv,axiom,
    ! [I_2: nat,J_1: nat,K: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_1),K)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_2),K)),J_1)) ) ).

tff(fact_910_add__diff__inverse,axiom,
    ! [M_1: nat,N: nat] :
      ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),N)) = M_1 ) ) ).

tff(fact_911_diff__less__mono,axiom,
    ! [C: nat,A_1: nat,B_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_1),B_2))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C),A_1))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,A_1),C)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,B_2),C))) ) ) ).

tff(fact_912_less__diff__iff,axiom,
    ! [Na: nat,K: nat,M: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),M))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),Na))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),K)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,Na),K)))
        <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),Na)) ) ) ) ).

tff(fact_913_less__eq__Suc__le__raw,axiom,
    ! [X_2: nat] : ( hAPP_n1699378549t_bool(ord_less_nat,X_2) = hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,X_2)) ) ).

tff(fact_914_mono__nat__linear__lb,axiom,
    ! [M: nat,K: nat,F: fun_nat_nat] :
      ( ! [M_2: nat,N_1: nat] :
          ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_2),N_1))
         => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(F,M_2)),hAPP_nat_nat(F,N_1))) )
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(F,M)),K)),hAPP_nat_nat(F,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),K)))) ) ).

tff(fact_915_inc__induct,axiom,
    ! [Pa: fun_nat_bool,I_2: nat,J_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),J_1))
     => ( hBOOL(hAPP_nat_bool(Pa,J_1))
       => ( ! [I: nat] :
              ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J_1))
             => ( hBOOL(hAPP_nat_bool(Pa,hAPP_nat_nat(suc,I)))
               => hBOOL(hAPP_nat_bool(Pa,I)) ) )
         => hBOOL(hAPP_nat_bool(Pa,I_2)) ) ) ) ).

tff(fact_916_less__imp__Suc__add,axiom,
    ! [M_1: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N))
     => ? [K_2: nat] : ( N = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),K_2)) ) ) ).

tff(fact_917_bounded__nat__set__is__finite,axiom,
    ! [Na: nat,N_2: fun_nat_bool] :
      ( ! [X_2: nat] :
          ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),N_2))
         => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_2),Na)) )
     => hBOOL(hAPP_f54304608l_bool(finite_finite_nat,N_2)) ) ).

tff(fact_918_less__mono__imp__le__mono,axiom,
    ! [I_2: nat,J_1: nat,F: fun_nat_nat] :
      ( ! [I: nat,J_2: nat] :
          ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J_2))
         => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(F,I)),hAPP_nat_nat(F,J_2))) )
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),J_1))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(F,I_2)),hAPP_nat_nat(F,J_1))) ) ) ).

tff(fact_919_lessE,axiom,
    ! [I_1: nat,K_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),K_1))
     => ( ( K_1 != hAPP_nat_nat(suc,I_1) )
       => ~ ! [J_2: nat] :
              ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),J_2))
             => ( K_1 != hAPP_nat_nat(suc,J_2) ) ) ) ) ).

tff(fact_920_Suc__lessE,axiom,
    ! [I_1: nat,K_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,I_1)),K_1))
     => ~ ! [J_2: nat] :
            ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),J_2))
           => ( K_1 != hAPP_nat_nat(suc,J_2) ) ) ) ).

tff(fact_921_less__zeroE,axiom,
    ! [N: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),zero_zero_nat)) ).

tff(fact_922_le0,axiom,
    ! [N: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),N)) ).

tff(fact_923_zero__less__Suc,axiom,
    ! [N: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(suc,N))) ).

tff(fact_924_le__0__eq,axiom,
    ! [Na: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Na),zero_zero_nat))
    <=> ( Na = zero_zero_nat ) ) ).

tff(fact_925_less__eq__nat_Osimps_I1_J,axiom,
    ! [N: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),N)) ).

tff(fact_926_diffs0__imp__equal,axiom,
    ! [M_1: nat,N: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),N) = zero_zero_nat )
     => ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),M_1) = zero_zero_nat )
       => ( M_1 = N ) ) ) ).

tff(fact_927_diff__self__eq__0,axiom,
    ! [M_1: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),M_1) = zero_zero_nat ) ).

tff(fact_928_minus__nat_Odiff__0,axiom,
    ! [M_1: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),zero_zero_nat) = M_1 ) ).

tff(fact_929_diff__0__eq__0,axiom,
    ! [N: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,zero_zero_nat),N) = zero_zero_nat ) ).

tff(fact_930_bot__nat__def,axiom,
    bot_bot_nat = zero_zero_nat ).

tff(fact_931_add__eq__self__zero,axiom,
    ! [M_1: nat,N: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),N) = M_1 )
     => ( N = zero_zero_nat ) ) ).

tff(fact_932_add__is__0,axiom,
    ! [M: nat,Na: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),Na) = zero_zero_nat )
    <=> ( ( M = zero_zero_nat )
        & ( Na = zero_zero_nat ) ) ) ).

tff(fact_933_Nat_Oadd__0__right,axiom,
    ! [M_1: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),zero_zero_nat) = M_1 ) ).

tff(fact_934_plus__nat_Oadd__0,axiom,
    ! [N: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),N) = N ) ).

tff(fact_935_gr0I,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) ) ).

tff(fact_936_gr__implies__not0,axiom,
    ! [M_1: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N))
     => ( N != zero_zero_nat ) ) ).

tff(fact_937_less__nat__zero__code,axiom,
    ! [N: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),zero_zero_nat)) ).

tff(fact_938_neq0__conv,axiom,
    ! [Na: nat] :
      ( ( Na != zero_zero_nat )
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na)) ) ).

tff(fact_939_not__less0,axiom,
    ! [N: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),zero_zero_nat)) ).

tff(fact_940_Suc__neq__Zero,axiom,
    ! [M_1: nat] : ( hAPP_nat_nat(suc,M_1) != zero_zero_nat ) ).

tff(fact_941_Zero__neq__Suc,axiom,
    ! [M_1: nat] : ( zero_zero_nat != hAPP_nat_nat(suc,M_1) ) ).

tff(fact_942_nat_Osimps_I3_J,axiom,
    ! [Nat_2: nat] : ( hAPP_nat_nat(suc,Nat_2) != zero_zero_nat ) ).

tff(fact_943_Suc__not__Zero,axiom,
    ! [M_1: nat] : ( hAPP_nat_nat(suc,M_1) != zero_zero_nat ) ).

tff(fact_944_nat_Osimps_I2_J,axiom,
    ! [Nat_1: nat] : ( zero_zero_nat != hAPP_nat_nat(suc,Nat_1) ) ).

tff(fact_945_Zero__not__Suc,axiom,
    ! [M_1: nat] : ( zero_zero_nat != hAPP_nat_nat(suc,M_1) ) ).

tff(fact_946_gr0__conv__Suc,axiom,
    ! [Na: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na))
    <=> ? [M_2: nat] : ( Na = hAPP_nat_nat(suc,M_2) ) ) ).

tff(fact_947_less__Suc0,axiom,
    ! [Na: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Na),hAPP_nat_nat(suc,zero_zero_nat)))
    <=> ( Na = zero_zero_nat ) ) ).

tff(fact_948_less__Suc__eq__0__disj,axiom,
    ! [M: nat,Na: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),hAPP_nat_nat(suc,Na)))
    <=> ( ( M = zero_zero_nat )
        | ? [J_2: nat] :
            ( ( M = hAPP_nat_nat(suc,J_2) )
            & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,J_2),Na)) ) ) ) ).

tff(fact_949_one__is__add,axiom,
    ! [M: nat,Na: nat] :
      ( ( hAPP_nat_nat(suc,zero_zero_nat) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),Na) )
    <=> ( ( ( M = hAPP_nat_nat(suc,zero_zero_nat) )
          & ( Na = zero_zero_nat ) )
        | ( ( M = zero_zero_nat )
          & ( Na = hAPP_nat_nat(suc,zero_zero_nat) ) ) ) ) ).

tff(fact_950_add__is__1,axiom,
    ! [M: nat,Na: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),Na) = hAPP_nat_nat(suc,zero_zero_nat) )
    <=> ( ( ( M = hAPP_nat_nat(suc,zero_zero_nat) )
          & ( Na = zero_zero_nat ) )
        | ( ( M = zero_zero_nat )
          & ( Na = hAPP_nat_nat(suc,zero_zero_nat) ) ) ) ) ).

tff(fact_951_add__gr__0,axiom,
    ! [M: nat,Na: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),Na)))
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M))
        | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na)) ) ) ).

tff(fact_952_zero__less__diff,axiom,
    ! [Na: nat,M: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,Na),M)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),Na)) ) ).

tff(fact_953_diff__less,axiom,
    ! [M_1: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M_1))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),N)),M_1)) ) ) ).

tff(fact_954_diff__add__0,axiom,
    ! [N: nat,M_1: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),M_1)) = zero_zero_nat ) ).

tff(fact_955_diff__is__0__eq,axiom,
    ! [M: nat,Na: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),Na) = zero_zero_nat )
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),Na)) ) ).

tff(fact_956_diff__is__0__eq_H,axiom,
    ! [M_1: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),N) = zero_zero_nat ) ) ).

tff(fact_957_One__nat__def,axiom,
    one_one_nat = hAPP_nat_nat(suc,zero_zero_nat) ).

tff(fact_958_diff__Suc,axiom,
    ! [M: nat,Na: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),hAPP_nat_nat(suc,Na)) = hAPP_nat_nat(nat_case_nat(zero_zero_nat,cOMBI_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),Na)) ) ).

tff(fact_959_Suc__pred,axiom,
    ! [N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
     => ( hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),hAPP_nat_nat(suc,zero_zero_nat))) = N ) ) ).

tff(fact_960_diff__Suc__less,axiom,
    ! [I_1: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),hAPP_nat_nat(suc,I_1))),N)) ) ).

tff(fact_961_nat__diff__split,axiom,
    ! [Pa: fun_nat_bool,A_2: nat,B_3: nat] :
      ( hBOOL(hAPP_nat_bool(Pa,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,A_2),B_3)))
    <=> ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_2),B_3))
         => hBOOL(hAPP_nat_bool(Pa,zero_zero_nat)) )
        & ! [D_2: nat] :
            ( ( A_2 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_3),D_2) )
           => hBOOL(hAPP_nat_bool(Pa,D_2)) ) ) ) ).

tff(fact_962_nat__diff__split__asm,axiom,
    ! [Pa: fun_nat_bool,A_2: nat,B_3: nat] :
      ( hBOOL(hAPP_nat_bool(Pa,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,A_2),B_3)))
    <=> ~ ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_2),B_3))
            & ~ hBOOL(hAPP_nat_bool(Pa,zero_zero_nat)) )
          | ? [D_2: nat] :
              ( ( A_2 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_3),D_2) )
              & ~ hBOOL(hAPP_nat_bool(Pa,D_2)) ) ) ) ).

tff(fact_963_card__less__Suc,axiom,
    ! [I_2: nat,M_3: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,zero_zero_nat),M_3))
     => ( hAPP_nat_nat(suc,hAPP_f22106695ol_nat(finite_card_nat,collect_nat(cOMBS_nat_bool_bool(cOMBB_1015721476ol_nat(fconj,hAPP_f800510211t_bool(cOMBC_226598744l_bool(cOMBB_1146692694ol_nat(member_nat,suc)),M_3)),hAPP_n1699378549t_bool(cOMBC_nat_nat_bool(ord_less_nat),I_2))))) = hAPP_f22106695ol_nat(finite_card_nat,collect_nat(cOMBS_nat_bool_bool(cOMBB_1015721476ol_nat(fconj,hAPP_f800510211t_bool(cOMBC_226598744l_bool(member_nat),M_3)),hAPP_n1699378549t_bool(cOMBC_nat_nat_bool(ord_less_nat),hAPP_nat_nat(suc,I_2))))) ) ) ).

tff(fact_964_card__less,axiom,
    ! [I_2: nat,M_3: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,zero_zero_nat),M_3))
     => ( hAPP_f22106695ol_nat(finite_card_nat,collect_nat(cOMBS_nat_bool_bool(cOMBB_1015721476ol_nat(fconj,hAPP_f800510211t_bool(cOMBC_226598744l_bool(member_nat),M_3)),hAPP_n1699378549t_bool(cOMBC_nat_nat_bool(ord_less_nat),hAPP_nat_nat(suc,I_2))))) != zero_zero_nat ) ) ).

tff(fact_965_card__less__Suc2,axiom,
    ! [I_2: nat,M_3: fun_nat_bool] :
      ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,zero_zero_nat),M_3))
     => ( hAPP_f22106695ol_nat(finite_card_nat,collect_nat(cOMBS_nat_bool_bool(cOMBB_1015721476ol_nat(fconj,hAPP_f800510211t_bool(cOMBC_226598744l_bool(cOMBB_1146692694ol_nat(member_nat,suc)),M_3)),hAPP_n1699378549t_bool(cOMBC_nat_nat_bool(ord_less_nat),I_2)))) = hAPP_f22106695ol_nat(finite_card_nat,collect_nat(cOMBS_nat_bool_bool(cOMBB_1015721476ol_nat(fconj,hAPP_f800510211t_bool(cOMBC_226598744l_bool(member_nat),M_3)),hAPP_n1699378549t_bool(cOMBC_nat_nat_bool(ord_less_nat),hAPP_nat_nat(suc,I_2))))) ) ) ).

tff(fact_966_Suc__diff__1,axiom,
    ! [N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
     => ( hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) = N ) ) ).

tff(fact_967_Suc__pred_H,axiom,
    ! [N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
     => ( N = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) ) ) ).

tff(fact_968_add__eq__if,axiom,
    ! [N: nat,M_1: nat] :
      ( ( ( M_1 = zero_zero_nat )
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),N) = N ) )
      & ( ( M_1 != zero_zero_nat )
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),N) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),one_one_nat)),N)) ) ) ) ).

tff(fact_969_ex__least__nat__less,axiom,
    ! [Na: nat,Pa: fun_nat_bool] :
      ( ~ hBOOL(hAPP_nat_bool(Pa,zero_zero_nat))
     => ( hBOOL(hAPP_nat_bool(Pa,Na))
       => ? [K_2: nat] :
            ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,K_2),Na))
            & ! [I: nat] :
                ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),K_2))
               => ~ hBOOL(hAPP_nat_bool(Pa,I)) )
            & hBOOL(hAPP_nat_bool(Pa,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_2),one_one_nat))) ) ) ) ).

tff(fact_970_ex__least__nat__le,axiom,
    ! [Na: nat,Pa: fun_nat_bool] :
      ( ~ hBOOL(hAPP_nat_bool(Pa,zero_zero_nat))
     => ( hBOOL(hAPP_nat_bool(Pa,Na))
       => ? [K_2: nat] :
            ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),Na))
            & ! [I: nat] :
                ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),K_2))
               => ~ hBOOL(hAPP_nat_bool(Pa,I)) )
            & hBOOL(hAPP_nat_bool(Pa,K_2)) ) ) ) ).

tff(fact_971_less__imp__add__positive,axiom,
    ! [I_1: nat,J: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),J))
     => ? [K_2: nat] :
          ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_2))
          & ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_1),K_2) = J ) ) ) ).

tff(fact_972_gr0__implies__Suc,axiom,
    ! [N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
     => ? [M_2: nat] : ( N = hAPP_nat_nat(suc,M_2) ) ) ).

tff(fact_973_nat_Oexhaust,axiom,
    ! [Y: nat] :
      ( ( Y != zero_zero_nat )
     => ~ ! [Nat: nat] : ( Y != hAPP_nat_nat(suc,Nat) ) ) ).

tff(fact_974_zero__induct,axiom,
    ! [Pa: fun_nat_bool,K: nat] :
      ( hBOOL(hAPP_nat_bool(Pa,K))
     => ( ! [N_1: nat] :
            ( hBOOL(hAPP_nat_bool(Pa,hAPP_nat_nat(suc,N_1)))
           => hBOOL(hAPP_nat_bool(Pa,N_1)) )
       => hBOOL(hAPP_nat_bool(Pa,zero_zero_nat)) ) ) ).

tff(fact_975_not0__implies__Suc,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ? [M_2: nat] : ( N = hAPP_nat_nat(suc,M_2) ) ) ).

tff(fact_976_nat__induct,axiom,
    ! [Na: nat,Pa: fun_nat_bool] :
      ( hBOOL(hAPP_nat_bool(Pa,zero_zero_nat))
     => ( ! [N_1: nat] :
            ( hBOOL(hAPP_nat_bool(Pa,N_1))
           => hBOOL(hAPP_nat_bool(Pa,hAPP_nat_nat(suc,N_1))) )
       => hBOOL(hAPP_nat_bool(Pa,Na)) ) ) ).

tff(fact_977_expand__Suc,axiom,
    ! [V: int] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),number_number_of_nat(V)))
     => ( number_number_of_nat(V) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,number_number_of_nat(V)),one_one_nat)) ) ) ).

tff(fact_978_mult__0,axiom,
    ! [N: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,zero_zero_nat),N) = zero_zero_nat ) ).

tff(fact_979_mult__0__right,axiom,
    ! [M_1: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),zero_zero_nat) = zero_zero_nat ) ).

tff(fact_980_mult__is__0,axiom,
    ! [M: nat,Na: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),Na) = zero_zero_nat )
    <=> ( ( M = zero_zero_nat )
        | ( Na = zero_zero_nat ) ) ) ).

tff(fact_981_mult__cancel1,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K),M) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K),Na) )
    <=> ( ( M = Na )
        | ( K = zero_zero_nat ) ) ) ).

tff(fact_982_mult__cancel2,axiom,
    ! [M: nat,K: nat,Na: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),K) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Na),K) )
    <=> ( ( M = Na )
        | ( K = zero_zero_nat ) ) ) ).

tff(fact_983_Suc__mult__cancel1,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,K)),M) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,K)),Na) )
    <=> ( M = Na ) ) ).

tff(fact_984_add__mult__distrib,axiom,
    ! [M_1: nat,N: nat,K_1: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),N)),K_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),K_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),K_1)) ) ).

tff(fact_985_add__mult__distrib2,axiom,
    ! [K_1: nat,M_1: nat,N: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),M_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),N)) ) ).

tff(fact_986_mult__le__mono,axiom,
    ! [K_1: nat,L_1: nat,I_1: nat,J: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),L_1))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I_1),K_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J),L_1))) ) ) ).

tff(fact_987_mult__le__mono2,axiom,
    ! [K_1: nat,I_1: nat,J: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),I_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),J))) ) ).

tff(fact_988_mult__le__mono1,axiom,
    ! [K_1: nat,I_1: nat,J: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I_1),K_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J),K_1))) ) ).

tff(fact_989_le__cube,axiom,
    ! [M_1: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),M_1)))) ).

tff(fact_990_le__square,axiom,
    ! [M_1: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),M_1))) ).

tff(fact_991_diff__mult__distrib,axiom,
    ! [M_1: nat,N: nat,K_1: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),N)),K_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),K_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),K_1)) ) ).

tff(fact_992_diff__mult__distrib2,axiom,
    ! [K_1: nat,M_1: nat,N: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),M_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),N)) ) ).

tff(fact_993_nat__mult__eq__1__iff,axiom,
    ! [M: nat,Na: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),Na) = one_one_nat )
    <=> ( ( M = one_one_nat )
        & ( Na = one_one_nat ) ) ) ).

tff(fact_994_nat__mult__1__right,axiom,
    ! [N: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),one_one_nat) = N ) ).

tff(fact_995_nat__1__eq__mult__iff,axiom,
    ! [M: nat,Na: nat] :
      ( ( one_one_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),Na) )
    <=> ( ( M = one_one_nat )
        & ( Na = one_one_nat ) ) ) ).

tff(fact_996_nat__mult__1,axiom,
    ! [N: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,one_one_nat),N) = N ) ).

tff(fact_997_mult__eq__1__iff,axiom,
    ! [M: nat,Na: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),Na) = hAPP_nat_nat(suc,zero_zero_nat) )
    <=> ( ( M = hAPP_nat_nat(suc,zero_zero_nat) )
        & ( Na = hAPP_nat_nat(suc,zero_zero_nat) ) ) ) ).

tff(fact_998_mult__less__mono2,axiom,
    ! [K_1: nat,I_1: nat,J: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),J))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_1))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),I_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),J))) ) ) ).

tff(fact_999_mult__less__mono1,axiom,
    ! [K_1: nat,I_1: nat,J: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),J))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_1))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I_1),K_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J),K_1))) ) ) ).

tff(fact_1000_mult__less__cancel2,axiom,
    ! [M: nat,K: nat,Na: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),K)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Na),K)))
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K))
        & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),Na)) ) ) ).

tff(fact_1001_mult__less__cancel1,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K),Na)))
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K))
        & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),Na)) ) ) ).

tff(fact_1002_nat__0__less__mult__iff,axiom,
    ! [M: nat,Na: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),Na)))
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M))
        & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na)) ) ) ).

tff(fact_1003_Suc__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,K)),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,K)),Na)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),Na)) ) ).

tff(fact_1004_mult__Suc__right,axiom,
    ! [M_1: nat,N: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),hAPP_nat_nat(suc,N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),N)) ) ).

tff(fact_1005_mult__Suc,axiom,
    ! [M_1: nat,N: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,M_1)),N) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),N)) ) ).

tff(fact_1006_Suc__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,K)),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,K)),Na)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),Na)) ) ).

tff(fact_1007_mult__eq__self__implies__10,axiom,
    ! [M_1: nat,N: nat] :
      ( ( M_1 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),N) )
     => ( ( N = one_one_nat )
        | ( M_1 = zero_zero_nat ) ) ) ).

tff(fact_1008_n__less__m__mult__n,axiom,
    ! [M_1: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),N))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),M_1))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),N))) ) ) ).

tff(fact_1009_n__less__n__mult__m,axiom,
    ! [M_1: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),N))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),M_1))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),M_1))) ) ) ).

tff(fact_1010_one__less__mult,axiom,
    ! [M_1: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),N))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),M_1))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),N))) ) ) ).

tff(fact_1011_one__le__mult__iff,axiom,
    ! [M: nat,Na: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,zero_zero_nat)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),Na)))
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,zero_zero_nat)),M))
        & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,zero_zero_nat)),Na)) ) ) ).

tff(fact_1012_mult__le__cancel1,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K),Na)))
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),Na)) ) ) ).

tff(fact_1013_mult__le__cancel2,axiom,
    ! [M: nat,K: nat,Na: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),K)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Na),K)))
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),Na)) ) ) ).

tff(fact_1014_mult__eq__if,axiom,
    ! [N: nat,M_1: nat] :
      ( ( ( M_1 = zero_zero_nat )
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),N) = zero_zero_nat ) )
      & ( ( M_1 != zero_zero_nat )
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),N) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),one_one_nat)),N)) ) ) ) ).

tff(fact_1015_nat__less__add__iff2,axiom,
    ! [U: nat,M: nat,Na: nat,I_2: nat,J_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),J_1))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I_2),U)),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J_1),U)),Na)))
      <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_1),I_2)),U)),Na))) ) ) ).

tff(fact_1016_nat__mult__commute,axiom,
    ! [M_1: nat,N: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),N) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),M_1) ) ).

tff(fact_1017_nat__mult__assoc,axiom,
    ! [M_1: nat,N: nat,K_1: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),N)),K_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),K_1)) ) ).

tff(fact_1018_less__number__of__int__code,axiom,
    ! [K: int,L: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(K)),number_number_of_int(L)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),L)) ) ).

tff(fact_1019_zmult__1,axiom,
    ! [Z: int] : ( hAPP_int_int(hAPP_int_fun_int_int(times_times_int,one_one_int),Z) = Z ) ).

tff(fact_1020_zmult__1__right,axiom,
    ! [Z: int] : ( hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z),one_one_int) = Z ) ).

tff(fact_1021_zmult__commute,axiom,
    ! [Z: int,W: int] : ( hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z),W) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,W),Z) ) ).

tff(fact_1022_times__numeral__code_I5_J,axiom,
    ! [V: int,W: int] : ( hAPP_int_int(hAPP_int_fun_int_int(times_times_int,number_number_of_int(V)),number_number_of_int(W)) = number_number_of_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V),W)) ) ).

tff(fact_1023_zmult__assoc,axiom,
    ! [Z1: int,Z2: int,Z3: int] : ( hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z1),Z2)),Z3) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z1),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z2),Z3)) ) ).

tff(fact_1024_less__eq__number__of__int__code,axiom,
    ! [K: int,L: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,number_number_of_int(K)),number_number_of_int(L)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K),L)) ) ).

tff(fact_1025_zdiff__zmult__distrib,axiom,
    ! [Z1: int,Z2: int,W: int] : ( hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Z1),Z2)),W) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z1),W)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z2),W)) ) ).

tff(fact_1026_zdiff__zmult__distrib2,axiom,
    ! [W: int,Z1: int,Z2: int] : ( hAPP_int_int(hAPP_int_fun_int_int(times_times_int,W),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Z1),Z2)) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,W),Z1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,W),Z2)) ) ).

tff(fact_1027_zadd__zmult__distrib2,axiom,
    ! [W: int,Z1: int,Z2: int] : ( hAPP_int_int(hAPP_int_fun_int_int(times_times_int,W),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z1),Z2)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,W),Z1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,W),Z2)) ) ).

tff(fact_1028_plus__numeral__code_I9_J,axiom,
    ! [V: int,W: int] : ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,number_number_of_int(V)),number_number_of_int(W)) = number_number_of_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V),W)) ) ).

tff(fact_1029_zadd__zmult__distrib,axiom,
    ! [Z1: int,Z2: int,W: int] : ( hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z1),Z2)),W) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z1),W)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z2),W)) ) ).

tff(fact_1030_zmult__zless__mono2,axiom,
    ! [K_1: int,I_1: int,J: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I_1),J))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),K_1))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_1),I_1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_1),J))) ) ) ).

tff(fact_1031_pos__zmult__eq__1__iff,axiom,
    ! [Na: int,M: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),M))
     => ( ( hAPP_int_int(hAPP_int_fun_int_int(times_times_int,M),Na) = one_one_int )
      <=> ( ( M = one_one_int )
          & ( Na = one_one_int ) ) ) ) ).

tff(fact_1032_odd__nonzero,axiom,
    ! [Z: int] : ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),Z)),Z) != zero_zero_int ) ).

tff(fact_1033_zadd__0__right,axiom,
    ! [Z: int] : ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z),zero_zero_int) = Z ) ).

tff(fact_1034_zadd__0,axiom,
    ! [Z: int] : ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,zero_zero_int),Z) = Z ) ).

tff(fact_1035_int__0__less__1,axiom,
    hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),one_one_int)) ).

tff(fact_1036_int__one__le__iff__zero__less,axiom,
    ! [Z_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),Z_2))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Z_2)) ) ).

tff(fact_1037_less__bin__lemma,axiom,
    ! [K: int,L: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),L))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,K),L)),zero_zero_int)) ) ).

tff(fact_1038_le__imp__0__less,axiom,
    ! [Z: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),Z))) ) ).

tff(fact_1039_odd__less__0,axiom,
    ! [Z_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),Z_2)),Z_2)),zero_zero_int))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_2),zero_zero_int)) ) ).

tff(fact_1040_zadd__left__mono,axiom,
    ! [K_1: int,I_1: int,J: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I_1),J))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,K_1),I_1)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,K_1),J))) ) ).

tff(fact_1041_zadd__assoc,axiom,
    ! [Z1: int,Z2: int,Z3: int] : ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z1),Z2)),Z3) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z1),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z2),Z3)) ) ).

tff(fact_1042_zadd__left__commute,axiom,
    ! [X: int,Y: int,Z: int] : ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Y),Z)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Y),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),Z)) ) ).

tff(fact_1043_zadd__commute,axiom,
    ! [Z: int,W: int] : ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z),W) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,W),Z) ) ).

tff(fact_1044_zle__refl,axiom,
    ! [W: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W),W)) ).

tff(fact_1045_zle__linear,axiom,
    ! [Z: int,W: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z),W))
      | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W),Z)) ) ).

tff(fact_1046_zle__trans,axiom,
    ! [K_1: int,I_1: int,J: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I_1),J))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,J),K_1))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I_1),K_1)) ) ) ).

tff(fact_1047_zle__antisym,axiom,
    ! [Z: int,W: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z),W))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W),Z))
       => ( Z = W ) ) ) ).

tff(fact_1048_zle__diff1__eq,axiom,
    ! [W_1: int,Z_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W_1),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Z_2),one_one_int)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),Z_2)) ) ).

tff(fact_1049_zless__linear,axiom,
    ! [X: int,Y: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),Y))
      | ( X = Y )
      | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y),X)) ) ).

tff(fact_1050_zless__le,axiom,
    ! [Z_2: int,W_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_2),W_1))
    <=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_2),W_1))
        & ( Z_2 != W_1 ) ) ) ).

tff(fact_1051_zadd__strict__right__mono,axiom,
    ! [K_1: int,I_1: int,J: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I_1),J))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,I_1),K_1)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,J),K_1))) ) ).

tff(fact_1052_zadd__zless__mono,axiom,
    ! [Z_4: int,Z: int,W_2: int,W: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_2),W))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_4),Z))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,W_2),Z_4)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,W),Z))) ) ) ).

tff(fact_1053_zle__add1__eq__le,axiom,
    ! [W_1: int,Z_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z_2),one_one_int)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W_1),Z_2)) ) ).

tff(fact_1054_add1__zle__eq,axiom,
    ! [W_1: int,Z_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,W_1),one_one_int)),Z_2))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),Z_2)) ) ).

tff(fact_1055_zless__imp__add1__zle,axiom,
    ! [W: int,Z: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W),Z))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,W),one_one_int)),Z)) ) ).

tff(fact_1056_zless__add1__eq,axiom,
    ! [W_1: int,Z_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z_2),one_one_int)))
    <=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),Z_2))
        | ( W_1 = Z_2 ) ) ) ).

tff(fact_1057_nat__mult__eq__cancel__disj,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K),M) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K),Na) )
    <=> ( ( K = zero_zero_nat )
        | ( M = Na ) ) ) ).

tff(fact_1058_left__add__mult__distrib,axiom,
    ! [I_1: nat,U_1: nat,J: nat,K_1: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I_1),U_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J),U_1)),K_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_1),J)),U_1)),K_1) ) ).

tff(fact_1059_nat__mult__eq__cancel1,axiom,
    ! [M: nat,Na: nat,K: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K))
     => ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K),M) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K),Na) )
      <=> ( M = Na ) ) ) ).

tff(fact_1060_nat__mult__less__cancel1,axiom,
    ! [M: nat,Na: nat,K: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K),Na)))
      <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),Na)) ) ) ).

tff(fact_1061_nat__mult__le__cancel1,axiom,
    ! [M: nat,Na: nat,K: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K),Na)))
      <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),Na)) ) ) ).

tff(fact_1062_nat__le__add__iff1,axiom,
    ! [U: nat,M: nat,Na: nat,J_1: nat,I_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J_1),I_2))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I_2),U)),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J_1),U)),Na)))
      <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I_2),J_1)),U)),M)),Na)) ) ) ).

tff(fact_1063_nat__diff__add__eq1,axiom,
    ! [U_1: nat,M_1: nat,N: nat,J: nat,I_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J),I_1))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I_1),U_1)),M_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J),U_1)),N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I_1),J)),U_1)),M_1)),N) ) ) ).

tff(fact_1064_nat__eq__add__iff1,axiom,
    ! [U: nat,M: nat,Na: nat,J_1: nat,I_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J_1),I_2))
     => ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I_2),U)),M) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J_1),U)),Na) )
      <=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I_2),J_1)),U)),M) = Na ) ) ) ).

tff(fact_1065_nat__le__add__iff2,axiom,
    ! [U: nat,M: nat,Na: nat,I_2: nat,J_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),J_1))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I_2),U)),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J_1),U)),Na)))
      <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_1),I_2)),U)),Na))) ) ) ).

tff(fact_1066_nat__diff__add__eq2,axiom,
    ! [U_1: nat,M_1: nat,N: nat,I_1: nat,J: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I_1),U_1)),M_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J),U_1)),N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J),I_1)),U_1)),N)) ) ) ).

tff(fact_1067_nat__eq__add__iff2,axiom,
    ! [U: nat,M: nat,Na: nat,I_2: nat,J_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),J_1))
     => ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I_2),U)),M) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J_1),U)),Na) )
      <=> ( M = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_1),I_2)),U)),Na) ) ) ) ).

tff(fact_1068_nat__less__add__iff1,axiom,
    ! [U: nat,M: nat,Na: nat,J_1: nat,I_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J_1),I_2))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I_2),U)),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J_1),U)),Na)))
      <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I_2),J_1)),U)),M)),Na)) ) ) ).

tff(fact_1069_zdiv__mono2__neg__lemma,axiom,
    ! [B_2: int,Q_2: int,R_2: int,B_1: int,Q_1: int,R_1: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),Q_2)),R_2) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_1),Q_1)),R_1) )
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_1),Q_1)),R_1)),zero_zero_int))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_2),B_2))
         => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_1))
           => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_1))
             => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_1),B_2))
               => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q_1),Q_2)) ) ) ) ) ) ) ).

tff(fact_1070_unique__quotient__lemma__neg,axiom,
    ! [B_2: int,Q_1: int,R_1: int,Q_2: int,R_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),Q_1)),R_1)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),Q_2)),R_2)))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,R_2),zero_zero_int))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),R_2))
         => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),R_1))
           => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q_2),Q_1)) ) ) ) ) ).

tff(fact_1071_number__of__is__id,axiom,
    ! [K_1: int] : ( number_number_of_int(K_1) = K_1 ) ).

tff(fact_1072_int__0__neq__1,axiom,
    zero_zero_int != one_one_int ).

tff(fact_1073_self__quotient__aux1,axiom,
    ! [R_2: int,Q_2: int,A_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_1))
     => ( ( A_1 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,R_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_1),Q_2)) )
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_2),A_1))
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),Q_2)) ) ) ) ).

tff(fact_1074_self__quotient__aux2,axiom,
    ! [R_2: int,Q_2: int,A_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_1))
     => ( ( A_1 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,R_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_1),Q_2)) )
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_2))
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q_2),one_one_int)) ) ) ) ).

tff(fact_1075_q__pos__lemma,axiom,
    ! [B_1: int,Q_1: int,R_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_1),Q_1)),R_1)))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_1),B_1))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_1))
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Q_1)) ) ) ) ).

tff(fact_1076_q__neg__lemma,axiom,
    ! [B_1: int,Q_1: int,R_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_1),Q_1)),R_1)),zero_zero_int))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_1))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_1))
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q_1),zero_zero_int)) ) ) ) ).

tff(fact_1077_unique__quotient__lemma,axiom,
    ! [B_2: int,Q_1: int,R_1: int,Q_2: int,R_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),Q_1)),R_1)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),Q_2)),R_2)))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_1))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_1),B_2))
         => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_2),B_2))
           => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q_1),Q_2)) ) ) ) ) ).

tff(fact_1078_zdiv__mono2__lemma,axiom,
    ! [B_2: int,Q_2: int,R_2: int,B_1: int,Q_1: int,R_1: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),Q_2)),R_2) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_1),Q_1)),R_1) )
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_1),Q_1)),R_1)))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_1),B_1))
         => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_2))
           => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_1))
             => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_1),B_2))
               => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q_2),Q_1)) ) ) ) ) ) ) ).

tff(fact_1079_int__less__induct,axiom,
    ! [Pa: fun_int_bool,I_2: int,K: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I_2),K))
     => ( hBOOL(hAPP_int_bool(Pa,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,K),one_one_int)))
       => ( ! [I: int] :
              ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I),K))
             => ( hBOOL(hAPP_int_bool(Pa,I))
               => hBOOL(hAPP_int_bool(Pa,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,I),one_one_int))) ) )
         => hBOOL(hAPP_int_bool(Pa,I_2)) ) ) ) ).

tff(fact_1080_int__le__induct,axiom,
    ! [Pa: fun_int_bool,I_2: int,K: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I_2),K))
     => ( hBOOL(hAPP_int_bool(Pa,K))
       => ( ! [I: int] :
              ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I),K))
             => ( hBOOL(hAPP_int_bool(Pa,I))
               => hBOOL(hAPP_int_bool(Pa,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,I),one_one_int))) ) )
         => hBOOL(hAPP_int_bool(Pa,I_2)) ) ) ) ).

tff(fact_1081_int__gr__induct,axiom,
    ! [Pa: fun_int_bool,K: int,I_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),I_2))
     => ( hBOOL(hAPP_int_bool(Pa,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,K),one_one_int)))
       => ( ! [I: int] :
              ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),I))
             => ( hBOOL(hAPP_int_bool(Pa,I))
               => hBOOL(hAPP_int_bool(Pa,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,I),one_one_int))) ) )
         => hBOOL(hAPP_int_bool(Pa,I_2)) ) ) ) ).

tff(fact_1082_int__ge__induct,axiom,
    ! [Pa: fun_int_bool,K: int,I_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K),I_2))
     => ( hBOOL(hAPP_int_bool(Pa,K))
       => ( ! [I: int] :
              ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K),I))
             => ( hBOOL(hAPP_int_bool(Pa,I))
               => hBOOL(hAPP_int_bool(Pa,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,I),one_one_int))) ) )
         => hBOOL(hAPP_int_bool(Pa,I_2)) ) ) ) ).

tff(fact_1083_Nat__Transfer_Otransfer__nat__int__function__closures_I2_J,axiom,
    ! [Y: int,X: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X),Y))) ) ) ).

tff(fact_1084_Nat__Transfer_Otransfer__nat__int__function__closures_I5_J,axiom,
    hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),zero_zero_int)) ).

tff(fact_1085_Nat__Transfer_Otransfer__nat__int__function__closures_I1_J,axiom,
    ! [Y: int,X: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),Y))) ) ) ).

tff(fact_1086_Nat__Transfer_Otransfer__nat__int__function__closures_I6_J,axiom,
    hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),one_one_int)) ).

tff(fact_1087_transfer__nat__int__set__cong,axiom,
    ! [P_1: fun_int_bool,Pa: fun_int_bool] :
      ( ! [X_2: int] :
          ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2))
         => ( hBOOL(hAPP_int_bool(Pa,X_2))
          <=> hBOOL(hAPP_int_bool(P_1,X_2)) ) )
     => ( collect_int(cOMBS_int_bool_bool(cOMBB_1652995168ol_int(fconj,hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int)),Pa)) = collect_int(cOMBS_int_bool_bool(cOMBB_1652995168ol_int(fconj,hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int)),P_1)) ) ) ).

tff(fact_1088_decr__mult__lemma,axiom,
    ! [K: int,Pa: fun_int_bool,D_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1))
     => ( ! [X_2: int] :
            ( hBOOL(hAPP_int_bool(Pa,X_2))
           => hBOOL(hAPP_int_bool(Pa,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_2),D_1))) )
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K))
         => ! [X_2: int] :
              ( hBOOL(hAPP_int_bool(Pa,X_2))
             => hBOOL(hAPP_int_bool(Pa,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K),D_1)))) ) ) ) ) ).

tff(fact_1089_conj__le__cong,axiom,
    ! [P_1: bool,Pa: bool,X_1: int] :
      ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
       => ( hBOOL(Pa)
        <=> hBOOL(P_1) ) )
     => ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
          & hBOOL(Pa) )
      <=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
          & hBOOL(P_1) ) ) ) ).

tff(fact_1090_imp__le__cong,axiom,
    ! [P_1: bool,Pa: bool,X_1: int] :
      ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
       => ( hBOOL(Pa)
        <=> hBOOL(P_1) ) )
     => ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
         => hBOOL(Pa) )
      <=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
         => hBOOL(P_1) ) ) ) ).

tff(fact_1091_incr__mult__lemma,axiom,
    ! [K: int,Pa: fun_int_bool,D_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1))
     => ( ! [X_2: int] :
            ( hBOOL(hAPP_int_bool(Pa,X_2))
           => hBOOL(hAPP_int_bool(Pa,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_2),D_1))) )
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K))
         => ! [X_2: int] :
              ( hBOOL(hAPP_int_bool(Pa,X_2))
             => hBOOL(hAPP_int_bool(Pa,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K),D_1)))) ) ) ) ) ).

tff(fact_1092_int__induct,axiom,
    ! [I_2: int,Pa: fun_int_bool,K: int] :
      ( hBOOL(hAPP_int_bool(Pa,K))
     => ( ! [I: int] :
            ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K),I))
           => ( hBOOL(hAPP_int_bool(Pa,I))
             => hBOOL(hAPP_int_bool(Pa,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,I),one_one_int))) ) )
       => ( ! [I: int] :
              ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I),K))
             => ( hBOOL(hAPP_int_bool(Pa,I))
               => hBOOL(hAPP_int_bool(Pa,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,I),one_one_int))) ) )
         => hBOOL(hAPP_int_bool(Pa,I_2)) ) ) ) ).

tff(fact_1093_minusinfinity,axiom,
    ! [Pa: fun_int_bool,P1: fun_int_bool,D_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1))
     => ( ! [X_2: int,K_2: int] :
            ( hBOOL(hAPP_int_bool(P1,X_2))
          <=> hBOOL(hAPP_int_bool(P1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),D_1)))) )
       => ( ? [Z_3: int] :
            ! [X_2: int] :
              ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_2),Z_3))
             => ( hBOOL(hAPP_int_bool(Pa,X_2))
              <=> hBOOL(hAPP_int_bool(P1,X_2)) ) )
         => ( ? [X1: int] : hBOOL(hAPP_int_bool(P1,X1))
           => ? [X1: int] : hBOOL(hAPP_int_bool(Pa,X1)) ) ) ) ) ).

tff(fact_1094_plusinfinity,axiom,
    ! [Pa: fun_int_bool,P_1: fun_int_bool,D_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1))
     => ( ! [X_2: int,K_2: int] :
            ( hBOOL(hAPP_int_bool(P_1,X_2))
          <=> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),D_1)))) )
       => ( ? [Z_3: int] :
            ! [X_2: int] :
              ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_3),X_2))
             => ( hBOOL(hAPP_int_bool(Pa,X_2))
              <=> hBOOL(hAPP_int_bool(P_1,X_2)) ) )
         => ( ? [X1: int] : hBOOL(hAPP_int_bool(P_1,X1))
           => ? [X1: int] : hBOOL(hAPP_int_bool(Pa,X1)) ) ) ) ) ).

tff(fact_1095_tsub__def,axiom,
    ! [Y: int,X: int] :
      ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y),X))
       => ( hAPP_int_int(hAPP_int_fun_int_int(nat_tsub,X),Y) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X),Y) ) )
      & ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y),X))
       => ( hAPP_int_int(hAPP_int_fun_int_int(nat_tsub,X),Y) = zero_zero_int ) ) ) ).

tff(fact_1096_zero__zle__int,axiom,
    ! [N: nat] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_nat_int(semiri1621563631at_int,N))) ).

tff(fact_1097_int__less__0__conv,axiom,
    ! [K_1: nat] : ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,K_1)),zero_zero_int)) ).

tff(fact_1098_int__1,axiom,
    hAPP_nat_int(semiri1621563631at_int,one_one_nat) = one_one_int ).

tff(fact_1099_int__0,axiom,
    hAPP_nat_int(semiri1621563631at_int,zero_zero_nat) = zero_zero_int ).

tff(fact_1100_int__eq__0__conv,axiom,
    ! [Na: nat] :
      ( ( hAPP_nat_int(semiri1621563631at_int,Na) = zero_zero_int )
    <=> ( Na = zero_zero_nat ) ) ).

tff(fact_1101_zless__int,axiom,
    ! [M: nat,Na: nat] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,M)),hAPP_nat_int(semiri1621563631at_int,Na)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),Na)) ) ).

tff(fact_1102_zle__int,axiom,
    ! [M: nat,Na: nat] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(semiri1621563631at_int,M)),hAPP_nat_int(semiri1621563631at_int,Na)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),Na)) ) ).

tff(fact_1103_zadd__int,axiom,
    ! [M_1: nat,N: nat] : ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(semiri1621563631at_int,M_1)),hAPP_nat_int(semiri1621563631at_int,N)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),N)) ) ).

tff(fact_1104_zadd__int__left,axiom,
    ! [M_1: nat,N: nat,Z: int] : ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(semiri1621563631at_int,M_1)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(semiri1621563631at_int,N)),Z)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),N))),Z) ) ).

tff(fact_1105_zle__iff__zadd,axiom,
    ! [W_1: int,Z_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W_1),Z_2))
    <=> ? [N_1: nat] : ( Z_2 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,W_1),hAPP_nat_int(semiri1621563631at_int,N_1)) ) ) ).

tff(fact_1106_zmult__int,axiom,
    ! [M_1: nat,N: nat] : ( hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(semiri1621563631at_int,M_1)),hAPP_nat_int(semiri1621563631at_int,N)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),N)) ) ).

tff(fact_1107_int__mult,axiom,
    ! [M_1: nat,N: nat] : ( hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),N)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(semiri1621563631at_int,M_1)),hAPP_nat_int(semiri1621563631at_int,N)) ) ).

tff(fact_1108_Nat__Transfer_Otransfer__int__nat__functions_I2_J,axiom,
    ! [X: nat,Y: nat] : ( hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(semiri1621563631at_int,X)),hAPP_nat_int(semiri1621563631at_int,Y)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X),Y)) ) ).

tff(fact_1109_transfer__int__nat__relations_I2_J,axiom,
    ! [X_1: nat,Y_1: nat] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,X_1)),hAPP_nat_int(semiri1621563631at_int,Y_1)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_1)) ) ).

tff(fact_1110_transfer__int__nat__relations_I3_J,axiom,
    ! [X_1: nat,Y_1: nat] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(semiri1621563631at_int,X_1)),hAPP_nat_int(semiri1621563631at_int,Y_1)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_1)) ) ).

tff(fact_1111_Nat__Transfer_Otransfer__int__nat__functions_I3_J,axiom,
    ! [X: nat,Y: nat] : ( hAPP_int_int(hAPP_int_fun_int_int(nat_tsub,hAPP_nat_int(semiri1621563631at_int,X)),hAPP_nat_int(semiri1621563631at_int,Y)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,X),Y)) ) ).

tff(fact_1112_Nat__Transfer_Otransfer__int__nat__functions_I1_J,axiom,
    ! [X: nat,Y: nat] : ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(semiri1621563631at_int,X)),hAPP_nat_int(semiri1621563631at_int,Y)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X),Y)) ) ).

tff(fact_1113_Nat__Transfer_Otransfer__nat__int__set__functions_I1_J,axiom,
    ! [A: fun_nat_bool] : ( hAPP_f22106695ol_nat(finite_card_nat,A) = hAPP_f957591787ol_nat(finite_card_int,image_nat_int(semiri1621563631at_int,A)) ) ).

tff(fact_1114_transfer__nat__int__set__relations_I1_J,axiom,
    ! [A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
    <=> hBOOL(hAPP_f448129468l_bool(finite_finite_int,image_nat_int(semiri1621563631at_int,A))) ) ).

tff(fact_1115_transfer__int__nat__numerals_I2_J,axiom,
    one_one_int = hAPP_nat_int(semiri1621563631at_int,one_one_nat) ).

tff(fact_1116_transfer__int__nat__numerals_I1_J,axiom,
    zero_zero_int = hAPP_nat_int(semiri1621563631at_int,zero_zero_nat) ).

tff(fact_1117_Nat__Transfer_Otransfer__int__nat__set__functions_I5_J,axiom,
    ! [Pa: fun_int_bool] : ( collect_int(cOMBS_int_bool_bool(cOMBB_1652995168ol_int(fconj,hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int)),Pa)) = image_nat_int(semiri1621563631at_int,collect_nat(cOMBB_int_bool_nat(Pa,semiri1621563631at_int))) ) ).

tff(fact_1118_Nat__Transfer_Otransfer__nat__int__function__closures_I9_J,axiom,
    ! [Z: nat] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_nat_int(semiri1621563631at_int,Z))) ).

tff(fact_1119_transfer__int__nat__quantifiers_I2_J,axiom,
    ! [Pa: fun_int_bool] :
      ( ? [X_2: int] :
          ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2))
          & hBOOL(hAPP_int_bool(Pa,X_2)) )
    <=> ? [X_2: nat] : hBOOL(hAPP_int_bool(Pa,hAPP_nat_int(semiri1621563631at_int,X_2))) ) ).

tff(fact_1120_transfer__int__nat__quantifiers_I1_J,axiom,
    ! [Pa: fun_int_bool] :
      ( ! [X_2: int] :
          ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2))
         => hBOOL(hAPP_int_bool(Pa,X_2)) )
    <=> ! [X_2: nat] : hBOOL(hAPP_int_bool(Pa,hAPP_nat_int(semiri1621563631at_int,X_2))) ) ).

tff(fact_1121_int__le__0__conv,axiom,
    ! [Na: nat] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(semiri1621563631at_int,Na)),zero_zero_int))
    <=> ( Na = zero_zero_nat ) ) ).

tff(fact_1122_int__Suc0__eq__1,axiom,
    hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(suc,zero_zero_nat)) = one_one_int ).

tff(fact_1123_zless__iff__Suc__zadd,axiom,
    ! [W_1: int,Z_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),Z_2))
    <=> ? [N_1: nat] : ( Z_2 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,W_1),hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(suc,N_1))) ) ) ).

tff(fact_1124_int__Suc,axiom,
    ! [M_1: nat] : ( hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(suc,M_1)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_nat_int(semiri1621563631at_int,M_1)) ) ).

tff(fact_1125_zdiff__int,axiom,
    ! [N: nat,M_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M_1))
     => ( hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_nat_int(semiri1621563631at_int,M_1)),hAPP_nat_int(semiri1621563631at_int,N)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),N)) ) ) ).

tff(fact_1126_zero__less__int__conv,axiom,
    ! [Na: nat] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_nat_int(semiri1621563631at_int,Na)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na)) ) ).

tff(fact_1127_zmult__zless__mono2__lemma,axiom,
    ! [K_1: nat,I_1: int,J: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I_1),J))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_1))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(semiri1621563631at_int,K_1)),I_1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(semiri1621563631at_int,K_1)),J))) ) ) ).

tff(fact_1128_Nat__Transfer_Otransfer__nat__int__function__closures_I3_J,axiom,
    ! [Y: int,X: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(nat_tsub,X),Y))) ) ) ).

tff(fact_1129_zdiff__int__split,axiom,
    ! [Pa: fun_int_bool,X_1: nat,Y_1: nat] :
      ( hBOOL(hAPP_int_bool(Pa,hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,X_1),Y_1))))
    <=> ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_1),X_1))
         => hBOOL(hAPP_int_bool(Pa,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_nat_int(semiri1621563631at_int,X_1)),hAPP_nat_int(semiri1621563631at_int,Y_1)))) )
        & ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_1))
         => hBOOL(hAPP_int_bool(Pa,zero_zero_int)) ) ) ) ).

tff(fact_1130_tsub__eq,axiom,
    ! [Y: int,X: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y),X))
     => ( hAPP_int_int(hAPP_int_fun_int_int(nat_tsub,X),Y) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X),Y) ) ) ).

tff(fact_1131_zero__less__imp__eq__int,axiom,
    ! [K_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),K_1))
     => ? [N_1: nat] :
          ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
          & ( K_1 = hAPP_nat_int(semiri1621563631at_int,N_1) ) ) ) ).

tff(fact_1132_int__int__eq,axiom,
    ! [M: nat,Na: nat] :
      ( ( hAPP_nat_int(semiri1621563631at_int,M) = hAPP_nat_int(semiri1621563631at_int,Na) )
    <=> ( M = Na ) ) ).

tff(fact_1133_transfer__nat__int__set__relations_I4_J,axiom,
    ! [A: fun_nat_bool,B: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,A),B))
    <=> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,image_nat_int(semiri1621563631at_int,A)),image_nat_int(semiri1621563631at_int,B))) ) ).

tff(fact_1134_transfer__nat__int__set__relations_I5_J,axiom,
    ! [A: fun_nat_bool,B: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B))
    <=> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,image_nat_int(semiri1621563631at_int,A)),image_nat_int(semiri1621563631at_int,B))) ) ).

tff(fact_1135_Nat__Transfer_Otransfer__int__nat__set__functions_I2_J,axiom,
    bot_bot_fun_int_bool = image_nat_int(semiri1621563631at_int,bot_bot_fun_nat_bool) ).

tff(fact_1136_transfer__nat__int__set__relations_I3_J,axiom,
    ! [A: fun_nat_bool,B: fun_nat_bool] :
      ( ( A = B )
    <=> ( image_nat_int(semiri1621563631at_int,A) = image_nat_int(semiri1621563631at_int,B) ) ) ).

tff(fact_1137_transfer__nat__int__set__relations_I2_J,axiom,
    ! [X_1: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
    <=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_nat_int(semiri1621563631at_int,X_1)),image_nat_int(semiri1621563631at_int,A))) ) ).

tff(fact_1138_int__if__cong,axiom,
    ! [X_1: nat,Y_1: nat,Pa: bool] :
      ( ( hBOOL(Pa)
       => ( hAPP_nat_int(semiri1621563631at_int,X_1) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(if_nat(Pa),X_1),Y_1)) ) )
      & ( ~ hBOOL(Pa)
       => ( hAPP_nat_int(semiri1621563631at_int,Y_1) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(if_nat(Pa),X_1),Y_1)) ) ) ) ).

tff(fact_1139_transfer__int__nat__relations_I1_J,axiom,
    ! [X_1: nat,Y_1: nat] :
      ( ( hAPP_nat_int(semiri1621563631at_int,X_1) = hAPP_nat_int(semiri1621563631at_int,Y_1) )
    <=> ( X_1 = Y_1 ) ) ).

tff(fact_1140_nonneg__int__cases,axiom,
    ! [K_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K_1))
     => ~ ! [N_1: nat] : ( K_1 != hAPP_nat_int(semiri1621563631at_int,N_1) ) ) ).

tff(fact_1141_nonneg__eq__int,axiom,
    ! [Z: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z))
     => ~ ! [M_2: nat] : ( Z != hAPP_nat_int(semiri1621563631at_int,M_2) ) ) ).

tff(fact_1142_zero__le__imp__eq__int,axiom,
    ! [K_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K_1))
     => ? [N_1: nat] : ( K_1 = hAPP_nat_int(semiri1621563631at_int,N_1) ) ) ).

tff(fact_1143_int__diff__cases,axiom,
    ! [Z: int] :
      ~ ! [M_2: nat,N_1: nat] : ( Z != hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_nat_int(semiri1621563631at_int,M_2)),hAPP_nat_int(semiri1621563631at_int,N_1)) ) ).

tff(fact_1144_decr__lemma,axiom,
    ! [X: int,Z: int,D: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,abs_abs_int(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X),Z))),one_one_int)),D))),Z)) ) ).

tff(fact_1145_zabs__less__one__iff,axiom,
    ! [Z_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,abs_abs_int(Z_2)),one_one_int))
    <=> ( Z_2 = zero_zero_int ) ) ).

tff(fact_1146_neg__def,axiom,
    ! [Z_1: int] :
      ( hBOOL(hAPP_int_bool(nat_neg,Z_1))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_1),zero_zero_int)) ) ).

tff(fact_1147_not__neg__eq__ge__0,axiom,
    ! [X_1: int] :
      ( ~ hBOOL(hAPP_int_bool(nat_neg,X_1))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1)) ) ).

tff(fact_1148_not__neg__1,axiom,
    ~ hBOOL(hAPP_int_bool(nat_neg,one_one_int)) ).

tff(fact_1149_not__neg__0,axiom,
    ~ hBOOL(hAPP_int_bool(nat_neg,zero_zero_int)) ).

tff(fact_1150_abs__zmult__eq__1,axiom,
    ! [M_1: int,N: int] :
      ( ( abs_abs_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,M_1),N)) = one_one_int )
     => ( abs_abs_int(M_1) = one_one_int ) ) ).

tff(fact_1151_not__neg__int,axiom,
    ! [N: nat] : ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_nat_int(semiri1621563631at_int,N))) ).

tff(fact_1152_abs__int__eq,axiom,
    ! [M_1: nat] : ( abs_abs_int(hAPP_nat_int(semiri1621563631at_int,M_1)) = hAPP_nat_int(semiri1621563631at_int,M_1) ) ).

tff(fact_1153_neg__imp__number__of__eq__0,axiom,
    ! [V: int] :
      ( hBOOL(hAPP_int_bool(nat_neg,number_number_of_int(V)))
     => ( number_number_of_nat(V) = zero_zero_nat ) ) ).

tff(fact_1154_eq__nat__number__of,axiom,
    ! [V_3: int,V_2: int] :
      ( ( number_number_of_nat(V_3) = number_number_of_nat(V_2) )
    <=> ( ( hBOOL(hAPP_int_bool(nat_neg,number_number_of_int(V_3)))
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,number_number_of_int(V_2)),zero_zero_int)) )
        & ( ~ hBOOL(hAPP_int_bool(nat_neg,number_number_of_int(V_3)))
         => ( ( hBOOL(hAPP_int_bool(nat_neg,number_number_of_int(V_2)))
             => ( number_number_of_int(V_3) = zero_zero_int ) )
            & ( ~ hBOOL(hAPP_int_bool(nat_neg,number_number_of_int(V_2)))
             => ( V_3 = V_2 ) ) ) ) ) ) ).

tff(fact_1155_nat__number__of__add__left,axiom,
    ! [V_1: int,K_1: nat,V: int] :
      ( ( hBOOL(hAPP_int_bool(nat_neg,number_number_of_int(V)))
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,number_number_of_nat(V)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,number_number_of_nat(V_1)),K_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,number_number_of_nat(V_1)),K_1) ) )
      & ( ~ hBOOL(hAPP_int_bool(nat_neg,number_number_of_int(V)))
       => ( ( hBOOL(hAPP_int_bool(nat_neg,number_number_of_int(V_1)))
           => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,number_number_of_nat(V)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,number_number_of_nat(V_1)),K_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,number_number_of_nat(V)),K_1) ) )
          & ( ~ hBOOL(hAPP_int_bool(nat_neg,number_number_of_int(V_1)))
           => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,number_number_of_nat(V)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,number_number_of_nat(V_1)),K_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,number_number_of_nat(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V),V_1))),K_1) ) ) ) ) ) ).

tff(fact_1156_int__nat__number__of,axiom,
    ! [V: int] :
      ( ( hBOOL(hAPP_int_bool(nat_neg,number_number_of_int(V)))
       => ( hAPP_nat_int(semiri1621563631at_int,number_number_of_nat(V)) = zero_zero_int ) )
      & ( ~ hBOOL(hAPP_int_bool(nat_neg,number_number_of_int(V)))
       => ( hAPP_nat_int(semiri1621563631at_int,number_number_of_nat(V)) = number_number_of_int(V) ) ) ) ).

tff(fact_1157_incr__lemma,axiom,
    ! [Z: int,X: int,D: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,abs_abs_int(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X),Z))),one_one_int)),D)))) ) ).

tff(fact_1158_int__val__lemma,axiom,
    ! [K: int,F: fun_nat_int,Na: nat] :
      ( ! [I: nat] :
          ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),Na))
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,abs_abs_int(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_nat_int(F,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),one_one_nat))),hAPP_nat_int(F,I)))),one_one_int)) )
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(F,zero_zero_nat)),K))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K),hAPP_nat_int(F,Na)))
         => ? [I: nat] :
              ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),Na))
              & ( hAPP_nat_int(F,I) = K ) ) ) ) ) ).

tff(fact_1159_nat0__intermed__int__val,axiom,
    ! [K: int,F: fun_nat_int,Na: nat] :
      ( ! [I: nat] :
          ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),Na))
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,abs_abs_int(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_nat_int(F,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),one_one_nat))),hAPP_nat_int(F,I)))),one_one_int)) )
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(F,zero_zero_nat)),K))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K),hAPP_nat_int(F,Na)))
         => ? [I: nat] :
              ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),Na))
              & ( hAPP_nat_int(F,I) = K ) ) ) ) ) ).

tff(fact_1160_nat__intermed__int__val,axiom,
    ! [K: int,F: fun_nat_int,Na: nat,M: nat] :
      ( ! [I: nat] :
          ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),I))
            & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),Na)) )
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,abs_abs_int(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_nat_int(F,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),one_one_nat))),hAPP_nat_int(F,I)))),one_one_int)) )
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),Na))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(F,M)),K))
         => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K),hAPP_nat_int(F,Na)))
           => ? [I: nat] :
                ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),I))
                & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),Na))
                & ( hAPP_nat_int(F,I) = K ) ) ) ) ) ) ).

tff(fact_1161_Suc__nat__number__of__add,axiom,
    ! [N: nat,V: int] :
      ( ( hBOOL(hAPP_int_bool(nat_neg,number_number_of_int(V)))
       => ( hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,number_number_of_nat(V)),N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,one_one_nat),N) ) )
      & ( ~ hBOOL(hAPP_int_bool(nat_neg,number_number_of_int(V)))
       => ( hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,number_number_of_nat(V)),N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,number_number_of_nat(succ(V))),N) ) ) ) ).

tff(fact_1162_succ__def,axiom,
    ! [K_1: int] : ( succ(K_1) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,K_1),one_one_int) ) ).

tff(fact_1163_Suc__nat__number__of,axiom,
    ! [V: int] :
      ( ( hBOOL(hAPP_int_bool(nat_neg,number_number_of_int(V)))
       => ( hAPP_nat_nat(suc,number_number_of_nat(V)) = one_one_nat ) )
      & ( ~ hBOOL(hAPP_int_bool(nat_neg,number_number_of_int(V)))
       => ( hAPP_nat_nat(suc,number_number_of_nat(V)) = number_number_of_nat(succ(V)) ) ) ) ).

tff(fact_1164_nat__number__of__Bit1,axiom,
    ! [W: int] :
      ( ( hBOOL(hAPP_int_bool(nat_neg,number_number_of_int(W)))
       => ( number_number_of_nat(bit1(W)) = zero_zero_nat ) )
      & ( ~ hBOOL(hAPP_int_bool(nat_neg,number_number_of_int(W)))
       => ( number_number_of_nat(bit1(W)) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,number_number_of_nat(W)),number_number_of_nat(W))) ) ) ) ).

tff(fact_1165_nat__1__add__number__of,axiom,
    ! [V: int] :
      ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V),pls))
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,one_one_nat),number_number_of_nat(V)) = one_one_nat ) )
      & ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V),pls))
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,one_one_nat),number_number_of_nat(V)) = number_number_of_nat(succ(V)) ) ) ) ).

tff(fact_1166_succ__Pls,axiom,
    succ(pls) = bit1(pls) ).

tff(fact_1167_neg__number__of__Bit1,axiom,
    ! [W_1: int] :
      ( hBOOL(hAPP_int_bool(nat_neg,number_number_of_int(bit1(W_1))))
    <=> hBOOL(hAPP_int_bool(nat_neg,number_number_of_int(W_1))) ) ).

tff(fact_1168_not__neg__number__of__Pls,axiom,
    ~ hBOOL(hAPP_int_bool(nat_neg,number_number_of_int(pls))) ).

tff(fact_1169_transfer__int__nat__numerals_I4_J,axiom,
    number_number_of_int(bit1(bit1(pls))) = hAPP_nat_int(semiri1621563631at_int,number_number_of_nat(bit1(bit1(pls)))) ).

tff(fact_1170_numeral__1__eq__Suc__0,axiom,
    number_number_of_nat(bit1(pls)) = hAPP_nat_nat(suc,zero_zero_nat) ).

tff(fact_1171_numeral__3__eq__3,axiom,
    number_number_of_nat(bit1(bit1(pls))) = hAPP_nat_nat(suc,hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))) ).

tff(fact_1172_Suc3__eq__add__3,axiom,
    ! [N: nat] : ( hAPP_nat_nat(suc,hAPP_nat_nat(suc,hAPP_nat_nat(suc,N))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,number_number_of_nat(bit1(bit1(pls)))),N) ) ).

tff(fact_1173_Numeral1__eq1__nat,axiom,
    one_one_nat = number_number_of_nat(bit1(pls)) ).

tff(fact_1174_nat__numeral__1__eq__1,axiom,
    number_number_of_nat(bit1(pls)) = one_one_nat ).

tff(fact_1175_one__is__num__one,axiom,
    one_one_int = number_number_of_int(bit1(pls)) ).

tff(fact_1176_zero__is__num__zero,axiom,
    zero_zero_int = number_number_of_int(pls) ).

tff(fact_1177_Bit1__def,axiom,
    ! [K_1: int] : ( bit1(K_1) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),K_1)),K_1) ) ).

tff(fact_1178_nat__number__of__Pls,axiom,
    number_number_of_nat(pls) = zero_zero_nat ).

tff(fact_1179_semiring__norm_I113_J,axiom,
    zero_zero_nat = number_number_of_nat(pls) ).

tff(fact_1180_Pls__def,axiom,
    pls = zero_zero_int ).

tff(fact_1181_rel__simps_I51_J,axiom,
    ! [K: int,L: int] :
      ( ( bit1(K) = bit1(L) )
    <=> ( K = L ) ) ).

tff(fact_1182_rel__simps_I46_J,axiom,
    ! [K_1: int] : ( bit1(K_1) != pls ) ).

tff(fact_1183_rel__simps_I39_J,axiom,
    ! [L_1: int] : ( pls != bit1(L_1) ) ).

tff(fact_1184_diff__bin__simps_I1_J,axiom,
    ! [K_1: int] : ( hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,K_1),pls) = K_1 ) ).

tff(fact_1185_add__Pls,axiom,
    ! [K_1: int] : ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,pls),K_1) = K_1 ) ).

tff(fact_1186_add__Pls__right,axiom,
    ! [K_1: int] : ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,K_1),pls) = K_1 ) ).

tff(fact_1187_mult__Pls,axiom,
    ! [W: int] : ( hAPP_int_int(hAPP_int_fun_int_int(times_times_int,pls),W) = pls ) ).

tff(fact_1188_Nat__Transfer_Otransfer__nat__int__function__closures_I8_J,axiom,
    hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),number_number_of_int(bit1(bit1(pls))))) ).

tff(fact_1189_rel__simps_I19_J,axiom,
    hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),pls)) ).

tff(fact_1190_rel__simps_I22_J,axiom,
    ! [K: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),bit1(K)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),K)) ) ).

tff(fact_1191_rel__simps_I34_J,axiom,
    ! [K: int,L: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,bit1(K)),bit1(L)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K),L)) ) ).

tff(fact_1192_less__eq__int__code_I16_J,axiom,
    ! [K1: int,K2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,bit1(K1)),bit1(K2)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K1),K2)) ) ).

tff(fact_1193_rel__simps_I2_J,axiom,
    ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),pls)) ).

tff(fact_1194_rel__simps_I12_J,axiom,
    ! [K: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,bit1(K)),pls))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),pls)) ) ).

tff(fact_1195_rel__simps_I17_J,axiom,
    ! [K: int,L: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,bit1(K)),bit1(L)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),L)) ) ).

tff(fact_1196_less__int__code_I16_J,axiom,
    ! [K1: int,K2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,bit1(K1)),bit1(K2)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K1),K2)) ) ).

tff(fact_1197_rel__simps_I29_J,axiom,
    ! [K: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,bit1(K)),pls))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),pls)) ) ).

tff(fact_1198_rel__simps_I5_J,axiom,
    ! [K: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),bit1(K)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),K)) ) ).

%----Helper facts (103)
tff(help_fNot_1_1_U,axiom,
    ! [P: bool] :
      ( ~ hBOOL(hAPP_bool_bool(fNot,P))
      | ~ hBOOL(P) ) ).

tff(help_fNot_2_1_U,axiom,
    ! [P: bool] :
      ( hBOOL(P)
      | hBOOL(hAPP_bool_bool(fNot,P)) ) ).

tff(help_fconj_1_1_U,axiom,
    ! [Q: bool,P: bool] :
      ( ~ hBOOL(P)
      | ~ hBOOL(Q)
      | hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fconj,P),Q)) ) ).

tff(help_fconj_2_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fconj,P),Q))
      | hBOOL(P) ) ).

tff(help_fconj_3_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fconj,P),Q))
      | hBOOL(Q) ) ).

tff(help_fdisj_1_1_U,axiom,
    ! [Q: bool,P: bool] :
      ( ~ hBOOL(P)
      | hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fdisj,P),Q)) ) ).

tff(help_fdisj_2_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ hBOOL(Q)
      | hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fdisj,P),Q)) ) ).

tff(help_fdisj_3_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fdisj,P),Q))
      | hBOOL(P)
      | hBOOL(Q) ) ).

tff(help_fFalse_1_1_U,axiom,
    ~ hBOOL(fFalse) ).

tff(help_fFalse_1_1_T,axiom,
    ! [P: bool] :
      ( ( P = fTrue )
      | ( P = fFalse ) ) ).

tff(help_fimplies_1_1_U,axiom,
    ! [Q: bool,P: bool] :
      ( hBOOL(P)
      | hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fimplies,P),Q)) ) ).

tff(help_fimplies_2_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ hBOOL(Q)
      | hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fimplies,P),Q)) ) ).

tff(help_fimplies_3_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fimplies,P),Q))
      | ~ hBOOL(P)
      | hBOOL(Q) ) ).

tff(help_fequal_1_1_fequal_000t__a_T,axiom,
    ! [X: x_a,Y: x_a] :
      ( ~ hBOOL(hAPP_a_bool(hAPP_a_fun_a_bool(fequal_a,X),Y))
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_000t__a_T,axiom,
    ! [X: x_a,Y: x_a] :
      ( ( X != Y )
      | hBOOL(hAPP_a_bool(hAPP_a_fun_a_bool(fequal_a,X),Y)) ) ).

tff(help_If_1_1_If_000tc__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(if_nat(fTrue),X),Y) = X ) ).

tff(help_If_2_1_If_000tc__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(if_nat(fFalse),X),Y) = Y ) ).

tff(help_If_3_1_If_000tc__Nat__Onat_T,axiom,
    ! [P: bool] :
      ( ( P = fTrue )
      | ( P = fFalse ) ) ).

tff(help_COMBI_1_1_COMBI_000tc__Nat__Onat_U,axiom,
    ! [P: nat] : ( hAPP_nat_nat(cOMBI_nat,P) = P ) ).

tff(help_fequal_1_1_fequal_000tc__Int__Oint_T,axiom,
    ! [X: int,Y: int] :
      ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(fequal_int,X),Y))
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_000tc__Int__Oint_T,axiom,
    ! [X: int,Y: int] :
      ( ( X != Y )
      | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(fequal_int,X),Y)) ) ).

tff(help_fequal_1_1_fequal_000tc__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] :
      ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(fequal_nat,X),Y))
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_000tc__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
      | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(fequal_nat,X),Y)) ) ).

tff(help_fequal_1_1_fequal_000tc__Com__Opname_T,axiom,
    ! [X: pname,Y: pname] :
      ( ~ hBOOL(hAPP_pname_bool(hAPP_p61793385e_bool(fequal_pname,X),Y))
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_000tc__Com__Opname_T,axiom,
    ! [X: pname,Y: pname] :
      ( ( X != Y )
      | hBOOL(hAPP_pname_bool(hAPP_p61793385e_bool(fequal_pname,X),Y)) ) ).

tff(help_COMBK_1_1_COMBK_000tc__HOL__Obool_000t__a_U,axiom,
    ! [P: bool,Q: x_a] : ( hAPP_a_bool(cOMBK_bool_a(P),Q) = P ) ).

tff(help_COMBK_1_1_COMBK_000t__a_000tc__Com__Opname_U,axiom,
    ! [P: x_a,Q: pname] : ( hAPP_pname_a(cOMBK_a_pname(P),Q) = P ) ).

tff(help_COMBC_1_1_COMBC_000t__a_000t__a_000tc__HOL__Obool_U,axiom,
    ! [P: fun_a_fun_a_bool,Q: x_a,R: x_a] : ( hAPP_a_bool(hAPP_a_fun_a_bool(cOMBC_a_a_bool(P),Q),R) = hAPP_a_bool(hAPP_a_fun_a_bool(P,R),Q) ) ).

tff(help_COMBK_1_1_COMBK_000tc__Int__Oint_000tc__Nat__Onat_U,axiom,
    ! [P: int,Q: nat] : ( hAPP_nat_int(cOMBK_int_nat(P),Q) = P ) ).

tff(help_COMBK_1_1_COMBK_000tc__HOL__Obool_000tc__Int__Oint_U,axiom,
    ! [P: bool,Q: int] : ( hAPP_int_bool(cOMBK_bool_int(P),Q) = P ) ).

tff(help_COMBK_1_1_COMBK_000tc__HOL__Obool_000tc__Nat__Onat_U,axiom,
    ! [P: bool,Q: nat] : ( hAPP_nat_bool(cOMBK_bool_nat(P),Q) = P ) ).

tff(help_fequal_1_1_fequal_000tc__fun_It__a_Mtc__HOL__Obool_J_T,axiom,
    ! [X: fun_a_bool,Y: fun_a_bool] :
      ( ~ hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(fequal_fun_a_bool,X),Y))
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_000tc__fun_It__a_Mtc__HOL__Obool_J_T,axiom,
    ! [X: fun_a_bool,Y: fun_a_bool] :
      ( ( X != Y )
      | hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(fequal_fun_a_bool,X),Y)) ) ).

tff(help_COMBC_1_1_COMBC_000tc__Int__Oint_000t__a_000tc__HOL__Obool_U,axiom,
    ! [P: fun_int_fun_a_bool,Q: x_a,R: int] : ( hAPP_int_bool(hAPP_a_fun_int_bool(cOMBC_int_a_bool(P),Q),R) = hAPP_a_bool(hAPP_int_fun_a_bool(P,R),Q) ) ).

tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__HOL__Obool_000t__a_U,axiom,
    ! [P: fun_bool_bool,Q: fun_a_bool,R: x_a] : ( hAPP_a_bool(cOMBB_bool_bool_a(P,Q),R) = hAPP_bool_bool(P,hAPP_a_bool(Q,R)) ) ).

tff(help_COMBS_1_1_COMBS_000t__a_000tc__HOL__Obool_000tc__HOL__Obool_U,axiom,
    ! [P: fun_a_fun_bool_bool,Q: fun_a_bool,R: x_a] : ( hAPP_a_bool(cOMBS_a_bool_bool(P,Q),R) = hAPP_bool_bool(hAPP_a_fun_bool_bool(P,R),hAPP_a_bool(Q,R)) ) ).

tff(help_COMBC_1_1_COMBC_000tc__Com__Opname_000t__a_000tc__HOL__Obool_U,axiom,
    ! [P: fun_pname_fun_a_bool,Q: x_a,R: pname] : ( hAPP_pname_bool(hAPP_a93125764e_bool(cOMBC_pname_a_bool(P),Q),R) = hAPP_a_bool(hAPP_p1534023578a_bool(P,R),Q) ) ).

tff(help_fequal_1_1_fequal_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_T,axiom,
    ! [X: fun_int_bool,Y: fun_int_bool] :
      ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(fequal_fun_int_bool,X),Y))
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_T,axiom,
    ! [X: fun_int_bool,Y: fun_int_bool] :
      ( ( X != Y )
      | hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(fequal_fun_int_bool,X),Y)) ) ).

tff(help_fequal_1_1_fequal_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_T,axiom,
    ! [X: fun_nat_bool,Y: fun_nat_bool] :
      ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(fequal_fun_nat_bool,X),Y))
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_T,axiom,
    ! [X: fun_nat_bool,Y: fun_nat_bool] :
      ( ( X != Y )
      | hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(fequal_fun_nat_bool,X),Y)) ) ).

tff(help_fequal_1_1_fequal_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_T,axiom,
    ! [X: fun_pname_bool,Y: fun_pname_bool] :
      ( ~ hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(fequal533582459e_bool,X),Y))
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_T,axiom,
    ! [X: fun_pname_bool,Y: fun_pname_bool] :
      ( ( X != Y )
      | hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(fequal533582459e_bool,X),Y)) ) ).

tff(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__HOL__Obool_000tc__Nat__Onat_U,axiom,
    ! [P: fun_int_bool,Q: fun_nat_int,R: nat] : ( hAPP_nat_bool(cOMBB_int_bool_nat(P,Q),R) = hAPP_int_bool(P,hAPP_nat_int(Q,R)) ) ).

tff(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__Int__Oint_000tc__HOL__Obool_U,axiom,
    ! [P: fun_int_fun_int_bool,Q: int,R: int] : ( hAPP_int_bool(hAPP_i1948725293t_bool(cOMBC_int_int_bool(P),Q),R) = hAPP_int_bool(hAPP_i1948725293t_bool(P,R),Q) ) ).

tff(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__Int__Oint_000tc__HOL__Obool_U,axiom,
    ! [P: fun_nat_fun_int_bool,Q: int,R: nat] : ( hAPP_nat_bool(hAPP_i418383825t_bool(cOMBC_nat_int_bool(P),Q),R) = hAPP_int_bool(hAPP_n1082236369t_bool(P,R),Q) ) ).

tff(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__Nat__Onat_000tc__HOL__Obool_U,axiom,
    ! [P: fun_nat_fun_nat_bool,Q: nat,R: nat] : ( hAPP_nat_bool(hAPP_n1699378549t_bool(cOMBC_nat_nat_bool(P),Q),R) = hAPP_nat_bool(hAPP_n1699378549t_bool(P,R),Q) ) ).

tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__HOL__Obool_000tc__Int__Oint_U,axiom,
    ! [P: fun_bool_bool,Q: fun_int_bool,R: int] : ( hAPP_int_bool(cOMBB_bool_bool_int(P,Q),R) = hAPP_bool_bool(P,hAPP_int_bool(Q,R)) ) ).

tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__HOL__Obool_000tc__Nat__Onat_U,axiom,
    ! [P: fun_bool_bool,Q: fun_nat_bool,R: nat] : ( hAPP_nat_bool(cOMBB_bool_bool_nat(P,Q),R) = hAPP_bool_bool(P,hAPP_nat_bool(Q,R)) ) ).

tff(help_COMBS_1_1_COMBS_000tc__Int__Oint_000tc__HOL__Obool_000tc__HOL__Obool_U,axiom,
    ! [P: fun_in531499254l_bool,Q: fun_int_bool,R: int] : ( hAPP_int_bool(cOMBS_int_bool_bool(P,Q),R) = hAPP_bool_bool(hAPP_i68813070l_bool(P,R),hAPP_int_bool(Q,R)) ) ).

tff(help_COMBS_1_1_COMBS_000tc__Nat__Onat_000tc__HOL__Obool_000tc__HOL__Obool_U,axiom,
    ! [P: fun_na1469252690l_bool,Q: fun_nat_bool,R: nat] : ( hAPP_nat_bool(cOMBS_nat_bool_bool(P,Q),R) = hAPP_bool_bool(hAPP_n1006566506l_bool(P,R),hAPP_nat_bool(Q,R)) ) ).

tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__HOL__Obool_000tc__Com__Opname_U,axiom,
    ! [P: fun_bool_bool,Q: fun_pname_bool,R: pname] : ( hAPP_pname_bool(cOMBB_647938656_pname(P,Q),R) = hAPP_bool_bool(P,hAPP_pname_bool(Q,R)) ) ).

tff(help_COMBS_1_1_COMBS_000tc__Com__Opname_000tc__HOL__Obool_000tc__HOL__Obool_U,axiom,
    ! [P: fun_pn250273176l_bool,Q: fun_pname_bool,R: pname] : ( hAPP_pname_bool(cOMBS_568398431l_bool(P,Q),R) = hAPP_bool_bool(hAPP_p393069232l_bool(P,R),hAPP_pname_bool(Q,R)) ) ).

tff(help_COMBC_1_1_COMBC_000tc__Com__Opname_000tc__Com__Opname_000tc__HOL__Obool_U,axiom,
    ! [P: fun_pn800050071e_bool,Q: pname,R: pname] : ( hAPP_pname_bool(hAPP_p61793385e_bool(cOMBC_1149511130e_bool(P),Q),R) = hAPP_pname_bool(hAPP_p61793385e_bool(P,R),Q) ) ).

tff(help_COMBB_1_1_COMBB_000t__a_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__Int__Oint,axiom,
    ! [P: fun_a_fun_a_bool,Q: fun_int_a,R: int] : ( hAPP_int_fun_a_bool(cOMBB_375279840ol_int(P,Q),R) = hAPP_a_fun_a_bool(P,hAPP_int_a(Q,R)) ) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__HOL__Obool_000t__,axiom,
    ! [P: fun_fun_a_bool_bool,Q: fun_a_fun_a_bool,R: x_a] : ( hAPP_a_bool(cOMBB_1273333599bool_a(P,Q),R) = hAPP_fun_a_bool_bool(P,hAPP_a_fun_a_bool(Q,R)) ) ).

tff(help_COMBC_1_1_COMBC_000t__a_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__HOL__Oboo,axiom,
    ! [P: fun_a_1255737515l_bool,Q: fun_a_bool,R: x_a] : ( hAPP_a_bool(hAPP_f2050579477a_bool(cOMBC_1355376034l_bool(P),Q),R) = hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(P,R),Q) ) ).

tff(help_COMBB_1_1_COMBB_000t__a_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__Com__Opna,axiom,
    ! [P: fun_a_fun_a_bool,Q: fun_pname_a,R: pname] : ( hAPP_p1534023578a_bool(cOMBB_1897541054_pname(P,Q),R) = hAPP_a_fun_a_bool(P,hAPP_pname_a(Q,R)) ) ).

tff(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__,axiom,
    ! [P: fun_in1106722302l_bool,Q: fun_a_bool,R: int] : ( hAPP_int_bool(hAPP_f1812805200t_bool(cOMBC_1753430397l_bool(P),Q),R) = hAPP_fun_a_bool_bool(hAPP_i888906320l_bool(P,R),Q) ) ).

tff(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__,axiom,
    ! [P: fun_na1632405922l_bool,Q: fun_a_bool,R: nat] : ( hAPP_nat_bool(hAPP_f282463732t_bool(cOMBC_1928494297l_bool(P),Q),R) = hAPP_fun_a_bool_bool(hAPP_n1414589940l_bool(P,R),Q) ) ).

tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Oboo,axiom,
    ! [P: fun_bo1549164019l_bool,Q: fun_a_bool,R: x_a] : ( hAPP_a_fun_bool_bool(cOMBB_1972296269bool_a(P,Q),R) = hAPP_b589554111l_bool(P,hAPP_a_bool(Q,R)) ) ).

tff(help_COMBS_1_1_COMBS_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__HOL__Obool_000tc_,axiom,
    ! [P: fun_fu554186387l_bool,Q: fun_fun_a_bool_bool,R: fun_a_bool] : ( hAPP_fun_a_bool_bool(cOMBS_1035972772l_bool(P,Q),R) = hAPP_bool_bool(hAPP_f198738859l_bool(P,R),hAPP_fun_a_bool_bool(Q,R)) ) ).

tff(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_,axiom,
    ! [P: fun_int_fun_int_bool,Q: fun_nat_int,R: nat] : ( hAPP_n1082236369t_bool(cOMBB_1628441366ol_nat(P,Q),R) = hAPP_i1948725293t_bool(P,hAPP_nat_int(Q,R)) ) ).

tff(help_COMBB_1_1_COMBB_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_,axiom,
    ! [P: fun_nat_fun_nat_bool,Q: fun_nat_nat,R: nat] : ( hAPP_n1699378549t_bool(cOMBB_800536526ol_nat(P,Q),R) = hAPP_n1699378549t_bool(P,hAPP_nat_nat(Q,R)) ) ).

tff(help_COMBC_1_1_COMBC_000t__a_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__fun_It__a,axiom,
    ! [P: fun_a_1752562919a_bool,Q: fun_a_bool,R: x_a] : ( hAPP_a_fun_a_bool(hAPP_f1038053735a_bool(cOMBC_394484100a_bool(P),Q),R) = hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(P,R),Q) ) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__HOL__Obo,axiom,
    ! [P: fun_fu819804446l_bool,Q: fun_int_fun_int_bool,R: int] : ( hAPP_int_bool(cOMBB_1418110531ol_int(P,Q),R) = hAPP_f448129468l_bool(P,hAPP_i1948725293t_bool(Q,R)) ) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__HOL__Obo,axiom,
    ! [P: fun_fu425979586l_bool,Q: fun_nat_fun_nat_bool,R: nat] : ( hAPP_nat_bool(cOMBB_955900739ol_nat(P,Q),R) = hAPP_f54304608l_bool(P,hAPP_n1699378549t_bool(Q,R)) ) ).

tff(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_,axiom,
    ! [P: fun_in938817141l_bool,Q: fun_int_bool,R: int] : ( hAPP_int_bool(hAPP_f1805168059t_bool(cOMBC_94739984l_bool(P),Q),R) = hAPP_f448129468l_bool(hAPP_i2112223885l_bool(P,R),Q) ) ).

tff(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_,axiom,
    ! [P: fun_in791894553l_bool,Q: fun_nat_bool,R: int] : ( hAPP_int_bool(hAPP_f183368031t_bool(cOMBC_1848398772l_bool(P),Q),R) = hAPP_f54304608l_bool(hAPP_i1718399025l_bool(P,R),Q) ) ).

tff(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_,axiom,
    ! [P: fun_na1583160273l_bool,Q: fun_int_bool,R: nat] : ( hAPP_nat_bool(hAPP_f274826591t_bool(cOMBC_620423604l_bool(P),Q),R) = hAPP_f448129468l_bool(hAPP_n609083369l_bool(P,R),Q) ) ).

tff(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_,axiom,
    ! [P: fun_na1436237685l_bool,Q: fun_nat_bool,R: nat] : ( hAPP_nat_bool(hAPP_f800510211t_bool(cOMBC_226598744l_bool(P),Q),R) = hAPP_f54304608l_bool(hAPP_n215258509l_bool(P,R),Q) ) ).

tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Oboo_059,axiom,
    ! [P: fun_bo1549164019l_bool,Q: fun_int_bool,R: int] : ( hAPP_i68813070l_bool(cOMBB_1652995168ol_int(P,Q),R) = hAPP_b589554111l_bool(P,hAPP_int_bool(Q,R)) ) ).

tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Oboo_060,axiom,
    ! [P: fun_bo1549164019l_bool,Q: fun_nat_bool,R: nat] : ( hAPP_n1006566506l_bool(cOMBB_1015721476ol_nat(P,Q),R) = hAPP_b589554111l_bool(P,hAPP_nat_bool(Q,R)) ) ).

tff(help_COMBS_1_1_COMBS_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__HOL__Obo,axiom,
    ! [P: fun_fu989146164l_bool,Q: fun_fu819804446l_bool,R: fun_int_bool] : ( hAPP_f448129468l_bool(cOMBS_1912792529l_bool(P,Q),R) = hAPP_bool_bool(hAPP_f1721100032l_bool(P,R),hAPP_f448129468l_bool(Q,R)) ) ).

tff(help_COMBS_1_1_COMBS_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__HOL__Obo,axiom,
    ! [P: fun_fu1016514960l_bool,Q: fun_fu425979586l_bool,R: fun_nat_bool] : ( hAPP_f54304608l_bool(cOMBS_1187019125l_bool(P,Q),R) = hAPP_bool_bool(hAPP_f1748468828l_bool(P,R),hAPP_f54304608l_bool(Q,R)) ) ).

tff(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__fun_Itc__Com__Opname_Mtc__HOL__Oboo,axiom,
    ! [P: fun_in711834963l_bool,Q: fun_pname_bool,R: int] : ( hAPP_int_bool(hAPP_f449020825t_bool(cOMBC_316919022l_bool(P),Q),R) = hAPP_f1664156314l_bool(hAPP_i1587699435l_bool(P,R),Q) ) ).

tff(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__fun_Itc__Com__Opname_Mtc__HOL__Oboo,axiom,
    ! [P: fun_na2122364079l_bool,Q: fun_pname_bool,R: nat] : ( hAPP_nat_bool(hAPP_f1066163005t_bool(cOMBC_386238098l_bool(P),Q),R) = hAPP_f1664156314l_bool(hAPP_n850744903l_bool(P,R),Q) ) ).

tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Oboo_061,axiom,
    ! [P: fun_bo1549164019l_bool,Q: fun_pname_bool,R: pname] : ( hAPP_p393069232l_bool(cOMBB_675860798_pname(P,Q),R) = hAPP_b589554111l_bool(P,hAPP_pname_bool(Q,R)) ) ).

tff(help_COMBS_1_1_COMBS_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_000tc__HOL__O,axiom,
    ! [P: fun_fu31783638l_bool,Q: fun_fu1430349052l_bool,R: fun_pname_bool] : ( hAPP_f1664156314l_bool(cOMBS_350070575l_bool(P,Q),R) = hAPP_bool_bool(hAPP_f1476298914l_bool(P,R),hAPP_f1664156314l_bool(Q,R)) ) ).

tff(help_COMBC_1_1_COMBC_000tc__Com__Opname_000tc__fun_Itc__Com__Opname_Mtc__HOL__Ob,axiom,
    ! [P: fun_pn422929397l_bool,Q: fun_pname_bool,R: pname] : ( hAPP_pname_bool(hAPP_f759274231e_bool(cOMBC_1058051404l_bool(P),Q),R) = hAPP_f1664156314l_bool(hAPP_p338031245l_bool(P,R),Q) ) ).

tff(help_COMBC_1_1_COMBC_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__fun_It__a_Mtc__HO,axiom,
    ! [P: fun_fu1471507361l_bool,Q: fun_a_bool,R: fun_a_bool] : ( hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(cOMBC_1732670874l_bool(P),Q),R) = hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(P,R),Q) ) ).

tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Oboo_062,axiom,
    ! [P: fun_bo1549164019l_bool,Q: fun_fun_a_bool_bool,R: fun_a_bool] : ( hAPP_f198738859l_bool(cOMBB_338059395a_bool(P,Q),R) = hAPP_b589554111l_bool(P,hAPP_fun_a_bool_bool(Q,R)) ) ).

tff(help_COMBB_1_1_COMBB_000tc__Nat__Onat_000tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HO,axiom,
    ! [P: fun_na1436237685l_bool,Q: fun_nat_nat,R: nat] : ( hAPP_n215258509l_bool(cOMBB_1146692694ol_nat(P,Q),R) = hAPP_n215258509l_bool(P,hAPP_nat_nat(Q,R)) ) ).

tff(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool__063,axiom,
    ! [P: fun_in719570932t_bool,Q: fun_int_bool,R: int] : ( hAPP_i1948725293t_bool(hAPP_f1048215610t_bool(cOMBC_1683390479t_bool(P),Q),R) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(P,R),Q) ) ).

tff(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool__064,axiom,
    ! [P: fun_na35521688t_bool,Q: fun_nat_bool,R: nat] : ( hAPP_n1699378549t_bool(hAPP_f1570313510t_bool(cOMBC_178881787t_bool(P),Q),R) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(P,R),Q) ) ).

tff(help_COMBC_1_1_COMBC_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__fun_Itc_,axiom,
    ! [P: fun_fu1151772339l_bool,Q: fun_int_bool,R: fun_int_bool] : ( hAPP_f448129468l_bool(hAPP_f284875647l_bool(cOMBC_605892544l_bool(P),Q),R) = hAPP_f448129468l_bool(hAPP_f284875647l_bool(P,R),Q) ) ).

tff(help_COMBC_1_1_COMBC_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__fun_Itc_,axiom,
    ! [P: fun_fu1217155507l_bool,Q: fun_nat_bool,R: fun_nat_bool] : ( hAPP_f54304608l_bool(hAPP_f103356543l_bool(cOMBC_1693257480l_bool(P),Q),R) = hAPP_f54304608l_bool(hAPP_f103356543l_bool(P,R),Q) ) ).

tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Oboo_065,axiom,
    ! [P: fun_bo1549164019l_bool,Q: fun_fu819804446l_bool,R: fun_int_bool] : ( hAPP_f1721100032l_bool(cOMBB_1974511970t_bool(P,Q),R) = hAPP_b589554111l_bool(P,hAPP_f448129468l_bool(Q,R)) ) ).

tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Oboo_066,axiom,
    ! [P: fun_bo1549164019l_bool,Q: fun_fu425979586l_bool,R: fun_nat_bool] : ( hAPP_f1748468828l_bool(cOMBB_444170502t_bool(P,Q),R) = hAPP_b589554111l_bool(P,hAPP_f54304608l_bool(Q,R)) ) ).

tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Oboo_067,axiom,
    ! [P: fun_bo1549164019l_bool,Q: fun_fu1430349052l_bool,R: fun_pname_bool] : ( hAPP_f1476298914l_bool(cOMBB_2095475776e_bool(P,Q),R) = hAPP_b589554111l_bool(P,hAPP_f1664156314l_bool(Q,R)) ) ).

tff(help_COMBC_1_1_COMBC_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_000tc__fun_It,axiom,
    ! [P: fun_fu802393907l_bool,Q: fun_pname_bool,R: fun_pname_bool] : ( hAPP_f1664156314l_bool(hAPP_f434788991l_bool(cOMBC_1284144636l_bool(P),Q),R) = hAPP_f1664156314l_bool(hAPP_f434788991l_bool(P,R),Q) ) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__fun_Itc__fun_It__,axiom,
    ! [P: fun_fu1471507361l_bool,Q: fun_int_fun_a_bool,R: int] : ( hAPP_i888906320l_bool(cOMBB_313729068ol_int(P,Q),R) = hAPP_f1631501043l_bool(P,hAPP_int_fun_a_bool(Q,R)) ) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__fun_Itc__fun_It___068,axiom,
    ! [P: fun_fu1471507361l_bool,Q: fun_nat_fun_a_bool,R: nat] : ( hAPP_n1414589940l_bool(cOMBB_1823939024ol_nat(P,Q),R) = hAPP_f1631501043l_bool(P,hAPP_nat_fun_a_bool(Q,R)) ) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__fun_Itc_,axiom,
    ! [P: fun_fu1151772339l_bool,Q: fun_int_fun_int_bool,R: int] : ( hAPP_i2112223885l_bool(cOMBB_1761098400ol_int(P,Q),R) = hAPP_f284875647l_bool(P,hAPP_i1948725293t_bool(Q,R)) ) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__fun_Itc__069,axiom,
    ! [P: fun_fu1151772339l_bool,Q: fun_nat_fun_int_bool,R: nat] : ( hAPP_n609083369l_bool(cOMBB_1123824708ol_nat(P,Q),R) = hAPP_f284875647l_bool(P,hAPP_n1082236369t_bool(Q,R)) ) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__fun_Itc_,axiom,
    ! [P: fun_fu1217155507l_bool,Q: fun_int_fun_nat_bool,R: int] : ( hAPP_i1718399025l_bool(cOMBB_516767136ol_int(P,Q),R) = hAPP_f103356543l_bool(P,hAPP_i418383825t_bool(Q,R)) ) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__fun_Itc__070,axiom,
    ! [P: fun_fu1217155507l_bool,Q: fun_nat_fun_nat_bool,R: nat] : ( hAPP_n215258509l_bool(cOMBB_2026977092ol_nat(P,Q),R) = hAPP_f103356543l_bool(P,hAPP_n1699378549t_bool(Q,R)) ) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_000tc__fun_It,axiom,
    ! [P: fun_fu802393907l_bool,Q: fun_in2065004473e_bool,R: int] : ( hAPP_i1587699435l_bool(cOMBB_607112096ol_int(P,Q),R) = hAPP_f434788991l_bool(P,hAPP_i7355787e_bool(Q,R)) ) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_000tc__fun_It_071,axiom,
    ! [P: fun_fu802393907l_bool,Q: fun_na936072029e_bool,R: nat] : ( hAPP_n850744903l_bool(cOMBB_2117322052ol_nat(P,Q),R) = hAPP_f434788991l_bool(P,hAPP_n1025906991e_bool(Q,R)) ) ).

tff(help_COMBC_1_1_COMBC_000tc__fun_Itc__fun_It__a_Mtc__HOL__Obool_J_Mtc__HOL__Obool,axiom,
    ! [P: fun_fu418465139l_bool,Q: fun_fun_a_bool_bool,R: fun_fun_a_bool_bool] : ( hAPP_f621171935l_bool(hAPP_f1434722111l_bool(cOMBC_331553030l_bool(P),Q),R) = hAPP_f621171935l_bool(hAPP_f1434722111l_bool(P,R),Q) ) ).

tff(help_COMBC_1_1_COMBC_000tc__fun_Itc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_Mtc__H,axiom,
    ! [P: fun_fu209342027l_bool,Q: fun_fu819804446l_bool,R: fun_fu819804446l_bool] : ( hAPP_f215623910l_bool(hAPP_f317860125l_bool(cOMBC_348868698l_bool(P),Q),R) = hAPP_f215623910l_bool(hAPP_f317860125l_bool(P,R),Q) ) ).

tff(help_COMBC_1_1_COMBC_000tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_Mtc__H,axiom,
    ! [P: fun_fu140186515l_bool,Q: fun_fu425979586l_bool,R: fun_fu425979586l_bool] : ( hAPP_f1637334154l_bool(hAPP_f1772781669l_bool(cOMBC_595898202l_bool(P),Q),R) = hAPP_f1637334154l_bool(hAPP_f1772781669l_bool(P,R),Q) ) ).

tff(help_COMBC_1_1_COMBC_000tc__fun_Itc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_Mtc_,axiom,
    ! [P: fun_fu1911931399l_bool,Q: fun_fu1430349052l_bool,R: fun_fu1430349052l_bool] : ( hAPP_f1935102916l_bool(hAPP_f510955609l_bool(cOMBC_7971162l_bool(P),Q),R) = hAPP_f1935102916l_bool(hAPP_f510955609l_bool(P,R),Q) ) ).

%----Conjectures (7)
tff(conj_0,hypothesis,
    hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,u)) ).

tff(conj_1,hypothesis,
    hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,g),image_pname_a(mgt_call,u))) ).

tff(conj_2,hypothesis,
    hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,na)),hAPP_fun_a_bool_nat(finite_card_a,image_pname_a(mgt_call,u)))) ).

tff(conj_3,hypothesis,
    hAPP_fun_a_bool_nat(finite_card_a,g) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_fun_a_bool_nat(finite_card_a,image_pname_a(mgt_call,u))),hAPP_nat_nat(suc,na)) ).

tff(conj_4,hypothesis,
    hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,pn),u)) ).

tff(conj_5,hypothesis,
    ~ hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,hAPP_pname_a(mgt_call,pn)),g)) ).

% tff(conj_6,conjecture,
%     hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,hAPP_pname_a(mgt_call,pn)),g)),image_pname_a(mgt_call,u))) ).

%------------------------------------------------------------------------------